Index [] · for optimal power flow problem, 197–198 outer approximation technique, 170–171,...

32
Index 0-1 knapsack problem, 51 0-1 programming problems, 50, 52 αBB branch-and-bound method, 283 α-ECP (solver), 288 AARC (affine adjustable robust counterpart) problem, 342 Accuracy, dynamic adjustment of, 523 “Act, learn, then act” approach, 458 “Act, then learn” approach, 458 Active-set method for linearly constrained optimization, 503 for nonlinear optimization, 224, 232–233 QP solvers using, 232–233 Adaptability, in recourse-based robust optimization, 342–343 Adaptive radiation therapy, 350 Adaptive robust UC model with net load uncertainty, 360–362 Adjoint method, 252 Adjustable robust solutions, 340–342 Advection equation, 125 Adverse selection, 474 Aerodynamic shape optimization, 252–254 Aerospace engineering, 249–250 Aerospace systems computational models, 250 described, 249–250 gradient computation, 251–252 multidisciplinary design optimization for, 249–257 optimization algorithms, 251 optimization problems, 250–251 satellite design and operation, 256–257 Aerostructural design optimization, 254–256 Affine adjustable robust counterpart (AARC) problem, 342 Affine adjustable robust solutions, 342 Affine policy–based robust optimization model, 365 AGC (automatic generation control) units, 364–365 Aggregated models, for process synthesis, 317–318 AIMMS modeling system, 91, 223, 287–288 Aircraft conflict avoidance, 294–301 history of approaches, 294–295 MINLP formulations, 295–300 solution approaches, 300–301 Aircraft deconfliction, 294 Air quality constraints, in building automation, 263 Air quality control systems, 260–261 Air separation systems distillation columns in, 244 distillation models for, 246, 247 production planning for, 78–82 Air traffic management (ATM), 293–301 aircraft conflict avoidance, 294–295 automation in, 293 MINLP formulations, 295–300 solution approaches, 300–301 Algebraic modeling languages, 287 ALGENCAN (solver), 235 Algorithmic differentiation, 252 Algorithm(s) global optimization, 168–172 linear optimization, 5–6 for MDO in aerospace systems, 251 nonlinear optimization, 229–235 augmented Lagrangian methods, 235 generalized reduced gradient method, 235 interior-point methods, 233–235 sequential quadratic programming, 229–233 POUNDERS, 533–535 quadratic optimization, 7 All-atomistic validation, 176 Allocation centers, in inventory modeling, 485 Alternate heuristic (ALT), for pooling problems, 212–213 AMBER energy values, 180–181 Ambiguity set, 344 AMPL modeling system, 223, 268–269, 287, 288, 290 665

Transcript of Index [] · for optimal power flow problem, 197–198 outer approximation technique, 170–171,...

Page 1: Index [] · for optimal power flow problem, 197–198 outer approximation technique, 170–171, 198–202, 277 piecewise-linear, 283–284 for pooling problem, 213–214 power, 446

Index

0-1 knapsack problem, 510-1 programming problems, 50, 52αBB branch-and-bound method, 283α-ECP (solver), 288AARC (affine adjustable robust counterpart)

problem, 342Accuracy, dynamic adjustment of, 523“Act, learn, then act” approach, 458“Act, then learn” approach, 458Active-set method

for linearly constrained optimization, 503for nonlinear optimization, 224, 232–233QP solvers using, 232–233

Adaptability, in recourse-based robustoptimization, 342–343

Adaptive radiation therapy, 350Adaptive robust UC model with net load

uncertainty, 360–362Adjoint method, 252Adjustable robust solutions, 340–342Advection equation, 125Adverse selection, 474Aerodynamic shape optimization, 252–254Aerospace engineering, 249–250Aerospace systems

computational models, 250described, 249–250gradient computation, 251–252multidisciplinary design optimization for,

249–257optimization algorithms, 251optimization problems, 250–251satellite design and operation, 256–257

Aerostructural design optimization, 254–256Affine adjustable robust counterpart (AARC)

problem, 342Affine adjustable robust solutions, 342Affine policy–based robust optimization model,

365AGC (automatic generation control) units,

364–365

Aggregated models, for process synthesis, 317–318AIMMS modeling system, 91, 223, 287–288Aircraft conflict avoidance, 294–301

history of approaches, 294–295MINLP formulations, 295–300solution approaches, 300–301

Aircraft deconfliction, 294Air quality constraints, in building automation,

263Air quality control systems, 260–261Air separation systems

distillation columns in, 244distillation models for, 246, 247production planning for, 78–82

Air traffic management (ATM), 293–301aircraft conflict avoidance, 294–295automation in, 293MINLP formulations, 295–300solution approaches, 300–301

Algebraic modeling languages, 287ALGENCAN (solver), 235Algorithmic differentiation, 252Algorithm(s)

global optimization, 168–172linear optimization, 5–6for MDO in aerospace systems, 251nonlinear optimization, 229–235

augmented Lagrangian methods, 235generalized reduced gradient method, 235interior-point methods, 233–235sequential quadratic programming, 229–233

POUNDERS, 533–535quadratic optimization, 7

All-atomistic validation, 176Allocation centers, in inventory modeling, 485Alternate heuristic (ALT), for pooling problems,

212–213AMBER energy values, 180–181Ambiguity set, 344AMPL modeling system, 223, 268–269, 287, 288,

290

665

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666 Index

Ancillary services markets, 462, 463Annual vaccine strain selection problem, 472ANTIGONE (solver), 289Approximation(s)

for biofuel production decisions, 465gradient, 512–513for nonlinear optimal control, 129–133for optimal power flow problem, 197–198outer approximation technique, 170–171,

198–202, 277piecewise-linear, 283–284for pooling problem, 213–214power, 446quasi-Newton, 227, 497robust, 31–32sample average, 382, 390stochastic, 381–382, 390successive approximation techniques, 170–171

Arc-flow formulation, for LNG inventoryrouting, 83

Asset risk contribution, 432ATM. See Air traffic managementATM (company), 75Attainable region, of reactor networks, 320Augmented Lagrangian function, 229, 235, 504Autocorrelation coefficients, 29Automatic differentiation techniques, 223Automatic generation control (AGC) units,

364–365Automation

in air traffic management, 293building, 259–269

building operations and inefficiencies,260–261

computational considerations, 268–269physical building model, 261–262predictive control, 262–268

HVAC system, 259plant, model predictive control in, 44–46

Automotive valve trains, 508Average occupation measures, 131–133Averaging, noise reduction via, 523

Backbone superpositions, 527Backorders, 441, 445Backward pass, for SDDP algorithm, 417, 418Balinski–Tucker procedure, 9BARON (solver), 217, 289, 313–314Barrier methods. See Interior-point methods

(IPMs)Barrier parameters, 233Barrier term, 233Base-stock policy, 445, 446, 449Basis pursuit, 32

Basis solution, interior vs., 9BC-DFO (solver), 498Benchmark tracking problem, 152Benders cuts, 417Benders decomposition, 71–72, 279, 283BFGS formula, 227, 231Big-M formulations, 69, 70Bilevel leader-follower game model, 464Bilevel optimization

for biofuel production, 464, 466in energy industry, 454for pooling problem, 212

Biofuel productionfeedstock procurement for, 463–465supply network design for refineries, 465–466

Biological applications of optimization, 175–176Biological constraints, in protein-DNA system

design, 178, 183Biorefinery locations, 465–466Black-box MIP solvers, 57–58Black-box optimization problems, 529Blenders, in biofuel supply chain, 465Blending equations, in scheduling models, 328Blending problem, 207Blocking, in no-wait job-shop scheduling, 69Blood and blood products, inventory management

for, 473, 484–487BMSs (building management systems), 259BOBYQA (solver), 503Bombardier Transportation, 75Bonmin (solver), 288Boolean variables, 316Bottlenecks, transportation network, 487–491Bounds

for branch-and-bound method, 169in global optimization, 169lower, 129, 274propagation of, 281for spherical codes, 34tightening of, 281–282upper, 274, 428, 431worst-case complexity, 448, 501–502

BPMPD (solver), 11Brachytherapy, 99–101Branch-and-bound methodαBB, 283bounds for, 169branching in, 169for global optimization, 169–170for integer optimization, 51, 52and lift-and-project tool, 55, 56for Lipschitz optimization, 172LP/NLP-based, 278–279

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Index 667

for mixed-integer nonlinear optimization, 274,283

nonlinear, 275–276pruning in, 169solving UFLPs with, 451spatial, 204–205, 280–281, 284termination of, 170

Branch-and-bound tree, 275Branch-and-cut method, 56, 102, 278Branch-and-reduce method, 281–282Branch callback, 63Branching, 169, 274, 280, 285Brand-name drugs, 475–477Breakdowns, corridor, 489Breast milk bank, 480–483Budgeted uncertainty set, 361Budget of uncertainty, 361Building automation, 259–269

building operations and inefficiencies, 260–261computational considerations, 268–269HVAC system automation, 259physical building model, 261–262predictive control, 262–268

Building management systems (BMSs), 259Building operations, 260–261Building-wide control strategy, 267–268Building-wide models, 261Building-wide nonlinear variable interactions,

259–260Bundle methods, 118, 452Buyback contracts, 454–455

CADANS (software), 59CAES (compressed-air energy storage), 462Callback functions, MIP solver, 62–63Cantilever beam, minimum volume problem with,

22, 24Capacity-based disruption, 478Capacity certainty, for power generators, 459–461Capacity expansion

in chemical engineering, 77in energy industry, 187, 365, 466in healthcare, 470and inventory optimization, 439

Capacity planningin energy industry, 457–458in healthcare, 469–470

Capacity uncertainty, 448Cap-and-trade mechanisms, 464–465Cardinality-constrained uncertainty set, 336–337Categorical variables, derivative-free optimization

for, 506Cauchy ordinary differential equation (ODE),

127–128

CCHP plants. See Combined cooling, heat, andpower plants

CCSO. See Chance constrained stochasticoptimization

Centralization, of distribution locations, 486–487Central path, 116Centroid-centroid force fields, 178–179, 182, 183CEOS (cubic equation of state), 242, 246Certificate of infeasibility, 277CGLP (cut-generating linear program), 56, 287Chance constrained stochastic optimization

(CCSO), 387–389, 391in chemical engineering, 393, 394distributionally robust models, 390, 391in financial engineering, 428, 429

Change parameter command, 61Chemical engineering

linear and quadratic optimization, 37–46design and operations applications, 37model predictive control, 41–46production planning, 37–41

mixed-integer linear optimization, 77–91chemical supply network optimization, 86–90LNG inventory routing, 82–86production planning for air separation plants,

78–82mixed-integer nonlinear programming, 291–292stochastic optimization, 393–404

CVaR-based method, 401–404operational planning in chemical plants,

393–394problem statement, 394–397robust optimization method, 397–400,

403–404Chemical plants

operational planning in, 393–394pooling problem in, 207–208

Chemical supply network optimization, 86–90Child nodes, 275CHP (combined heat and power) plants, 303Chvátal rank, 54Circuit placement, 33–34Classification problems, 94, 96–99CLO. See Conic linear optimizationClosed-form solutions

for EOQ problem, 441for infinite-horizon problems, 446for multiechelon inventory optimization, 449for single-reservoir model, 412

CLP (solver), 11COBYLA (solver), 505Coefficient reduction, 285Cogeneration energy systems, 303–314

computational experiments for, 310–314

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668 Index

data-driven vs. first-principles approaches to,304–305

described, 303–304MINLP formulations for, 307–310unit commitment vs., 305–307variations in, 305

Cogeneration power plants, 303Coherent risk measures, 390Collaborative R&D, 474–475Column-and-constraint generation algorithm, 362Combinatorial optimization, 52–53, 165Combined cooling, heat, and power (CCHP)

plantscomputational experiments, 310–314MINLP formulation for, 307–310unit commitment in, 306–307

Combined heat and power (CHP) plants, 303Comfort overrelaxation strategy, 266–269Committed service time (CST), 449–450Communication networks, 36Competitive feedstock procurement, 463–465Complementarity conditions

for linear optimization, 4for nonlinear optimization, 225for second-order cone optimization, 116

Complementarity constraintsMPCCs, 235

column optimization with phase changes,242–243

distillation column optimal design, 243–245distillation systems, 240–241

in semidefinite optimization, 116Complementarity pivot algorithms, 7Complementarity problems, 167, 535Complementary optimal solutions, 4Completely positive cones, 112–113Complete polling, 499Complete quadratic models, 497Complete water networks (CWNs), 326Complexity

for dynamic economic lot-sizing problem, 442of global optimization, 166–167of LO and QO, 8–9and MINLO software, 288–291of nonlinear optimization problems, 224worst-case complexity bounds, 448, 501–502

