An Extended Mathematical Programming Framework Extended Mathematical Programming Framework ......
Embed Size (px)
Transcript of An Extended Mathematical Programming Framework Extended Mathematical Programming Framework ......

An Extended Mathematical Programming Framework
Michael C. Ferris
University of Wisconsin, Madison
Computing with Uncertainty, IMA, October 21, 2010
Ferris (Univ. Wisconsin) EMP IMA, October 2010 1 / 35

Optimal Power Flow
OPF(): miny energy dispatch cost (y , )s.t. conservation of power flow at nodes
Kirchoffs voltage law, and simple bound constraints
are (given) price bids, parametric optimization
Leader(i ): maxi ,y , firm i s profit (i , y , )s.t. 0 i i
y solves OPF(i , i )
Note that objective involves multiplier from OPF problem
Leader(i ): maxi ,y , firm i s profit (y , , )s.t. 0 i i
(y , ) solves KKT(OPF(i , i ))
This is an example of an MPCC since KKT form complementarityconstraints
Ferris (Univ. Wisconsin) EMP IMA, October 2010 2 / 35

Optimal Power Flow
OPF(): miny energy dispatch cost (y , )s.t. conservation of power flow at nodes
Kirchoffs voltage law, and simple bound constraints
are (given) price bids, parametric optimization
Leader(i ): maxi ,y , firm i s profit (i , y , )s.t. 0 i i
y solves OPF(i , i )
Note that objective involves multiplier from OPF problem
Leader(i ): maxi ,y , firm i s profit (y , , )s.t. 0 i i
(y , ) solves KKT(OPF(i , i ))
This is an example of an MPCC since KKT form complementarityconstraints
Ferris (Univ. Wisconsin) EMP IMA, October 2010 2 / 35

Optimal Power Flow
OPF(): miny energy dispatch cost (y , )s.t. conservation of power flow at nodes
Kirchoffs voltage law, and simple bound constraints
are (given) price bids, parametric optimization
Leader(i ): maxi ,y , firm i s profit (i , y , )s.t. 0 i i
y solves OPF(i , i )
Note that objective involves multiplier from OPF problem
Leader(i ): maxi ,y , firm i s profit (y , , )s.t. 0 i i
(y , ) solves KKT(OPF(i , i ))
This is an example of an MPCC since KKT form complementarityconstraints
Ferris (Univ. Wisconsin) EMP IMA, October 2010 2 / 35

Multiplayer EPEC and security constraints
(1, 2, . . . , m) is an equilibrium if
i solves Leader(i ), i
(Nonlinear) Nash Equilibrium where each player solves an MPCC
MPCC is hard (lacks a constraint qualification)
Nash Equilibrium is PPADcomplete (Chen et al, Papadimitriou et al)
In practice, also require contingency (scenario) constraints imposed inthe OPF problem
Model involves: complementarity, hierarchy, interacting agents, riskmeasures
How to convey and exploit such model structure
Ferris (Univ. Wisconsin) EMP IMA, October 2010 3 / 35

Multiplayer EPEC and security constraints
(1, 2, . . . , m) is an equilibrium if
i solves Leader(i ), i
(Nonlinear) Nash Equilibrium where each player solves an MPCC
MPCC is hard (lacks a constraint qualification)
Nash Equilibrium is PPADcomplete (Chen et al, Papadimitriou et al)
In practice, also require contingency (scenario) constraints imposed inthe OPF problem
Model involves: complementarity, hierarchy, interacting agents, riskmeasures
How to convey and exploit such model structure
Ferris (Univ. Wisconsin) EMP IMA, October 2010 3 / 35

Complementarity Problems via Graphs
T = NR+ = (R+ {0})
({0} R)
y T () (,y) T 0 y 0
By approximating (smoothing) graph can generate interior pointalgorithms for example y = , y , > 0
F (x) NRn+(x) (x ,F (x)) Tn 0 x F (x) 0
Ferris (Univ. Wisconsin) EMP IMA, October 2010 4 / 35

Complementarity Systems (DVI)
dxdt (t) = f (x(t), (t))
y(t) = h(x(t), (t))
0 y(t) (t) 0
Ferris (Univ. Wisconsin) EMP IMA, October 2010 5 / 35