Complex-step method, 252Compliance function, 138Component mass balances, distillation column,

239Composite step-based SQP, 505Compressed-air energy storage (CAES), 462Compressed sensing, 32, 497Computational complexity. See Complexity

Computational methods, LO and QO, 9–10Computational models for aerospace systems, 250Computed tomography (CT) images, 346Computer systems

optimization algorithms, xxixpower of, xxix–xxx

Concave minimization (concave optimization)problems, 164, 170–171

Concentration model. See P-formulationConditional distribution, stochastic optimization

with, 384Conditional probability, from scenario trees, 385Conditional value-at-risk (CVaR), 394

in chemical engineering, 401–404and distributional robustness, 350for investment portfolio construction, 430–432marginal CVaR contribution for assets, 434maximization return problem with upper

bound on, 431mean-CVaR model, 431minimization of CVaR problems, 431and z transformations, 401

Conflict-free trajectory planning, 295Conflict resolution, for aircraft, 294Conflicts

between aircraft, 294between objectives, 479, 480in train dispatching, 68

Congestion delays, 490–491ConicBundle (software), 118Conic constraints, 235, 338Conic linear optimization (CLO), 107–120

with convex uncertainty set, 337–338duality and constraint qualification, 113–115financial engineering, 149–160

equal-risk contribution portfolios, 157–160linear optimization models, 149–150portfolio optimization problems, 151–153robust mean-variance optimization, 155–157SOCO problems, 150–151transaction costs with market impact,

153–155general conic optimization, 111–113nonlinear optimal control, 121–133

approximation results, 129–130control over set of initial conditions, 130–133history of optimal control, 121–122LP formulation, 125–129polynomial optimal control, 122–124

optimality conditions and interior-pointalgorithms, 115–116

polynomial optimization, 119–120second-order cone optimization, 110–111semidefinite optimization, 107–110

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Index 669

software, 117–119truss topology design, 135–147

applications, 144integer variables, problems with, 145–147nonlinear optimization formulation, 136–139SDO formulation, 141–143SOCO formulation, 139–141truss notation, 135–136vibration constraints, 144–145

CONOPT (solver), 196, 235, 242, 244Conservation of mass, equation of, 127Conservativeness, of SCUC model, 363–364Constraint disaggregation, MINLP problems

with, 285Constraint qualification (CQ)

linear-independence, 225, 226Mangasarian–Fromowitz, 225Slater’s, 114–115

Constraints. See also Complementarity constraintsfor aircraft conflict avoidance, 296, 299basic distillation model, 240building automation, 263–264column-and-constraint generation algorithm,

362complementarity, 116conic, 235, 338derivative-free optimization, 495, 504–505dose-volume, 94equilibrium, 465hidden constraint problems, 507–517

convergence results, 510–514imfil.m code, 514–517implicit filtering, 508–510with random directions, 515–517

investment portfolio construction with,432–435

knapsack, 430LB, 58linear, 335–338, 511marginal VaR and CVaR, 432–435nonanticipativity, 386nonlinear

general uncertainty set, 338–339MINLP model for cogeneration systems,

309–310robust optimization with uncertainty,

338–339optimal power flow problem, 189periodic event-scheduling problem, 59polyhedral, 266, 269pooling problem, 210portfolio optimization, 151–153positive semidefiniteness, 56protein-DNA system design, 178–180, 182, 183

reformulation-linearization technique, 180relaxable, 504–505spectral mask, 29TE, 152–153thermal comfort, 263–264in train-dispatching problem, 68–69unrelaxable, 504, 505, 535vibration, 144–145

Constraint softening, 44Construction heuristics, 84Consumers, strategic behavior by, 471–472Continuity equation, 125Continuous-demand problems, 446–448Continuous location models, 450–451Continuous operating decisions, 78–79Continuous relaxation, in branch-and-bound

method, 274Continuous-review models, 440, 447–448Continuous variables

in GDP models, 316in global optimization, 164–165

Contractsbuyback, 454–455cost-sharing, 471decentralized supply chain optimization for,

453–455fail-to-supply, 477–478fee-for-service, 475–477in healthcare, 473–478licensing, 474–475wholesale price, 454

Control(s). See also Model predictive control(MPC); Optimal control

building automation, 260, 261, 265–266dynamic matrix, 41, 42hybrid process, 317inventory control policy, 462, 473LO and QO for, 27over set of initial conditions, 130–133process, 37, 317, 328–329relaxed, 125–126subliminal, 294tumor control probability, 99–101

Convergenceglobal convergence

defined, 496derivative-free optimization with, 501–502nonlinear optimization with, 227–228

for hidden constraint problems, 510–514geometry of D and necessary conditions, 510gradient approximation, 512–513limitations, 514Lipschitz-continuous f, 513–514search direction sets, 511–512

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670 Index

for implicit filtering, 509–510for nonsmooth functions, 500and sampling using simplex sets, 498and SDDP algorithm, 417–418uniform, 130, 132–133

Conversion loss, 462Convex analysis, cutting-plane approach with,

54–55Convex cost network flow model, 487, 490–491Convex envelope, 167Convex hull search technique, 213–214Convexity

identifying, 166–167in nonlinear optimization, 163

Convex MINLP, 274–279defined, 274LP/NLP-based branch-and-bound approach for,

278–279nonlinear branch-and-bound approach for,

275–276outer approximation algorithm for, 277polyhedral relaxations for, 276–277solver software for, 288–289

Convex nonlinear optimization problems,222–223

Convex objective functiondeterministic inventory optimization with, 441multiechelon inventory optimization with, 449stochastic inventory optimization with, 444

Convex optimization problems, 107, 382. See alsoConic linear optimization

Convex outer approximation (OA)for optimal power flow problem, 198–202via McCormick envelopes, 196–197via piecewise-linear envelopes, 199–202

Convex quadratic constraints, 151–153Convex uncertainty set, 337–338Coordinated supply chains, 453–455Coordinate search, 509–511Coordination collar, 45–46Copositive cone, 112–113Corners, in breast milk bank system, 481Corrective actions, SCUC model with, 363Corridor breakdowns, 489Corridor delays, 487–489Cosine measure of PSSs, 499Cost functions, 40, 296Cost-sharing contracts, 471Cost-to-go functions

stochastic optimization with, 384–385supply chain optimization with, 442, 445, 450

COUENNE (solver), 289, 313–314Coupled modeling approach, 467Coupled systems, 250, 261

Covering models, 453CPLEX (solver), 10, 22–25, 56, 82, 85–86, 202, 217,

399, 402, 450CQ. See Constraint qualificationCreditMetrics, 432Crew scheduling problem, 59Criss-cross type algorithms, 5Critical fractile (critical ratio), 444CSPD (solver), 117CST (committed service time), 449–450CT (computed tomography) images, 346Cubic equation of state (CEOS), 242, 246Cut-generating linear program (CGLP), 56, 287Cuts

Benders, 417flow cover, 450rounds of, 56

Cutting-plane methodextended, 279for integer optimization problems, 53–54and lift-and-project tool, 55–56for MINLP problems, 286–287and SDDP algorithm, 417

CVaR. See Conditional value-at-riskCVX (environment), 117, 119CWNs (complete water networks), 326Cycling cost, 460

�, geometry of, 510Daily production profile, chemical plant, 393DAMIP (discriminant analysis via MIP), 96–99Data-driven black box approaches, 304–305Data set construction, protein-binding cavity,

525–526Day-ahead electricity markets, 405D.C. (difference of convex) functions, 166, 168DC (direct current) power flow linearization,

197–198DDF (differentiable distribution function),

243–244DDNA3 energy, 181Decentralized supply chains, 453–455Decision-hazard problems, 413Decision-making problems, in power system

operations, 357–358Decision support tools, 482–483, 492Decision Tree for Optimization Software Website,

222Decision variables

for aircraft conflict avoidance, 295–296in optimal power flow problem, 189for short-term cogeneration systems planning,

307–309Decoding techniques, 35

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Index 671

DecompositionBenders, 71–72, 279, 283nested, 386, 389–391for nonconvex MINLP problems, 283in process flowsheet synthesis, 318scenario, 386, 389, 391stochastic, 391for train dispatching applications, 70–72, 74

Defender-attacker-defender model, 341–342Degenerate LO problems, 5–6Degrees of freedom, 224, 531DEL (dynamic economic lot-sizing problem),

442–443Delay modeling

in delivery routing, 489–491for port and corridor delays, 487–489

Delivery routing, 489–491DELTA Supply Chain, 60–61Demand distribution, 444–448Demand uncertainty

in chemical engineering, 393–394in robust optimization method, 397

Derivative-based methods, 251Derivative-free optimization (DFO), 495–506

aerospace applications, 251described, 529–530extensions, 506global, 169linearly constrained optimization, 502–503nonlinearly constrained optimization, 503–506

extensions, 506model-based trust-region methods, 505–506relaxable constraints, 504–505unrelaxable constraints, 504, 505

POUNDERS solver, 529–539algorithm underlying POUNDERS, 533–535and DFO described, 529–530energy density functional calibration,

536–539smooth residual models, 530–533Toolkit for Advanced Optimization software,

535–536protein-binding cavity alignment via

DFO-VASP, 519–528computational experiments, 525–527DFO method, 521–522electrostatic data, 521noise-handling strategies for VASP, 522–525protein specificities, 519–520VASP method, 520

unconstrained optimization, 496–502noisy functions, 500–501nonsmooth functions, 499–500probabilistic models, 502

smooth functions, 496–499worst-case complexity and global

convergence, 501–502uses, 495

Derivatives, availability of, 223Descent direction, 225Design template, protein-DNA system, 177–178,

182Deterministic approaches

for aircraft conflict avoidance, 300for electric power systems, 358for inventory optimization, 440–443for security-constrained unit commitment,

359–360Deterministic convex problems, 388DFO. See Derivative-free optimizationDFO (solver), 498, 503, 505DFO-VASP protein-binding cavity alignment,

519–528computational experiments, 525–527DFO method, 521–522electrostatic data, 521noise-handling strategies for VASP, 522–525protein specificities, 519–520VASP method, 520

Diagnosis, integer optimization in, 93–94Dicopt (solver), 289Difference gradient, 512Difference of convex (D.C.) functions, 166, 168Differentiable distribution function (DDF),

243–244DIRECT (solver), 172, 506, 508Direct current (DC) power flow linearization,

197–198Directions, sampling along, 502–503Direct methods for evaluating derivatives, 252Direct noise level reduction, 523Direct-search methods

linearly constrained optimization, 502–503nonlinearly constrained optimization, 504probabilistic descent, 502unconstrained optimization, 498–499worst-case complexity bounds, 501–502

Disaster mitigation, 480Disaster preparedness, 480, 483–484Disaster recovery, 491Disaster response, 483–484Discontinuities, aerospace system, 251Discounting

and finite-horizon problem, 445for marginal water values, 412–414

Discrete-continuous optimization, 316Discrete facility location model, 465–466Discrete location models, 451

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672 Index

Discrete operating decisions, 78–79Discrete optimization, 164. See also Integer

optimizationDiscrete stochastic optimization problems, 389Discretization

in building automation, 268–269in conic linear optimization, 121, 129in mixed-integer linear programming, 213–214,

217of radiation doses, 347–348in supply chain optimization, 445of three-dimensional PDEs, 224, 231

Discriminant, in aircraft conflict avoidance, 297Discriminant analysis via MIP (DAMIP), 96–99Disease detection, 93–94Disjunctions, GDP model, 316Disjunctive precedence constraint, 68–69Disjunctive programming, 69, 286–287. See also

Generalized disjunctive programming(GDP)

Dispatching, train. See Train dispatchingDisplacement-like variables, in truss topology

design, 19Disruptions, supply, 448, 477, 478Distillation column

basic model of, 238–240MPCC formulation of, 240–241optimal design of, 243–245with phase changes, 242–243

Distillation sequences, 323–324Distillation systems, 237–247

basic distillation model, 238–240case studies of, 241–245described, 237–238extensions of nonlinear optimization for,

245–247MPCC formulation for, 240–241

Distributed parameters, 62Distributionally robust optimization (DRO), 344Distributional robustness

chance constrained SO models with, 390, 391linear optimization for, 348–350in radiation therapy, 348–350stochastic optimization models with, 390, 391

Distributional uncertainties, in wind farm layout,368

Distribution locations, in inventory modeling,485–487

DMC (dynamic matrix control), 41, 42DNA. See Protein-DNA system designDOASA software, 419Dose-volume constraints, 94Dow Chemical Company, 78, 86–90DP (dynamic programming) formulation, 442, 450

Drag coefficient, 253DRO (distributionally robust optimization), 344Drugs. See PharmaceuticalsDrug shortages, mitigating, 477–478DSDP (solver), 118Dual capacity sourcing problem, 469–470Dual cone, 112Duality, xxx, 113–114. See also Strong duality

conditionDUALOC algorithm, 452Dual problem

in nonlinear optimization formulation, 139for optimal control applications, 129in second-order cone optimization, 140–141in semidefinite optimization, 108, 142–143in truss topology design, 139–142

Dual-response manufacturing, 458Dual simplex method pivot rules, 5–6Dynamic adjustment of accuracy, 523Dynamic distillation systems, 246–247Dynamic economic lot-sizing (DEL) problem,

442–443Dynamic matrix control (DMC), 41, 42Dynamic programming (DP) formulation, 442,

450Dynamic stochastic optimization, 382–387

electricity prices from, 406in energy industry, 461formulations for, 384–387supply chain engineering application, 382–384

ε-subdifferential, 168Echelon base-stock level, 449Echelon base-stock policy, 449Echelon holding costs, 449Echelons, inventory, 449Economic curtailment policy, 461Economic dispatch (ED) problem

electricity prices from, 406and optimal power flow problem, 187and power system operations, 459robust optimization model for, 364security-constrained, 191

Economic lot-sizing problem (ELSP), 442–443Economic order quantity (EOQ) problem,

440–442Economic production quantity (EPQ) problem,

442ECP (extended cutting plane) method, 279EDFs (energy density functionals), 536–539ED problem. See Economic dispatch problemEfficiency–responsiveness trade-off, in strategic

sourcing, 457–458Efficient frontier, 149

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Index 673

Elastic equilibrium equations, 15, 16Electrical engineering

communication networks, 36filter design, 28–30information and coding theory, 34–35linear and quadratic optimization, 27–36norm optimization, 31–34pattern classification, 30–31

Electric power systemsmarginal water valuation in, 419–425reliability of, 357robust optimization, 357–365

decision-making problems in power systemoperations, 357–358

extensions, 362–364real-time operation and long-term planning,

364–365security-constrained unit commitment model,

359–362Electrostatic isopotentials, 521Ellipsoidal uncertainty set, 335–336, 338, 339ELSP (economic lot-sizing problem), 442–443Embedded optimization, 27Emergency facility location problem, 51Energy arbitrage, 463Energy balances, in physical building model, 262Energy consumption, by distillation systems, 237Energy density functionals (EDFs), 536–539Energy industry

feedstock procurement, 463–465inventory management, 461–463power system management, 457–461supply chain optimization, 457–467supply network design, 465–467

Energy metric, protein-DNA system design, 176,181–182

Energy optimization, xxx. See also Short-termplanning in cogeneration energy systems

Energy output, wind farm layout and, 367Energy storage facilities, 461–463Enhancement techniques

MINLO, 284–287cutting planes, 286–287presolve, branching, and reformulations,

284–285primal heuristics, 285–286

stability enhancement, 44, 372–374Enolase superfamily, 525, 526Enthalpy, 239Enumerative approach, 52, 170. See also

Branch-and-bound methodEnvelope-based relaxations, 214–215Environmental Protection Agency (EPA), 208,

458

EOQ (economic order quantity) problem,440–442

EOQ with backorders problem, 441EPA (Environmental Protection Agency), 208,

458EPQ (economic production quantity) problem,

442Equality-constrained NLO problems, 230–231Equal-risk contribution (ERC) portfolios, 157–160Equilibrium constraints, 465Equilibrium demand, 472Equilibrium relations, distillation column, 239Equity objectives, in humanitarian applications,

480–483ERASMUS, 294, 295ERC (equal-risk contribution) portfolios, 157–160Euler’s theorem, 433Exact, maximally complementary primal-dual

optimal solution pair, 9Exact, strictly complementary primal-dual

optimal solution pair, 9Exact penalty function, 228–229Expectation objective, 381Extended cutting plane (ECP) method, 279Extreme barriers, 504ExxonMobil, 78

Facial programs, 55Facility location problem

in energy industry, 463, 465–466in healthcare logistics, 95in humanitarian applications, 480–483integer optimization for, 50–51, 95in supply chain optimization, 450–453

Factorable programming, 279–280Fail-to-supply (FTS) contracts, 477–478Fairness, in humanitarian supply chain

optimization, 480, 483, 491Fathoming, 169, 275FBBT (feasibility-based bound tightening), 281FCFS (first-come, first-served) usage, 485–487FDA (Food and Drug Administration) approval,

469, 474Feasibility-based bound tightening (FBBT), 281Feasibility pump (FP), 57, 286Feasible IPMs, 6Feasible methods for nonlinearly constrained

optimization, 504Feasible set, for branch-and-bound method, 274Feasible solution, from polyhedral relaxations, 277Feedstock procurement, 463–465Feed tray location, distillation column, 320–323Fee-for-service (FFS) contracts, 475–477FFS (fee-for-service) contracts, 475–477

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674 Index

FILMINT (solver), 289Filter design, LO and QO for, 28–30Filter methods, 229, 505FILTERSQP (solver), 233Financial engineering

conic linear optimization models, 149–160equal-risk contribution portfolios, 157–160linear optimization models, 149–150portfolio optimization problems, 151–153robust mean-variance optimization, 155–157SOCO problems, 150–151transaction costs with market impact,

153–155optimization applications, xxx

Financial optimization, 149, 427Finite-horizon inventory problem, 445–446Finite-horizon stochastic dynamic programming

model, 470Finite impulse response (FIR) filters, 28FIR (finite impulse response) filters, 28First-come, first-served (FCFS) usage, 485–487First-order conditions

for distillation models, 246in inventory optimization, 441, 444, 448, 454,

455First-order methods

for global optimization, 169for unconstrained optimization of smooth

functions, 496–497First-principles approaches, 304–305Fixed block signaling systems, 66Fixed charge problem, 50, 164Fixed cost(s)

in chemical engineering, 87, 90in dynamic economic lot-sizing problem, 442in energy systems, 308, 359in EOQ problem, 440–442in facility location problem, 451, 453in finite-horizon problem, 445in infinite-horizon problem, 447in multiechelon inventory optimization, 449

Flexible capacity, in healthcare, 470Flexible mode, QP solver, 233Flight level, changes in, 294Flow cover cuts, 450Flow model. See P-formulationFold specificity, protein-DNA system, 180–181Food and Drug Administration (FDA) approval,

469, 474Food-energy-environment trilemma, 464Forward pass, for SDDP algorithm, 417, 418FP (feasibility pump), 57, 286Fractional linear optimization problem, 167Franz Edelman Award, 58

Freight trains, dispatching of, 74FTS (fail-to-supply) contracts, 477–478Fuel burn, 254–256Fukushima tragedy, 458Full-Newton model, 533Fully adjustable robust solutions, 340–341Fully flexible power generators, 460Fully linear models, 496–497Fully quadratic models, 497

Game theory, contract analysis in, 453–455, 474GAMS (software), 195–196, 205, 223, 244, 399, 402Gasoline blending, 208Gauss–Newton model, 533GBD (generalized Benders decomposition), 279,

283GDP. See Generalized disjunctive programmingGegenbauer polynomials, 34General conic optimization problem, 111–113Generalized Benders decomposition (GBD), 279,

283Generalized disjunctive programming (GDP),

315–329general model, 316linear, 317molecular computing, 329planning and scheduling, 327–328process control, 328–329process synthesis, 317–327superstructure, 315–316types of models, 316–317

Generalized pattern search, 499Generalized pooling problem, 211Generalized reduced gradient method, 235General purpose solvers, 233–235General uncertainty set, 338–339Generation contingencies, SCUC model with, 364Generic drugs, fail-to-supply contracts for,

477–478Gibbs–Duhem relation, 241Gibbs free energy, 240Gini coefficient, 480Global convergence

defined, 496derivative-free optimization with, 501–502nonlinear optimization with, 227–228

Globally optimal solutions, 222–223Global minima

in aerospace applications, 251and CLO problems with integer variables, 145for concave minimization problems, 164, 170,

172for convex vs. nonconvex problems, 163and termination of branch-and-bound

algorithms, 170

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Index 675

Global optimality conditions, 167–168Global optimization, 163–173

algorithms, 168–172computational complexity, 166–167derivative-free optimization for, 506described, 163in medical treatment design, 95MIP in, 60optimality conditions, 167–168optimal power flow, 187–205

convex outer approximation, 198–202example data and formulation, 192–197formulation, 189–190future research directions, 205Lagrangian relaxation, 202–203linear approximation with direct current flow,

197–198McCormick envelopes, 198–199moment sum of squares, 204OPF problem described, 187–188piecewise-linear envelopes, 199–202problem statement, 188solution methods, 191–205spatial branch-and-bound algorithm, 204–205sufficient strong duality condition, 203–204variants and extensions, 190–191

pooling problem, 214–217protein-DNA system design, 175–185

biological applications of optimization,175–176

energy metric, 181–182fold specificity, 180–181input parameters and constraints, 177–179methods, 176–177PFV integrase enzyme, 182–184protein-DNA sequence selection, 179–180

software, 173types of problems, 164–166

GLOMIQO (solver), 196, 203GloptiPoly, 120GLPK (solver), 11Goldfeld, Quandt, and Trotter (GQT) routine,

536Governing equations, 250Government intervention, supply chain

optimization and, 464–465, 472GQT (Goldfeld, Quandt, and Trotter) routine,

536Gradient approximation, 512–513Gradient-based methods, 169, 235, 501

Gradientsfor aerospace systems, 251–252difference, 512simplex, 500, 512

Grey-box optimization problems, 529Ground structure approach, 13–14Guaranteed-service model, 449Gurobi (solver), 10, 56, 146

Hakimi property, 451Half-wheel minimum volume problem, 22, 24Hamilton–Jacobi–Bellman equation, 122Hazard-decision assumption, 413HDA (hydrodealkylation), 318–320HDR (high-dose rate) brachytherapy, 99–101Heading, changes in aircraft, 294Healthcare

humanitarian issues in global, 479, 491integer optimization applications, 93–103

diagnosis and detection, 93–94discriminant analysis, 96–99health logistics and operations, 95open challenges, 102public health, 95solution strategies, 102TCP-driven PET-image-guided

treatment-planning model, 99–101treatment design, 94–95

optimization applications, xxxrecourse-based robust optimization in, 344supply chain optimization, 469–478

capacity planning, 469–470inventory management, 473production planning, 470–472supply chain contracting, 473–478

Healthcare facilities, inventory management at,473

Health logistics, 95Heat balances, distillation column, 239Heat exchange networks (HENs)

MINLO and GDP process synthesisapplications, 324, 325

models for, 317Heat exchangers, in distillation systems, 238,

246Heating, ventilation, and air-conditioning

(HVAC) system, 259–261HENs. See Heat exchange networksHere-and-now decisions

in marginal water value calculations, 413, 414robust optimization with, 340stochastic optimization with, 381

Heuristic callback, 63Heuristic mode, MIP solvers, 61–62

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676 Index

Heuristicsfor integer optimization, 57–58for LNG inventory routing, 84–85for MINLP model of aircraft conflict avoidance,

300–301for MINLP problems, 285–286for pooling problem, 212–214in train-dispatching problem, 70

Hewlett Packard, 458Hidden action, in licensing contract optimization,

474Hidden constraint problems, 507–517

convergence results, 510–514imfil.m code, 514–517implicit filtering, 508–510with random directions, 515–517

Hidden constraints, 507Hidden information, 474Hierarchical approach to planning and scheduling

under uncertainty, 394High-dose rate (HDR) brachytherapy, 99–101Hölder condition, 172Holding costs, 449

in dynamic economic lot-sizing problem, 442in dynamic stochastic optimization, 382–383in EOQ problem, 440–441in finite-horizon problem, 445in infinite-horizon problem, 448local vs. echelon, 449in newsvendor problem, 443

Horn Rev wind farm, 368Hospital readmissions, 96–99Huber penalty function, 31Humanitarian applications of supply chain

optimization, 479–491bottlenecks in transportation networks,

487–491challenges, 479–480facility location, 480–483future perspectives, 491inventory modeling, 483–487