Complementarity Systems (DVI)
dxdt (t) = f (x(t), (t))
y(t) = h(x(t), (t))
0 y(t) (t) 0
Ferris (Univ. Wisconsin) EMP IMA, October 2010 5 / 35

Complementarity Systems (DVI)
dxdt (t) = f (x(t), (t))
y(t) = h(x(t), (t))
((t),y(t)) T
Ferris (Univ. Wisconsin) EMP IMA, October 2010 6 / 35

Operators and Graphs (X = [1, 1])
xi = 1,Fi (x) 0 or xi (1, 1),Fi (x) = 0 or xi = 1,Fi (x) 0
ComplementaritySystems
Ferris(Univ.Wisconsin)
EMP
NonsmoothSchool,June2010
2/42 Complementarity Systems
Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 2 / 42
Complementarity Systems
Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 2 / 42
T () T 1(y) (I + T )1(y) = PT (y)
PT (y) is the projection of y onto [`, u]
Ferris (Univ. Wisconsin) EMP IMA, October 2010 7 / 35

Generalized Equations
Suppose T is a maximal monotone operator
0 F (z) + T (z) (GE )
Define PT = (I + T )1
If T is polyhedral (graph of T is a finite union of convex polyhedralsets) then PT is piecewise affine (continous, singlevalued,nonexpansive)
0 F (z) + T (z) z F (z) + I(z) + T (z) z F (z) (I + T )(z) PT (z F (z)) = z
Use in fixed point iterations (cf projected gradient methods)
Ferris (Univ. Wisconsin) EMP IMA, October 2010 8 / 35

Splitting Methods
Suppose T is a maximal monotone operator
0 F (z) + T (z) (GE )
Can devise Newton methods (e.g. SQP) that treat F via calculus andT via convex analysisAlternatively, can split F (z) = A(z) + B(z) (and possibly T also) sowe solve solve (GE) by solving a sequence of problems involving just
T1(z) = A(z) and T2(z) = B(z) + T (z)
where each of these is simpler
ForwardBackward splitting:
zk+1 = (I + ckT2)1 (I ckT1)
(zk),
Ferris (Univ. Wisconsin) EMP IMA, October 2010 9 / 35

Normal Map
Suppose T is a maximal monotone operator
0 F (z) + T (z) (GE )
Define PT = (I + T )1
0 F (z) + T (z) z F (z) + I(z) + T (z) z F (z) = y and y (I + T )(z) z F (z) = y and PT (y) = z PT (y) F (PT (y)) = y 0 = F (PT (y)) + y PT (y)
This is the socalled Normal Map Equation
Ferris (Univ. Wisconsin) EMP IMA, October 2010 10 / 35

Normal manifold = {Fi + NFi}
Ferris (Univ. Wisconsin) EMP IMA, October 2010 11 / 35

C = {z Bz b},F (z) = Mz + q
Ferris (Univ. Wisconsin) EMP IMA, October 2010 12 / 35

Cao/Ferris Path (Eaves)
Start in cell that has interior(face is an extreme point)
Move towards a zero ofaffine map in cell
Update direction when hitboundary (pivot)
Solves or determinesinfeasible if M iscopositiveplus on rec(C )
Solves 2person bimatrixgames, 3person games too,but these are nonlinear
Cao/Ferris Path (Eaves) Start in cell that has
interior (face is an extreme point)
Move towards a zero of affine map in cell
Update direction when hit boundary
Solves or determines infeasible if M is copositiveplus on rec(C)
Nails 2person game
But algorithm has exponential complexity (von Stengel et al)
Ferris (Univ. Wisconsin) EMP IMA, October 2010 13 / 35

Extensions and Computational Results
Embed AVI solver in a Newton Method  each Newton step solves anAVI
Compare performance of PathAVI with PATH on equivalent LCP
PATH the most widely used code for solving MCP
AVIs constructed to have solution with Mnn symmetric indefinite
PathAVI PATHSize (m,n) Resid Iter Resid Iter
(180, 60) 3 1014 193 0.9 10176(360, 120) 3 1014 1516 2.0 10594
2  10x speedup in Matlab using sparse LU instead of QR
2  10x speedup in C using sparse LU updates
Ferris (Univ. Wisconsin) EMP IMA, October 2010 14 / 35