HVAC (heating, ventilation, and air-conditioning)system, 259–261

Hybrid process control, 317Hydrodealkylation (HDA), 318–320Hydroelectric generators, marginal water

valuation for, 405–406Hypergraphs, 102

IB (investment buying) model, 475–477IDEAS (Infinite DimEnsion A1 State-space), 320IFFCO (solver), 500IIR (infinite impulse response) filters, 29Image deblurring, 32

imfil.m code, 507, 514–517Implicit filtering

applications of, 508convergence theorem for, 509–510imfil.m code for, 514–517

Implicit function theorem, 252IMPT. See Intensity-modulated proton therapyIMRT. See Intensity-modulated radiation therapyIncremental risk contributions, 432Independent system operator (ISO), 358Index tracking problem, 152Indistinguishable scenarios, 385Inequalities, information theory, 34–35Infeasibility, certificate of, 277Infeasible IPMs, 6, 8Infimum, 124, 125Infinite DimEnsion A1 State-space (IDEAS), 320Infinite-horizon inventory problem, 446–448Infinite impulse response (IIR) filters, 29Inflow spreading, 420–421Influenza vaccine, production planning for,

470–472Information and coding theory, 34–35Information-theoretic inequality prover (ITTP),

35Informative callback, 62INFORMS (organization), 58Infrafraction motion, 348Infrastructure deterioration, 465–467Initial conditions, control over set of, 130–133Inner approximation approach, 171Integer optimization, 49–63

for CCSO problems, 388–389combinatorial optimization, 52–53heuristics, 57–58impact of MIP technology, 58–61lift-and-project tool, 55–56medical and healthcare applications, 93–103

diagnosis and detection, 93–94discriminant analysis, 96–99health logistics and operations, 95open challenges, 102public health, 95solution strategies, 102TCP-driven PET-image-guided

treatment-planning model, 99–101treatment design, 94–95

protein-DNA system design, 176–180input parameters and constraints, 177–179protein-DNA sequence selection, 179–180

scope and applicability, 50–51solution methods, 53–55train dispatching applications, 65–75

basic MILP models, 68–70

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Index 677

decomposition principle, 70–72dispatching in railway systems, 65–68real-life implementation, 72–75

using MIP codes, 61–63Integer rounding, 53–54Integer variables

derivative-free optimization for problems with,506

truss topology design problems with, 145–147Integrated facility location model, 466Intensity-modulated proton therapy (IMPT), 345,

346, 354–356Intensity-modulated radiation therapy (IMRT)

integer optimization in, 94–95robust optimization in, 345–347, 355treatment planning for, 346–347

Interfraction motion, 350Interior-point condition (IPC), 6Interior-point methods (IPMs)

active-set SQP methods vs., 224complexity of, 8–9for conic linear optimization, 116–118extensions of, 11implementation of, 9–10interior optimal solution from, 9for linear optimization, 6–11for nonlinear optimization, 233–235for quadratic optimization, 7

Interior solution, basis vs., 9Interlocking routes, 66Intermediate-capacity power generators, 460Intermittent power generation, 459–461Inventory control policy, 462, 473Inventory level

in EOQ problem, 440–441in finite-horizon problem, 445in infinite-horizon problem, 446–448in MILO for chemical engineering, 80, 84

Inventory managementin energy industry, 461–463in healthcare, 473

Inventory modeling, for humanitarianapplications, 483–487

Inventory optimization, 439–450deterministic inventory optimization, 440–443

dynamic economic lot-sizing problem,442–443

EOQ problem, 440–442multiechelon inventory optimization, 449–450stochastic inventory optimization, 443–448

finite-horizon problem, 445–446infinite-horizon problem, 446–448newsvendor problem, 443–444supply uncertainty, 448

Inventory policies, 445–447Inventory routing, liquid natural gas, 82–86Investment buying (IB) model, 475–477Investment portfolio construction, 427–435

CVaR models, 430–432financial optimization, 427with marginal VaR and CVaR constraints,

432–435VaR models, 428–430

Iowa, energy industries in, 466IPC (interior-point condition), 6IPFILTER (solver), 234IPMs. See Interior-point methodsIPOPT (solver), 11, 196, 234, 247, 268–269ISO (independent system operator), 358Isocenter optimization, 94Italy

regional train dispatching in, 72–73train dispatching at large stations in, 74train dispatching for mass transit in, 75

ITTP (information-theoretic inequality prover),35

Jensen model, 368–369Job-shop scheduling problems, 69Joint replenishment problem (JRP), 442

Karush–Kuhn–Tucker (KKT) conditionsin equality-constrained NLO problems, 230in global optimization, 163, 167in interior-point methods, 234in MPCC formulation, 241nonlinear optimization methods that satisfy,

163, 225, 226in optimal power flow problem, 190, 192

K*DN A energy metric, 181–182Kernel trick, 31k-hypergraphs, 102KKT conditions. See Karush–Kuhn–Tucker

conditionsKnapsack constraint, 430Knapsack problem, 51KNITRO (solver), 233, 234–235, 289Kreisselmeier–Steinhauser (KS) function, 255

LaGO (solver), 289, 290Lagrangian function, 225Lagrangian multipliers, 139, 225, 452Lagrangian relaxation

in optimal power flow problem, 202–203for pooling problem, 215, 216solving UFLPs with, 452–453in supply network design for energy industry,

466LANCELOT (solver), 235

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678 Index

Large neighborhood search, 58Large-scale cogeneration system, short-term

planning in, 311–312Lasso method, 33Latin-hypercube sampling (LHS) technique, 525Latvia, freight train dispatching in, 74Lazy cut callback, 62–63LB (local branching) constraint, 58LB heuristic, 58, 286LCP (linear complementary programming), 317Leader-follower game model for biofuel

production decisions, 464Lead time

and committed service time, 449–450in dynamic economic lot-sizing problem, 442,

443in energy industry, 457–459in EOQ problem, 440, 441in infinite-horizon problem, 447in pharmaceutical capacity planning, 469, 470

Lead-time demand, 447Lead-time uncertainty, 448“Learn, then act” approach, 458Least-squares regression techniques, 500–501Leibniz’s rule, 444LGDP (linear generalized disjunctive

programming), 317LHS (Latin-hypercube sampling) technique, 525Licensing contracts, for new drugs, 474–475LICQ (linear-independence constraint

qualification), 225, 226Lift-and-project tool, 55–56Limited-memory quasi-Newton methods, 227Limiting growth of a state, modeling, 124LINDOGlobal (solver), 290Linear approximation, for OPF problem, 197–198Linear blending indices, 39Linear complementarity problem, 167Linear complementary programming (LCP), 317Linear constraints

direction sets for hidden vs., 511robust optimization with uncertainty in,

335–338Linear generalized disjunctive programming

(LGDP), 317Linear-independence constraint qualification

(LICQ), 225, 226Linearizations, for mixed-integer nonlinear

programming, 276Linearly constrained optimization, 502–503Linear matrix inequalities (LMIs), 110Linear model predictive control, 317Linear optimization (LO), 5–11. See also integer

optimization

algorithmic concepts, 5–6available solver software, 10–11basis vs. interior solution, 9chemical engineering, 37–46

design and operations applications, 37model predictive control, 41–46production planning, 37–41

complementarity conditions in SOCO vs., 116complexity, 8–9computational methods and software, 9–10and CVaR problems, 431–432distributional robustness, 348–350electrical engineering, 27–36

communication networks, 36filter design, 28–30information and coding theory, 34–35norm optimization, 31–34pattern classification, 30–31

financial engineering, 149–150general problem, 3–4and global optimization, 165integer optimization vs., 49, 51IPM extensions, 11and optimal control problems, 122, 125–129and polynomial optimization, 119for pooling problems, 212process systems engineering, 317and robust optimization, 337and semidefinite optimization, 108truss topology design, 13–25

ground structure approach, 13–14limitations, 23, 25LO formulations, 16–19minimum compliance problem, 19–21numerical experiments, 22–25structural analysis of trusses, 15–16

voxelwise worst-case roubustness via, 354–355Linear optimization bounds, for spherical codes,

34Linear phase filter design, 28–29Linear placement methods, 34Linear programming (LP) relaxation, 451Linear quadratic regulator (LQR), 123–124Linear relaxations, for pooling problem, 214–216Line-search methods, 228Liouville PDE, 127–128Liouville’s equation, 125Lipschitz condition, 171Lipschitz-continuous f, 513–514Lipschitz functions, 171Lipschitz optimization, 171–172LIPSOL (solver), 11Liquid natural gas (LNG) inventory routing,

82–86

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Index 679

LMIs (linear matrix inequalities), 110l1-norm optimization, 31–34

in circuit placement problems, 33–34in robust approximation problems, 31–32in sparse optimization problems, 32–33

LNG (liquid natural gas) inventory routing, 82–86LO. See Linear optimizationLocal branching (LB) heuristic, 58, 286Local holding costs, 449Local minima

for convex problems, 163and nonlinear optimization methods, 226, 251for quadratic problems, 166, 167

Local search approach, in healthcare applications,102

Local solution methodsfor nonlinear optimization, 163, 222, 251for pooling problem, 212–214

Log-concave distribution, CC problems with, 388LogicNet Plus, 88Logic propositions, in GDP models, 316Long-term development issues, 479, 491Long-term planning, in electric power systems,

364–365LOQO (solver), 11, 117, 234Lorentz cone, 107Loss function

in decentralized supply chain optimization, 454in infinite-horizon problems, 448in newsvendor problem, 444overproduction, 401underproduction, 401–402

Loss-of-goodwill cost, 443, 445Lost load, value of, 411Lost sales, 443, 445Lower bound

in branch-and-bound method, 274of value function, 129

LP (linear programming) decoding techniques, 35LP relaxation, 451LP/NLP-based branch-and-bound approach,

278–279LQR (linear quadratic regulator), 123–124

Macroscopic/microscopic approach, 70–71MADS. See Mesh adaptive direct searchMagnitude filter design, 29Main line railway systems, 66Mali, delivery routing in, 489Mangasarian–Fromowitz constraint qualification

(MFCQ), 225Marching cube algorithms, 520Marginal benefit, in inventory modeling, 485Marginal CVaR constraints, 432–435

Marginal risk contributions, 432Marginal VaR constraints, 432–435Marginal water valuation, 405–425

hydroelectric generators with reservoirs,405–406

multiple-reservoir model, 416–419New Zealand electricity system, 419–425observed electricity prices vs. model outputs,

424–425single-reservoir model, 411–416social planning problem formulation, 406–411

Marginal water values, 409Margin-based planning, 353Mass, in aerospace systems, 249Mass balances

for distillation column, 239in physical building model, 261–262

Mass-equilibrium-summation-heat (MESH)equations, 238–239

Mass transit railway systems, 66, 75Master model for nonlinear least squares, 532–533Master problem, in Benders decomposition, 71Matching problem, 52Material balances, in physical building model,

261–262Mathematical programs with complementarity

constraints (MPCCs), 235column optimization with phase changes

formulated as, 242–243formulation of distillation systems as, 240–241optimal design of distillation columns

formulated as, 243–245Mathematical programs with equilibrium

constraints (MPECs), 465Matrix-free approach to nonlinear optimization

problems, 224Maximal covering location problems, 453Maximally complementary optimal solutions, 4Maximization of expected portfolio return with

upper bound on CVaR problem, 431Maximization of expected portfolio return with

upper bound on VaR problem, 428Maximum clique problems, 165Maximum comfort tracking strategy, 265, 268Maximum independent set problems, 165Maximum stiffness problem. See Minimum

compliance problemMaximum volume assumption, in truss topology

design, 137McCormick envelopes, 198–199, 214MCS (solver), 506MDO. See Multidisciplinary design optimization

for aerospace systemsMean-CVaR model, 431

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680 Index

Mean-variance portfolio optimization problemwith convex quadratic constraints, 151–153robust optimization in, 155–157with transaction costs, 153–155

Mean-VaR model, 429Medicine

diagnosis and detection, 93–94discriminant analysis in, 96–99health logistics and operations, 95integer optimization applications, 93–103open challenges for, 102public health, 95solution strategies in, 102TCP-driven PET-image-guided

treatment-planning model, 99–101treatment design in, 94–95

Mehrotra’s predictor-corrector algorithm, 6Merit functions, 228–229, 505Mesh adaptive direct search (MADS), 500, 503, 504MESH (mass-equilibrium-summation-heat)

equations, 238–239Metaheuristics

for integer optimization, 57, 58for mixed-integer nonlinear optimization, 301,

327for optimal power flow problem, 192

MFCQ (Mangasarian–Fromowitz constraintqualification), 225

MH (moral hazard), 474Microgeneration system, short-term planning in,

310–311MIDO (mixed-integer dynamic optimization), 328MILANO (solver), 289Milano Underground System, 75Military, min-max-min models in, 341–342Milk banks, South African, 481MILO. See Mixed-integer linear optimizationMILP. See Mixed-integer linear programmingMin-gen penalty, 460Minima

global minimain aerospace applications, 251and CLO problems with integer variables, 145for concave minimization problems, 164, 170,

172for convex vs. nonconvex problems, 163and termination of branch-and-bound

algorithms, 170local minima

for convex problems, 163and nonlinear optimization methods, 226, 251for quadratic problems, 166, 167

in MINLP model of aircraft conflict avoidance,297

Minimax stochastic optimization, 344Minimization of CVaR problem, 431Minimization of VaR problem, 428Minimum compliance problem, 19–21, 137–138Minimum Frobenius norm models, 497Minimum volume problems, 16–18, 22–23MINLO. See Mixed-integer nonlinear

optimizationMINLP. See Mixed-integer nonlinear optimizationMINLPBB (solver), 289Min-max-min models, 341–342MINOS (solver), 235MINOTAUR (solver), 289MIP. See Mixed-integer programmingMIQCP (mixed-integer quadratically constrained

programming), 273–274, 282MISOCP (mixed-integer second-order cone

programming), 274, 286Michell beam, 22, 23Mixed-integer dynamic optimization (MIDO), 328Mixed-integer linear optimization (MILO)

for aircraft conflict avoidance, 294–295. See alsoMixed-integer linear programming (MILP)

and chance constrained stochastic optimization,389

chemical engineering applications, 77–91chemical supply network optimization, 86–90LNG inventory routing, 82–86production planning for air separation plants,

78–82for generalized pooling problem, 212in optimal power flow problem, 199–202, 205process systems engineering applications, 317train dispatching applications, 68–70

Mixed-integer linear programming (MILP). Seealso Mixed-integer linear optimization(MILO)

discretization, 213–214, 217piecewise relaxations, 215–216relaxations, 216–217

Mixed-integer nonlinear optimization (MINLO),273–292

air traffic management, 293–301aircraft conflict avoidance, 294–295automation in air traffic management, 293MINLP formulations, 295–300solution approaches, 300–301

applications, 291–292branch-and-bound method, 274convex, 275–279enhancement techniques, 284–287expression of problem, 273for generalized pooling problem, 212general model, 316

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Index 681

nonconvex, 279–284process systems engineering, 315–329

molecular computing, 329planning and scheduling, 327–328process control, 328–329process synthesis, 317–327superstructure, 315–316types of models, 316–317

for SCUC problem, 191short-term planning in cogeneration energy

systems, 303–314computational experiments, 310–314data-driven vs. first-principles approaches,

304–305described, 303–304energy system variations, 305MINLP formulations, 307–310unit commitment vs., 305–307

solver software, 287–291special cases, 273–274

Mixed-integer programming (MIP)codes, 61–63defined, 49discriminant analysis via, 96–99fixed charge problem in, 50heuristics for, 57–58impact of, 58–61in medical diagnosis and detection, 94

Mixed-integer quadratically constrainedprogramming (MIQCP), 273–274, 282

Mixed-integer second-order cone programming(MISOCP), 274, 286

Model-based trust-region methods, 505–506, 521Model predictive control (MPC)

building automation, 259, 262–268case studies, 266–268constraints, 263–264control strategies, 265–266multiple objectives, 265objective functions, 264

LO and QO models, 27, 41–46MPC formulations, 42–43plant automation structure, 44–46variants and extensions, 43–44

nonlinear, 247, 317Modes of operation, 78–79Molecular computing, 317, 329Moment sum of squares, 204Monfalcone, Italy, 74–75Monte Carlo sampling, 388–389Moore’s law, xxixMoral hazard (MH), 474MOSEK (solver), 10, 117Motzkin–Straus QP, 165

MPC. See Model predictive controlMPCCs. See Mathematical programs with

complementarity constraintsMPECs (mathematical programs with equilibrium

constraints), 465MSSLP (multistage stochastic linear

programming) problems, 384–385Multicolumn distillation systems, 246Multicommodity flow formulation, 211–212Multidisciplinary design optimization (MDO) for

aerospace systems, 249–257aerodynamic shape optimization, 252–254aerostructural design optimization, 254–256computational models, 250described, 249–250gradient computation, 251–252optimization algorithms, 251optimization problems, 250–251satellite design and operation, 256–257

Multiechelon inventory optimization, 449–450,463–465

Multigroup classification problems, 94Multiobjective optimization, 506Multiobjective studies, nonlinear optimization in,

260, 265Multiperiod dispatch problem, 364Multiperiod problems, 382–384Multiple countries, vaccine issues involving, 472Multiple-reservoir model for marginal water

valuation, 416–419Multistage robust optimization, 343, 364Multistage stochastic linear programming

(MSSLP) problems, 384–385Multistage stochastic optimization model, 394Mutation constraints, protein-DNA system, 178,

182Myopic adaptive reoptimization, 343–344

Nash competition models, 464, 465Natural disasters, 479Natural gas industry

infrastructure impacted by, 466–467inventory management for storage facilities in,

462Neighborhood, 58Neighborhood search

in integer optimization, 57, 58large, 58relaxation-enforced, 286relaxation-induced, 58variable, 212, 213, 301

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682 Index

Nelder–Mead algorithm, 498NEOS Guide, 222NEOS (Network-Enabled Optimization System)

server, 119, 288Nested decomposition method, 386, 389–391Netherlands Railways, 58–59Net load, worst-case, 362Net load uncertainty, 360–362Network design

energy supply, 465–467and facility location problems, 450

Network-Enabled Optimization System (NEOS)server, 119, 288

Network flow model, convex cost, 487, 490–491Network location models, 451Network optimization, MILP models for, 86–90Network utility maximization (NUM) problem,

36NEWUOA (solver), 498, 503Newsvendor problem, 381

and decentralized supply chain optimization,453

for donated blood units in humanitarianapplications, 484

for fee-for-service contracts for brand-namedrugs, 476

and influenza vaccine supply chain, 470and multiechelon inventory optimization, 449as stochastic inventory optimization, 443–444with yield uncertainty, 448

Newton–Krylov method, 255Newton’s method, 227Newton system, 6, 7New Zealand electricity system, 419–425Next Generation Air Transportation System

(NextGen), 293NGBD (nonconvex generalized Benders

decomposition), 283NLO. See Nonlinear optimizationNodal formulation, for stochastic optimization,

386Node callback, 63No duality gap, polynomial optimal control

problems with, 129Noise analysis, 522–523Noise-handling strategies, 522–525Noise level reduction, 523–524Noisy functions, in derivative-free optimization,

500–501NOMAD (solver), 500Nominal problem

in robust optimization, 334

Nominal wind farm layout modelsperformance of robust vs., 371–374problem formulation for, 369–370

Nonanticipativity condition, 364Nonanticipativity constraints, 386Nonconvex generalized Benders decomposition

(NGBD), 283Nonconvex MINLP, 279–284

defined, 274domain propagation and bound tightening for,

281–282factorable programming for, 279–280piecewise-linear approximations and relaxations

for, 283–284relaxations of structured sets for, 282–283solver software for, 288–291spatial branch-and-bound for, 280–281

Nonconvex nonlinear optimization problems,222–223

Nonconvex optimization problems,computational complexity of, 167

Nonideal phase equilibrium, 246Nonlinear blending, 40Nonlinear branch-and-bound approach, 275–276,

279Nonlinear classifiers, 31Nonlinear constraints

with general uncertainty set, 338–339for short-term cogeneration systems planning,

309–310uncertainty in, robust optimization with,

338–339Nonlinearity, in refining planning problems, 40Nonlinear least squares, 529–530, 532–533Nonlinearly constrained optimization, 503–506

extensions, 506model-based trust-region methods, 505–506relaxable constraints, 504–505unrelaxable constraints, 504, 505

Nonlinear model predictive control, 247, 317Nonlinear optimal control, 121–133

approximation results, 129–130control over set of initial conditions, 130–133history of optimal control, 121–122LP formulation, 125–129polynomial optimal control, 122–124

Nonlinear optimization (NLO), 221–235aerospace system applications, 250–251algorithm frameworks, 229–235availability of derivatives, 223building automation, 259–269

building operations and inefficiencies,260–261

computational considerations, 268–269

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Index 683

HVAC system automation, 259physical building model, 261–262predictive control, 262–268

characteristics of problems, 222–224convexity in, 163convex vs. nonconvex problems, 222–223distillation systems, 237–247

basic distillation model, 238–240case studies, 241–245described, 237–238extensions, 245–247MPCC formulation, 240–241

for equal-risk contribution portfolios, 158–159extensions, 235general problem statement, 221interior-point methods, 233–235line-search and trust-region globalization

methods, 227–228merit functions and filters, 228–229Newton and quasi-Newton methods, 227optimality conditions, 224–226and polynomial optimization, 119process systems engineering applications, 317refinery planning problems, 41sequential quadratic programming methods,

229–233truss topology design, 16, 136–139

dual problem, 139existence of solution, 137–139primal problem, 137and semidefinite optimization formulation,

143voxelwise worst-case roubustness via, 354–355for worst-case robustness, 352–354

Nonpolyhedral envelopes, 283Non-Shannon-type inequalities, 35Nonsmooth functions

linearly constrained optimization, 503sampling along directions for, 503unconstrained optimization, 499–500worst-case complexity bounds, 501

No relaxation gap assumption, 125Normal distribution, 388, 398, 429, 448Norm optimization, 31–34

circuit placement, 33–34robust approximation, 31–32sparse optimization, 32–33

Norway, train dispatching in, 74NOWPAC (solver), 506NP-completeness, theory of, 52NP-hard problem(s)

conic linear optimization, 113facility location, 450, 451global optimization, 166–167

integer optimization, 96, 99inventory optimization, 442mixed-integer nonlinear optimization, 274optimal power flow problems as, 191, 203pooling problem as, 207in protein-DNA system design, 175robust optimization, 343, 362

NUM (network utility maximization) problem,36

Nurse scheduling, 95

OA technique. See Outer approximationtechnique

OBBT (optimality-based bound tightening),281–282

Objective functionsbuilding automation, 264convex, 441, 444, 449evaluation of, in TAO, 535–536pooling problem, 209

Observed electricity prices, 424–425Occupation measures, 125–128, 131–133ODE (ordinary differential equation), Cauchy,

127–128Offtake delays, 488–490OMEGA technology, 208One-degree-of-freedom cogeneration units, 305On-hand inventory, 383, 440, 445, 476Operating reserve markets, 462, 463Operating room scheduling, 95Operational planning, 393–394OPF problem. See Optimal power flow problemOpportunistic polling, 499Optimal control. See also Polynomial optimal

control problems (POCPs)nonlinear, 121–133

approximation results, 129–130control over set of initial conditions, 130–133history of optimal control, 121–122LP formulation, 125–129polynomial optimal control, 122–124

polynomial, 122–124examples of, 123–124general description, 123history of, 121–122

Optimal ordering policies, in healthcare, 473Optimal policy, for marginal water values,

414–415Optimal power flow (OPF) problem, 187–205

convex outer approximation for, 198–202described, 187–188example data for, 192–197formulation for, 189–190future research directions, 205

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684 Index

Lagrangian relaxation for, 202–203linear approximation with direct current flow,

197–198McCormick envelopes in, 198–199moment sum of squares for, 204OPF problem described, 187–188piecewise-linear envelopes in, 199–202problem statement for, 188solution methods, 191–205spatial branch-and-bound algorithm, 204–205sufficient strong duality condition, 203–204variants and extensions for, 190–191

Optimal reactive power flow (ORPF) problem,191

Optimality-based bound tightening (OBBT),281–282

Optimality conditionsin conic linear optimization, 115–116for global optimization, 167–168for nonlinear optimization, 224–226Pareto robust, 340

Optimality tolerance, of branch-and-boundmethod, 274

OQNLP (solver), 290Order-up-to level (decision variable), 445, 446Ordinary differential equation (ODE), Cauchy,

127–128ORPF (optimal reactive power flow) problem, 191Outer approximation (OA) technique

for concave minimization problem, 170–171convex, 198–202for convex MINLP problems, 277for optimal power flow problem, 198–202