But who cares?
Why arent you using my *********** algorithm?(Michael Ferris, Boulder, CO, 1994)
Show me on a problem like mine
Must run on defaults
Must deal graciously with poorly specified cases
Must be usable from my environment (Matlab, R, GAMS, ...)
Must be able to model my problem easily
EMP provides annotations to an existing optimization model that conveynew model structures to a solver
Ferris (Univ. Wisconsin) EMP IMA, October 2010 15 / 35

But who cares?
Why arent you using my *********** algorithm?(Michael Ferris, Boulder, CO, 1994)
Show me on a problem like mine
Must run on defaults
Must deal graciously with poorly specified cases
Must be usable from my environment (Matlab, R, GAMS, ...)
Must be able to model my problem easily
EMP provides annotations to an existing optimization model that conveynew model structures to a solver
Ferris (Univ. Wisconsin) EMP IMA, October 2010 15 / 35

EMP(i): MPCC: complementarity constraints
minx ,s
f (x , s)
s.t. g(x , s) 0,0 s h(x , s) 0
g , h model engineering expertise: finite elements, etc
models complementarity, disjunctionsComplementarity constraints available in AIMMS, AMPL andGAMS
NLPEC: use the convert tool to automatically reformulate as aparameteric sequence of NLPs
Solution by repeated use of standard NLP softwareI Problems solvable, local solutions, hard
Ferris (Univ. Wisconsin) EMP IMA, October 2010 16 / 35

EMP(i): MPCC: complementarity constraints
minx ,s
f (x , s)
s.t. g(x , s) 0,0 s h(x , s) 0
g , h model engineering expertise: finite elements, etc
models complementarity, disjunctionsComplementarity constraints available in AIMMS, AMPL andGAMS
NLPEC: use the convert tool to automatically reformulate as aparameteric sequence of NLPs
Solution by repeated use of standard NLP softwareI Problems solvable, local solutions, hard
Ferris (Univ. Wisconsin) EMP IMA, October 2010 16 / 35

EMP(ii): Hierarchical models
Bilevel programs:
minx,y
f (x, y)
s.t. g(x, y) 0,y solves min
yv(x, y) s.t. h(x, y) 0
model bilev /deff,defg,defv,defh/;empinfo: bilevel min v y defv defh
EMP tool automatically creates the MPCC
minx,y,
f (x, y)
s.t. g(x, y) 0,0 v(x, y) + Th(x, y) y 00 h(x, y) 0
Ferris (Univ. Wisconsin) EMP IMA, October 2010 17 / 35

Large scale example: bioreactor (with Niebel)
Challenge
Formulating an optimization problem that allows the estimation of thedynamic changes in intracellular fluxes based on measured externalbioreactor concentrations.
Approach
Using existing constraintbased stoichiometric models of the cellularmetabolism to formulate a bilevel dynamic optimization problem.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 18 / 35

Dynamic optimizationApproach:The different timescales of the metabolism (fast) and the reactor growth(slow), allows to assume steadystate for the metabolism.
minimize / maximize Objective (eg. parameter tting)
s. t.
s. t.
bioreactor dynamics
maximize growth rate
stoichiometric constraints
ux constraints
constraints on exchange uxes
Different mathematical programming techniques are used to transform theproblem to a nonlinear program. The differential equations aretransformed into nonlinear constraints using collocation methods.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 19 / 35

The overall scheme!
Collection of algebraic equations
Form a bilevel program via emp
EMP tool automatically creates the MPCC (model transformation)
NLPEC tool automatically creates (a series of) NLP models (modeltransformation)
GAMS automatically rewrites NLP models for global solution viaBARON (model transformation)
Is this global? Whats the hitch?
Note that heirarchical structure is available to solvers for analysis orutilization
Ferris (Univ. Wisconsin) EMP IMA, October 2010 20 / 35

The overall scheme!
Collection of algebraic equations
Form a bilevel program via emp
EMP tool automatically creates the MPCC (model transformation)
NLPEC tool automatically creates (a series of) NLP models (modeltransformation)
GAMS automatically rewrites NLP models for global solution viaBARON (model transformation)
Is this global? Whats the hitch?
Note that heirarchical structure is available to solvers for analysis orutilization
Ferris (Univ. Wisconsin) EMP IMA, October 2010 20 / 35