Output, cost-sharing contracts based on, 471Outsourcing of pharmaceutical production,

469–470Overage cost, 381, 443Overproduction demand constraints, 396Overproduction loss function, 401

Parallel deterministic parameters, 62Parallelism, in TAO, 535–536Parallelizing DFO approaches, 506Parallel opportunistic parameters, 62Parallel processing, xxix–xxxParameterization, of uncertainty set, 339Parent nodes, branch-and-bound tree, 275Pareto front, 264Pareto robust optimization paradigm, 340Partial differential equations (PDEs)

in aerospace engineering, 250–252direct methods for evaluating derivatives of, 252discretization of three-dimensional, 224, 231Liouville, 127–128

Partial-load penalty, 460Partitioning of uncertainty, 342–343PATHNLP (solver), 196Pattern classification, 30–31Pavement rehabilitation, 466PcX (solver), 11PDEs. See Partial differential equationsPeaking power generators, 460Peaking premium, 460Peano curves, 172Penalties

exact penalty function, 228–229Huber penalty function, 31min-gen, 460partial-load, 460stockout, 380–381sum-of-squares, 31worst-case, 353–354

Penalty-steering methods for nonlinearoptimization, 229

PENNON (solver), 235Pennsylvania, infrastructure and energy industry

in, 466PENOPT solvers, 118Periodic event-scheduling problem (PESP)

constraints, 59Periodic review

in dynamic economic lot sizing problem,442–443

infinite-horizon problem with, 446Perishable goods

in healthcare settings, 473in humanitarian applications, 483–487in newsvendor problem, 443

PESP (periodic event-scheduling problem)constraints, 59

PET (positron emissiontomography)-image-guidedtreatment-planning model, 99–101

P-formulation, pooling problem, 209–210, 214,215

PFV integrase enzyme, 182–184constraints and force fields, 182–183protein-DNA design results, 183–184template generation, 182

Pharmaceutical capacity planning, 469–470Pharmaceuticals

fail-to-supply contracts for generic drugs,477–478

fee-for-service contracts for brand-name drugs,475–477

global healthcare logistics for, 491licensing contracts for new drugs, 474–475

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Index 685

Phase changes, distillation column optimizationwith, 242–243

Phase equilibrium, 237, 238, 246Phase-shifting and tap-changing transformers, 190Physical coupling of building control systems, 261Pickup and delivery process, MIP for, 60Piecewise-constant functions for uncertainty,

342–343Piecewise-linear approximations, 283–284Piecewise-linear envelopes, 199–202Piecewise-linear outer estimators, 284Piecewise-linear relaxations, 284Pivot algorithms

complexity of, 8for linear optimization, 5–6optimal basis solution from, 9for quadratic optimization, 7

Pivot rules, 5–6Pivots, defined, 5Piyavskii’s algorithm, 172Planning

capacity planningenergy industry, 457–458healthcare, 469–470

in chemical plants, 393–394long-term planning in electric power systems,

364–365margin-based, 353MINLO and GDP applications, 327operational, 393–394production planning, 470–472

air separation plants, 78–82healthcare, 470–472linear optimization, 37–41

refinery planning, 37–41models, 327planning model extensions, 40–41process description and LO formulation,

38–40short-term planning in cogeneration energy

systems, 303–314computational experiments, 310–314data-driven vs. first-principles approaches,

304–305described, 303–304energy system variations, 305MINLP formulations, 307–310unit commitment vs., 305–307

social planning problems, 406–411supply chain planning model, 393–394TCP-driven PET-image-guided model, 99–101trajectory planning, 295treatment planning

described, 346–348

distributional robustness, 348–350integer optimization applications, 94–95probabilistic robustness, 350–352robust optimization, 346–355voxelwise worst-case roubustness, 354–355worst-case roubustness, 352–354

types of models for, 317urban, xxxi

Planning horizonand adaptive robust UC model with net load

uncertainty, 361for dynamic economic lot-sizing problem, 442and inventory management for energy storage

facilities, 461and LNG inventory routing, 83–85and marginal water values, 407, 409for optimal power flow problem, 187, 188, 206and pharmaceutical capacity planning, 470for scenarios, 385

Plant automation, 44–46Platelet inventory management, 473P-median problems, 453PMV (predicted mean vote), 263–264PMV constrained strategy, 266, 268, 269POCPs. See Polynomial optimal control problemsPointed cone, 112Poisson distribution, 444, 447, 486Polar coordinates, 189–190Policy, defined, 385Polling

linearly constrained optimization, 503probabilistic descent, 502unconstrained optimization, 498–499

Polyhedral annexation, 171Polyhedral combinatorics, 52Polyhedral constraints strategy, 266, 269Polyhedral envelopes, 282Polyhedral relaxations, 276–277Polyhedral uncertainty set, 335Polynomial models for unconstrained

optimization, 497Polynomial optimal control, 122–124

examples of, 123–124general description, 123history of, 121–122

Polynomial optimal control problems (POCPs)approximation results for, 129–130with control over initial conditions, 130–133examples of, 123–124general description, 123LP problem formulation, 125–129

dual problem, 129occupation measure, 126–128

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686 Index

primal problem, 128–129relaxed controls, 125–126

Polynomial optimization, 119–120Polynomial optimization problems (POPs),

119–121Pooling, in refining planning problems, 40Pooling problem, 207–217

blending problem vs., 207computational advancements, 216–217described, 207–208formulation, 208–212

P-formulation, 209–210PQ-formulation, 211problem statement, 208–209Q-formulation, 210TP- and STP-formulations, 211variants and extensions, 211–212

global solution methods, 214–216local solution methods and heuristics, 212–214variants and extensions, 211–212

POPs (polynomial optimization problems),119–121

p-order cone, 112Port delays, 487–489Portfolio optimization problems

conic linear optimization models for, 151–153typical formulation for, 149–150

Portfolio selection, 149Positive duality gap, conic optimization with,

113–114Positive semidefinite (PSD) matrices, 108Positive semidefiniteness constraints, 56PSD relaxations, 205Positive spanning sets (PSSs), 498–499Positron emission tomography

(PET)-image-guided treatment-planningmodel, 99–101

POUNDERS (Practical Optimization Using NoDerivatives for Sums of Squares) solver,529–539

algorithm underlying, 533–535and DFO described, 529–530energy density functional calibration, 536–539inputs, 536smooth residual models, 530–533Toolkit for Advanced Optimization software,

535–536Power approximation, 446Power generators, capacity certainty and volume

flexibility of, 459–461Power market competition, 461Power system management, 457–461

efficient and responsive sourcing in capacityplanning, 457–458

random capacity and volume flexibility, 459–461Power system operations, decision-making

problems in, 357–358PPD (predicted percentage dissatisfied), 263–264PQ-formulation, pooling problem, 211, 214–216Practical Optimization Using No Derivatives for

Sums of Squares solver. See POUNDERSPraxair, 77–82Predicted mean vote (PMV), 263–264Predicted percentage dissatisfied (PPD), 263–264Presolve technique, for MINLP problems, 284Pressure control system, 260Primal-dual approach for nonlinear optimization,

234–235Primal heuristics, 285–286Primal problem

in nonlinear optimization, 137for optimal control applications, 128–129in second-order cone optimization, 139–140in semidefinite optimization, 108, 141in truss topology design, 137, 139–141

Primal simplex method, 22–25Primal simplex method pivot rules, 5–6Primitive uncertainty sets, 342Principal-agent problems, 474Principal minors, 108Prioritized usage, inventory modeling with,

485–487Priority dispatch policy, 460–461Private-sector facility location, 450Probabilistic descent, 502Probabilistic Markowitz model, 429Probabilistic models, derivative-free optimization

for, 502Probabilistic robustness, 350–352Probabilistically constrained stochastic

optimization, 387Probability of technical success (PTS), 474Probing method, 282Process control. See also Model predictive control

(MPC)LO and QO for, 37MINLO and GDP applications, 328–329types of models for, 317

Process flowsheet synthesis, 317–319Processing time, 395–396, 450, 489Process synthesis, 317–327

aggregated models, 317–318distillation sequences, 323–324heat exchange networks, 324, 325process flowsheet synthesis, 318–319reactor networks, 319–320rigorous models, 318shortcut models, 318

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Index 687

single distillation columns, 320–323types of models for, 317utility systems, 324–326water networks, 326–327

Process systems engineering (PSE)MINLO and GDP applications, 315–329

molecular computing, 329planning and scheduling, 327–328process control, 328–329process synthesis, 317–327superstructure, 315–316types of models, 316–317

MIP in, 77Process yields, refining, 40Production planning

air separation plants, 78–82healthcare applications, 470–472linear optimization, 37–41

planning model extensions, 40–41process description and LO formulation,

38–40Production volume, 471Progressive barrier method, 505Projected aggregate production profiles, 393Proper cone, 112Proportional market impact cost model, 153–154Proportional model. See Q-formulationProtein-binding cavity alignment via DFO-VASP,

519–528computational experiments, 525–527DFO method, 521–522electrostatic data, 521noise-handling strategies for VASP, 522–525protein specificities, 519–520VASP method, 520

Protein-DNA sequence selection, 179–180Protein-DNA system design, 175–185

biological applications of optimization, 175–176energy metric, 181–182fold specificity, 180–181global optimization methods, 176–177input parameters and constraints, 177–179for PFV integrase enzyme, 182–184protein-DNA sequence selection, 179–180

Protein families, 525–526Protein specificities, 519–520Pruning (fathoming), 169, 275PSD (positive semidefinite) matrices, 108PSD relaxations, 205PSE. See Process systems engineeringPseudocost branching, 285PSSs (positive spanning sets), 498–499PTS (probability of technical success), 474Public health, 95

Public-sector facility location, 450, 453Purchase costs, 440, 442, 445pyOpt interface, 251

QCO (quadratically constrained optimization),152–153

QCQO (quadratically constrained quadraticoptimization) problem, 139, 211

Q-formulation, pooling problem, 210, 214, 215QO. See Quadratic optimizationQP. See Quadratic optimization problemQuadratically constrained optimization (QCO),

152–153Quadratically constrained quadratic optimization

(QCQO) problem, 139, 211Quadratic functions, relaxations of, 282Quadratic interpolation models, 530–531Quadratic models for smooth functions, 497Quadratic optimization (QO). See also Quadratic

optimization problemalgorithmic concepts, 7available solver software, 10–11basis vs. interior solution, 9chemical engineering, 37–46

design and operations applications, 37model predictive control, 41–46production planning, 37–41

complexity, 8–9computational methods and software, 9–10electrical engineering applications, 27–36

communication networks, 36filter design, 28–30information and coding theory, 34–35norm optimization, 31–34pattern classification, 30–31

financial engineering applications, 149and global optimization, 165–166IPM extensions, 11for portfolio optimization problems, 151–152process systems engineering applications, 317

Quadratic optimization problem (QP). See alsoQuadratic optimization

active-set solvers, 232–233equality-constrained, 230–231general, 4, 231–232general purpose solvers, 233SQP methods for, 229

Quadratic placement methods, 34Quadratic problems, convexity of, 166–167Quadratic unconstrained binary optimization

(QUBO) problems, 164–165, 168Quasi-Newton approximation, 227, 497

Radial basis functions (RBFs), 497–498Radiation therapy

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688 Index

integer optimization, 94–95robust optimization, 345–356

intensity-modulated radiation therapy,345–346

LO for distributional robustness, 348–350NLO for worst-case robustness, 352–354scenario doses, 347–348SOCO for probabilistic robustness, 350–352treatment planning, 346–347uncertainties, 347voxelwise worst-case robustness, 354–355

Railway systems, dispatching in, 65–68Railway timetables, 58–59Railway transportation, 65Random capacity, in power system operations,

459–461Random directions, hidden constraint problems

with, 515–517Random-start approach, 525Random yield, in vaccine planning, 471RANS (Reynolds-averaged Navier–Stokes)

equations, 253RBFs (radial basis functions), 497–498R&D (research and development), collaborative,

474–475RdDS approach, 500, 503, 504Reactive power planning (RPP) problem, 191Reactor networks, 317, 319–320Read command, MIP, 61Realized scenarios, 334Real options, 462Real-time operation, in electric power systems,

364–365Real-time optimization (RTO), 44–46, 317Real-time pricing (RTP), 78Reboilers

in distillation system optimization, 238, 242,246, 247

in MINLP for single distillation column,321–323

RECIPE heuristic, 286Recourse-based robust optimization, 340–344

affine adjustable robust solutions from, 342finite adaptability with, 342–343fully adjustable robust solutions in, 340–341min-max-min models in, 341–342myopic adaptive reoptimization with, 343–344

Rectangular coordinates, 189–190Reduced-cost bound tightening, 281–282Reduced Hessian matrix, 231Reduced-order models (ROMs), 318Reduced-space methods

in aerospace systems, 250–251for nonlinear optimization, 224, 230

Reference solutions, 58Refinery planning, 37–41

models for, 327planning model extensions, 40–41process description and LO formulation, 38–40

Reformulation-linearization technique (RLT)constraints in, 180generating stronger relaxations with, 282and PQ-formulation of pooling problem, 211

Regional trains, 72–74Relative humidity control system, 260, 261Relaxable constraints, 504–505Relaxation-enforced neighborhood search

(RENS), 286Relaxation-induced neighborhood search (RINS),

58Relaxations

in branch-and-bound method, 274envelope-based, 214–215Lagrangian relaxations

optimal power flow problem, 202–203pooling problem, 215, 216solving UFLPs, 452–453supply network design for energy industry,

466linear, for pooling problem, 214–216linear programming, 451MILP, 216–217piecewise-linear, 284polyhedral, 276–277positive semidefinite, 205of quadratic functions, 282of structured sets, 282–283

Relaxed controls, LP problem with, 125–126Reliability, of electric power systems, 357Reliability branching, 285Renewable energy. See also Wind farms

infrastructure and transportation issues relatedto, 466

intermittent sourcing of, 459–461Renewable Identification Number (RIN) system,

465RENS (relaxation-enforced neighborhood search),

286Reorder point, 446Research and development (R&D), collaborative,

474–475Reserves, electric power system, 360Reservoirs, hydroelectric generators with,

405–406Residuals, modeling, 531–532Resource allocation problems, 95, 96Responsiveness–efficiency trade-off, in strategic

sourcing, 457–458

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Index 689

Restoration methods, in derivative-freeoptimization, 505

Reverse mode algorithmic differentiation, 252Reynolds-averaged Navier–Stokes (RANS)

equations, 253Rich direction sets, 512Rigid templates, protein-DNA system, 178Rigorous models, for process synthesis, 318RINS (relaxation-induced neighborhood search),

58RIN (Renewable Identification Number) system,

465Risk-averse SO models, 389–391Risk budgeting, 435Risk management, 149RiskMetrics framework, 428Risk neutrality, 425RLT. See Reformulation-linearization techniqueRobust approximation, 31–32Robust counterpart, 334Robust counterpart optimization, 394Robust mean-variance optimization, 155–157Robust optimization, 333–344

applicability, 333–334distributionally robust optimization, 344electric power systems, 357–365

decision-making problems in power systemoperations, 357–358

extensions, 362–364real-time operation and long-term planning,

364–365security-constrained unit commitment model,

359–362and linear optimization, 337Pareto paradigm, 340problem formulations, 334–340radiation therapy, 345–356

intensity-modulated radiation therapy,345–346

LO for distributional robustness, 348–350NLO for worst-case robustness, 352–354scenario doses, 347–348SOCO for probabilistic robustness, 350–352treatment planning, 346–347uncertainties, 347voxelwise worst-case robustness, 354–355

recourse-based robust optimization, 340–344and stochastic optimization, 397–400, 403–404train dispatching applications of, 66with uncertainty in linear constraints, 335–338with uncertainty in nonlinear constraints,

338–339uncertainty set parameterizations, 339wind farm layout, 367–375

history of layout optimization, 367–368performance of nominal vs. robust models,

371–374problem formulations, 369–371wake models, 368–369

Robust portfolio selection, 156–157Robust problems, 334Robust solutions

adjustable, 340–342affine adjustable, 342defined, 334fully adjustable, 340–341

Robust wind farm layout modelsperformance of nominal vs., 371–374problem formulation for, 370–371

Rolling stock scheduling, 59ROMs (reduced-order models), 318ROSA system, 59RosettaDock, 181Rounding heuristic, 286Rounds, of cuts, 56Route optimization, 60Routes, train, 67RPP (reactive power planning) problem, 191(r,Q) policy, 446, 447RTO (real-time optimization), 44–46, 317RTP (real-time pricing), 78Run/optimize command, MIP, 61

SA (stochastic approximation), 381–382, 390SAA (sample average approximation), 382, 390SABR (South African Breastmilk Reserve), 481Safety-first optimization problem, 429Sample average approximation (SAA), 382, 390Sampling

in linearly constrained optimization, 502–503for smooth functions, 496–499

Sampling error, 424Satellites, 256–257SBB (solver), 289SCED (security-constrained economic dispatch)

problem, 191Scenario-based models, 153, 401–404Scenario decomposition, 386, 389, 391Scenario doses, 347–348Scenario trees, 385, 387, 407–408Scenarios, 334, 385, 386Scheduling

in healthcare, 95integer optimization applications, 95MINLO and GDP applications, 328models for, 317in refinery planning, 40, 41train, 68

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690 Index

Schur complement theorem, 141, 142SCIP (solver), 290, 313–314SCUC model. See Security-constrained unit

commitment modelSDDP (stochastic dual dynamic programming),

406SDDP algorithm, 416–419, 424–425SDO. See Semidefinite optimizationSDPA (solver), 118SDPLR (solver), 118SDPNAL (solver), 118SDPT3 (solver), 117Search

convex hull, 213–214coordinate, 509–511direct-search methods

linearly constrained optimization, 502–503nonlinearly constrained optimization, 504probabilistic descent, 502unconstrained optimization, 498–499worst-case complexity bounds, 501–502

generalized pattern search, 499line-search methods, 228local search approach, 102mesh adaptive direct search, 500, 503, 504neighborhood search

in integer optimization, 57, 58large, 58relaxation-enforced, 286relaxation-induced, 58variable, 212, 213, 301

tabu, 57Search direction sets, 511–512Search step, in DFO algorithms, 506SEC (Securities and Exchange Commission), 475Second-order cone optimization (SOCO)

and chance constrained stochastic optimization,429–430

and conic optimization, 110–111filter design, 29–30financial engineering applications, 150–160

equal-risk contribution portfolios, 157–160portfolio optimization problems, 151, 153transaction costs with market impact,

153–155optimality conditions for, 116for probabilistic robustness, 350–352truss topology design, 139–141

dual problem, 140–141primal problem, 139–140reformulation of problems with integer

variables, 145–147Securities and Exchange Commission (SEC), 475

Security-constrained economic dispatch (SCED)problem, 191

Security-constrained unit commitment (SCUC)model, 191, 359–362

computational study, 361–362conservativeness of, 363–364with corrective actions, 363and deterministic model, 359–360extensions, 362–364robust model with net load uncertainty,

360–361solution method, 361–362

SeDuMi (solver), 11, 117Self-dual embedding model, 8Self-scaled cones, 112Selling season, 443Semidefinite optimization (SDO). See also

Second-order cone optimization (SOCO)in conic linear optimization, 107–110defined, 107examples, 109–110financial engineering applications, 150, 160generating stronger relaxations with, 282–283optimality conditions, 115–116for optimal power flow problem, 203–204for polynomial optimization problems, 119–121with positive duality gap, 113–114and robust optimization, 342software, 118standard formulation, 108truss topology design, 141–143

dual problem, 142–143nonlinear optimization formulation, 143primal problem, 141

with weak infeasibility, 114SEN (state equipment network) model, 323Sensitivity analysis, 374, 441Separable functions, 279Separation maneuvers, aircraft, 294, 295, 298Separations, models for, 317Sequentially convexifiable programs, 55Sequential quadratic programming (SQP)

composite step-based, 505with nonlinear branch-and-bound, 279nonlinear optimization, 229–233

active-set QP solvers, 232–233active-set SQP, 224equality-constrained NLO problems, 230–231general NLO problems, 231–232general purpose solvers, 233

Sequential synthesis method, 324Serial systems, 449–450Serine protease superfamily, 525, 526SESAR (Single European Sky ATM Research), 293

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Index 691

Set-back relaxation strategy, 265–268Set covering location problems, 453Set-partitioning problem, 51Set recovering problems, 51Setup costs. See Fixed costsSetup errors, in radiation therapy, 353–354Shifting bottleneck procedure, 57Shipment routing, in biofuel production, 466Shortages, drug, 477–478Shortcut models for process synthesis, 318Shortest-path problem, 442–443SHORTREC (Tactical Planning in Pickup and

Delivery) program, 60Short-term planning in cogeneration energy

systems, 303–314computational experiments, 310–314data-driven vs. first-principles approaches,

304–305described, 303–304energy system variations, 305MINLP formulations for, 307–310unit commitment vs., 305–307

Sifting method, 22–25Simplex gradients, 500, 512Simplex sets, 498Simultaneous synthesis method, 324Single European Sky ATM Research (SESAR), 293Single-period inventory model, 381Single-reservoir model for marginal water

valuation, 411–416Stage and Larsson vs. discounting methods,

411–413with thermal generation, 413–416

Single-server model assumption, 489Single-stage robust solutions, 340Site selection, wind farm, 367Slater’s constraint qualification, 114–115Slave problem, in Benders decomposition, 71SLP (successive linear programming) techniques,

212Smooth functions, 496–499Smoothing functions, 500Smooth problems, NLO problems as, 223Smooth residual models, 530–533

master model for nonlinear least squares,532–533

modeling residuals in DFO, 531–532quadratic interpolation models, 530–531

SO. See stochastic optimizationSNOPT (solver), 196, 233, 251Socially optimal demand, 472Social planning problem formulation, 406–411Social welfare, 470, 472, 477, 478SOCO. See Second-order cone optimization

Solve callback, 63SoPLex (solver), 10SOS2 (special-ordered sets of type 2), 284SOSTOOLS (software), 120Sources, in STP-formulation, 211South Africa, breast milk bank in,

480–483South African Breastmilk Reserve (SABR), 481South Dakota, infrastructure and energy industry

in, 466Space-filling curves, 172Sparse optimization problems, 32–33Sparse solution recovery theory, 497Sparsity

of building automation NLO problems, 268,269

and constrained optimization problems, 224of Hessian, 497and reduced-space approach, 224, 230

Spatial branch-and-boundfor nonconvex MINLP, 280–281for optimal power flow problem, 204–205piecewise-linear relaxations with, 284

Spatial branching, 280Spatial location equilibrium, 464Special-ordered sets of type 2 (SOS2), 284Spectral factorization method, 29Spectral mask constraints, 29Spherical codes, 34SQP. See Sequential quadratic programmingSSP (stochastic social planning) problems, 407–411(s,S) policy, 445–446Stability enhancement

MPC formulations for, 44and wind farm layout optimization, 372–374

Stackelberg game model, 454, 464–465Stage and Larsson method

for single-reservoir model, 411–412with thermal generation, 413–414

Stages, inventory chain, 449Stagewise independent data, 384Standalone risk approach, 432Standard robust optimization, 341State equipment network (SEN) model, 323States, wind, 368State-task-network (STN) model, 77, 323, 395State variables, 250–254Statically determinate structures, 18Static stochastic optimization, 380–382Stationary interval property, 440STATIONS (solver), 59Stations, railway, 66, 74Statistics-based approaches for noisy functions,

501

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692 Index

Stavanger–Moi line, 74Steepest descent method, 227Stencil failure, 509Stereotactic radiation, 94STN model. See State-task-network modelStochastic approximation (SA), 381–382, 390Stochastic control, 125–126Stochastic decomposition, 391Stochastic dual dynamic programming (SDDP),

406Stochastic dynamic programming model, 461, 462,

472Stochastic failures, 488–489Stochastic integer problems, 389Stochastic optimization (SO), 379–391

chance constrained, 387–389chemical engineering, 393–404

CVaR-based method, 401–404operational planning in chemical plants,

393–394problem statement, 394–397robust optimization method, 397–400,

403–404and distributionally robust optimization, 344dynamic, 382–387extensions, 389–390history, 390–391inventory optimization, 443–448

finite-horizon problem, 445–446infinite-horizon problem, 446–448newsvendor problem, 443–444supply uncertainty, 448

investment portfolio construction, 427–435CVaR models, 430–432financial optimization, 427with marginal VaR and CVaR constraints,

432–435VaR models, 428–430

marginal water valuation, 405–425hydroelectric generators with reservoirs,

405–406multiple-reservoir model, 416–419New Zealand electricity system, 419–425observed electricity prices vs. model outputs,

424–425single-reservoir model, 411–416social planning problem formulation,

406–411static, 380–382train dispatching applications of, 66uncertain parameters, 379

Stochastic programming. See Stochasticoptimization

Stochastic social planning (SSP) problems,407–411

Stockout costs, 443, 445, 447, 448, 453Stockout penalties, 380–381Stockouts, 84, 440–444Storage conversion loss, 462Storage efficiency, 462STP-formulation, pooling problem, 211, 214, 215Strategic behavior

by healthcare consumers, 471–472and marginal water values, 425

Strategic sourcing, 457–458Stress limits, 16Strict complementarity, 226Strong duality condition

for conic optimization, 113for optimal power flow problem, 203–204and solution of adaptive robust model, 361–362

Strong duality theorem, 3–4Structural engineering, xxxStructural optimization, 13Structural topology optimization, 13Structured sets, relaxations of, 282–283Subgradient optimization, 452Subliminal control, 294Substitution, at healthcare facilities, 473Successive approximation techniques, 170–171Successive linear programming (SLP) techniques,

212Successive partitioning, 169Successive underestimation methods, 171Sufficient strong duality condition, 203–204SUmb flow solver, 253Sum of squares, moment, 204Sum-of-squares certificate approach, 119–121Sum-of-squares penalty, 31Superpositions, protein, 519–520, 526–527Superstructure optimization, 315–316, 319–320Supply chain contracting, 473–478Supply chain coordination, 453–455Supply chain engineering

chance-constrained stochastic optimization in,387

dynamic stochastic optimization in, 382–384static stochastic optimization in, 380–381two-stage model for, 393–394

Supply Chain Guru, 88Supply chain network design, 465–467Supply chain optimization, 439–455

decentralized supply chains, 453–455energy industry applications, 457–467

feedstock procurement, 463–465inventory management, 461–463

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Index 693

power system management, 457–461supply network design, 465–467

facility location problems, 450–453healthcare applications, 469–478

capacity planning, 469–470inventory management, 473production planning, 470–472supply chain contracting, 473–478

humanitarian applications, 479–491bottlenecks in transportation networks,

487–491challenges, 479–480facility location, 480–483future perspectives, 491inventory modeling, 483–487

and inventory optimization, 439–450MIP for, 60–61mixed-integer linear optimization models for,

86–90Supply chains, decentralized, 453–455Supply disruptions, 448, 477, 478Supply uncertainty, 448, 459–461Support vector machines, 30, 98Surrogate models, 318Sustainable buildings, 259Symmetric cones, 112, 117–118Symmetry, MINLP problems with, 285

Tabu search, 57Tactical-level aircraft conflict avoidance, 294–295Tactical Planning in Pickup and Delivery

(SHORTREC) program, 60Take-off gross weight, 254–256Tandem queuing model, 488–489TAO (Toolkit for Advanced Optimization)

software, 535–536Targeting, reactor network, 320Taylor-like conditions, 531TCP. See tumor control probability (TCP)-driven

PET-image-guided treatment-planningmodel, 99–101

TE (tracking error) constraints, 152–153Temperature control system, 260, 261Terminal nodes, in TP-formulation, 211Terminal value function, 445Termination of branch-and-bound algorithm, 170Termination tests, 226Ternary mixture, distillation of, 242–243Thermal comfort constraints, 263–264Thermal generation, 413–416Thermally coupled systems, 323–324Threshold policy for marginal water values, 414Time configurations, for aircraft, 298Time-indexed formulations for train dispatching,

69

Time-of-use (TOU) rates, 78Time-sensitive electricity prices, 78–82Time-space network representation, 83Timetables, railway, 58–59, 67Time value of money, 445Time-window improvement heuristic, 85Time windows, in aircraft conflict avoidance,

298–299TMSs (train management systems), 67–68, 72–75TNT Express, 60–61TNT Express Routing and Network Scheduling

(TRANS) program, 60Toolkit for Advanced Optimization (TAO)

software, 535–536Topology design. See Truss topology designTotal mass balance, distillation column, 239Total probability, 385Total variation regularization, in image

deblurring, 32Total water networks (TWNs), 326–327TOU (time-of-use) rates, 78TP-formulation, pooling problem, 211, 214Traffic equilibrium, 466Train dispatching, 65–75

basic MILP models, 68–70decomposition principle, 70–72dispatching in railway systems, 65–68real-life implementation, 72–75

Train-dispatching problem, 67–70Train management systems (TMSs), 67–68, 72–75Trajectory planning, aircraft, 293, 294Transaction costs, with market impact, 153–155Transformers, phase-shifting and tap-changing,

190Transmission network planning problem, 365Transportation congestion, 465–467Transportation costs

in biofuel supply chain design, 466for breast milk bank, 480–483and facility location, 451, 480–483

Transportation networksbottlenecks in, 487–491delay modeling in delivery routing, 489–491port and corridor delays in, 487–489

TRANS (TNT Express Routing and NetworkScheduling) program, 60

Traveling salesman problem (TSP), 52–53Treatment planning

integer optimization applications in, 94–95robust optimization, 346–355

described, 346–348distributional robustness, 348–350probabilistic robustness, 350–352voxelwise worst-case robustness, 354–355

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694 Index

worst-case robustness, 352–354TCP-driven PET-image-guided model, 99–101

Tree networks, 450branch-and-bound tree, 275scenario trees, 385, 387, 407–408

Trento–Bassano del Grappa line, 72–73Trondheim–Dombås line, 74Truncation, in finite-horizon problem, 445Trusses, structural analysis of, 15–16Truss notation, 135–136Truss topology design

conic linear optimization models, 135–147applications, 144integer variables, problems with, 145–147nonlinear optimization formulation, 136–139SDO formulation, 141–143SOCO formulation, 139–141truss notation, 135–136vibration constraints, 144–145

linear optimization, 13–25ground structure approach, 13–14limitations, 23, 25LO formulations, 16–19minimum compliance problem, 19–21numerical experiments, 22–25structural analysis of trusses, 15–16

Trust-funnel methods, 505Trust-region methods

derivative-free optimizationlinearly constrained optimization, 503noisy functions, 501nonlinearly constrained optimization,

505–506probabilistic models, 502smooth functions, 496–498unconstrained optimization, 496–498, 501,

502model-based trust-region methods

derivative-free optimization, 505–506in protein-binding cavity volumetric

alignment, 521for nonlinear optimization, 228

TSP (traveling salesman problem), 52–53Tumor control probability (TCP)-driven

PET-image-guided treatment-planningmodel, 99–101

TURNI (software), 59TWNs (total water networks), 326–327Two-degrees-of-freedom cogeneration units, 305Two-ship improvement heuristic, 85Two-stage adaptable optimization problem,

340–341Two-stage approach to protein-DNA system

design, 176

Two-stage fully adaptive robust optimizationmodel

computational study, 362for electric power systems, 360–364extensions of, 362–364formulation, 360–361solution method, 361–362

Two-stage stochastic linear optimization approach,394

Two-stage stochastic planning under uncertaintymodel, 393

Two-stage stochastic supply chain planning model,393–394

Two-station, tandem queuing model, for port andcorridor delays, 488–489

Type-1 service level, 444

UC (unit commitment), 359UFLPs (uncapacitated fixed-charge location

problems), 451–453Uncapacitated fixed-charge location problems

(UFLPs), 451–453Uncertain parameters, stochastic optimization

with, 379Uncertainty(-ies)

budget of, 361capacity, 448demand, 393–394, 397distributional, 368in electric power systems, 358hierarchical approach to planning and

scheduling under, 394lead-time, 448in linear constraints, 335–338and MILP models, 91net load, 360–362in nonlinear constraints, 338–339partitioning of, 342–343piecewise-constant functions for, 342–343in radiation therapy, 347in refinery planning problems, 40–41and robust optimization, 333supply, 448, 459–461two-stage stochastic planning under, 393in utility capacity planning problem, 458in wind farm layout optimization, 368, 370yield, 448

Uncertainty set(s)budgeted, 361cardinality-constrained, 336–337convex, 337–338defined, 334for electric power application, 361ellipsoidal, 335–336, 338, 339

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Index 695

general, 338–339parameterizations of, 339polyhedral, 335primitive, 342for wind farm layout optimization, 371

Unconstrained optimizationderivative-free optimization, 496–502

noisy functions, 500–501nonsmooth functions, 499–500probabilistic models, 502smooth functions, 496–499worst-case complexity and global

convergence, 501–502nonlinear optimization, 223–224,

226–227protein-binding cavity volumetric alignment,

521–522Underage cost, 381, 443Undercover heuristic, 286Underproduction demand constraints,

396Underproduction loss function, 401–402UNEDF (Universal Nuclear Energy Density

Function) low-energy physics project,536–539

Uniform convergence, 130, 132–133Unit commitment (UC). See UC (unit

commitment) problemsUC (unit commitment) problems

in cogeneration energy systems, 304–305and power system operations, 459robust optimization for, 358–362security-constrained, 191short-term planning problems vs., 305–307

Universal Nuclear Energy Density Function(UNEDF) low-energy physics project,536–539

Unrelaxable constraintsderivative-free optimization with, 504, 505and POUNDERS, 535

UN World Food Programme, 488Upper bound

in branch-and-bound method, 274on CVaR, 431on VaR, 428

Urban planning, xxxiUsage policy, 484–487User cut callback, 63Utility, in vaccine supply chain problem, 471Utility systems, MINLO and GDP for,

324–326

Vaccine production planning, 470–472Vaccines, 96, 470–472

Value-at-risk (VaR), 149, 150, 394CVaR vs., 431for investment portfolio construction,

428–430marginal VaR contribution for assets, 433–434

Value function, 129Value of lost load, 411Valve trains, automotive, 508VaR. See Value-at-riskVariable costs, 359Variable-depth interchange heuristic, 57Variable neighborhood search (VNS), 212, 213,

301VASP. See Volumetric analysis of surface

propertiesVelocity regulation, aircraft, 294Vertex cover problem, 52, 453Vibration constraints, 144–145Violation transfer, 281, 285Virtual constraints. See Hidden constraintsVNS. See Variable neighborhood searchVolume flexibility, of power generators,

459–461Volumetric analysis of surface properties (VASP).

See also DFO-VASP protein-binding cavityalignment

described, 520noise-handling strategies for, 522–525

Voxels, 347Voxelwise worst-case robustness, 354–355

Wagner–Whitin algorithm, 442Wagner–Whitin problem, 442–443Wait-and-see decisions

in marginal water value calculations, 413, 419,424

robust optimization for, 340–341Wake models, 368–369Wake region, wind turbine, 367, 368Warehouse location problem, 51Warm-start approach, 524–525Warm-start feature, of QP solvers, 232Wastewater treatment networks (WWTNs),

326–327Water networks, 317, 326–327Water resource policy, 508Water treatment network problem, 216Water-using networks (WUNs), 326–327Water valuation, 405–425

and hydroelectric generators with reservoirs,405–406

multiple-reservoir model for, 416–419in New Zealand electricity system, 419–425observed electricity prices vs. model outputs for,

424–425

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696 Index

single-reservoir model of, 411–416social planning problem formulation for,

406–411Watson project, xxxWCC bounds. See Worst-case complexity boundsWeak duality condition, 113Weak infeasibility, conic optimization with, 114Weber problem, 450–451Weiszfeld procedure, 451Welfare-maximizing solution to SSP, 410Wholesale electricity markets, 405Wholesale price contracts, 454Wholesale prices, 454, 455Wind direction, 368, 371Wind farms

layout optimization, 367–375history, 367–368performance of nominal vs. robust models,

371–374problem formulations, 369–371wake models, 368–369

prevalence of, 367Wind states, 368Wing box, 256

Wing designaerodynamic shape optimization, 252–254aerostructural design optimization, 254–256

Working set, for active-set QP solvers, 232Worst-case complexity (WCC) bounds

in derivative-free optimization, 501–502in supply chain optimization, 448

Worst-case dose distributions, 354–355Worst-case net load, 362Worst-case penalty, 353–354Worst-case robustness, 352–355Worst-case scenarios, 333WUNs (water-using networks), 326–327WWTNs (wastewater treatment networks),

326–327

XPRESS (solver), 10, 82Xpress-SLP (solver), 289

YALMIP modeling language, 119Yes-no constraints, 507. See also Hidden

constraintsYield optimization problem, 508Yield uncertainty, 448Young measure. See Relaxed control

Zero-inventory ordering (ZIO) property, 440, 442Zero-order methods, 169z transformations, in CVaR method, 401