Variational inequalities
Find z X such that
0 F (z) +NX (z)
Many applications where F is not the derivative of some f
model vi / F, g /;empinfo: vifunc F z
Convert problem into complementarity problem by introducingmultipliers on representation of X
Can now do MPEC (as opposed to MPCC)!
Projection algorithms, robustness (evaluate F only at points in X )
Ferris (Univ. Wisconsin) EMP IMA, October 2010 21 / 35

EMP(iii): Embedded modelsModel has the format:
Agent o: minx
f (x , y)
s.t. g(x , y) 0 ( 0)
Agent v: H(x , y , ) = 0 ( y free)
Difficult to implement correctly (multiple optimization models)Can do automatically  simply annotate equationsempinfo: equilibriummin f x defgvifunc H y dualvar defgEMP tool automatically creates an MCP
x f (x , y) + Tg(x , y) = 00 g(x , y) 0
H(x , y , ) = 0
Ferris (Univ. Wisconsin) EMP IMA, October 2010 22 / 35

World Bank Project (Uruguay Round)
24 regions, 22 commoditiesI Nonlinear complementarity
problemI Size: 2200 x 2200
Short term gains $53 billion p.a.I Much smaller than previous
literature
Long term gains $188 billion p.a.I Number of less developed
countries loose in short term
Unpopular conclusions  forcedconcessions by World Bank
Region/commodity structure notapparent to solver
Application: Uruguay Round World Bank Project with
Harrison and Rutherford 24 regions, 22 commodities
2200 x 2200 (nonlinear) Short term gains $53 billion p.a.
Much smaller than previous literature
Long term gains $188 billion p.a. Number of less developed
countries loose in short term Unpopular conclusions forced
concessions by World Bank
Ferris (Univ. Wisconsin) EMP IMA, October 2010 23 / 35

Nash Equilibria
Nash Games: x is a Nash Equilibrium if
xi arg minxiXi
`i (xi , xi , q),i I
xi are the decisions of other players.
Quantities q given exogenously, or via complementarity:
0 H(x , q) q 0
empinfo: equilibriummin loss(i) x(i) cons(i)vifunc H q
Applications: DiscreteTime FiniteState Stochastic Games.Specifically, the Ericson & Pakes (1995) model of dynamiccompetition in an oligopolistic industry.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 24 / 35

Key point: models generated correctly solve quicklyHere S is mesh spacing parameter
S Var rows nonzero dense(%) Steps RT (m:s)
20 2400 2568 31536 0.48 5 0 : 0350 15000 15408 195816 0.08 5 0 : 19100 60000 60808 781616 0.02 5 1 : 16200 240000 241608 3123216 0.01 5 5 : 12
Convergence for S = 200 (with new basis extensions in PATH)
Iteration Residual
0 1.56(+4)1 1.06(+1)2 1.343 2.04(2)4 1.74(5)5 2.97(11)
Ferris (Univ. Wisconsin) EMP IMA, October 2010 25 / 35

General Equilibrium models
(C ) : maxxkXk
Uk(xk) s.t. pT xk ik(y , p)
(I ) :ik(y , p) = pTk +
j
kjpTgj(yj)
(P) : maxyjYj
pTgj(yj)
(M) : maxp0
pT
k
xk k
k
j
gj(yj)
s.t. l
pl = 1
Can reformulate as embedded problem (Ermoliev et al):
maxxX ,yY
k
tkk
log Uk(xk)
s.t.k
xk k
k +
j
gj(yj)
tk = ik(y , p) where p is multiplier on NLP constraint
Leads to sequential joint maximization algorithm (Rutherford)
Ferris (Univ. Wisconsin) EMP IMA, October 2010 26 / 35

General Equilibrium models
(C ) : maxxkXk
Uk(xk) s.t. pT xk ik(y , p)
(I ) :ik(y , p) = pTk +
j
kjpTgj(yj)
(P) : maxyjYj
pTgj(yj)
(M) : maxp0
pT
k
xk k
k
j
gj(yj)
s.t. l
pl = 1
Can reformulate as embedded problem (Ermoliev et al):
maxxX ,yY
k
tkk
log Uk(xk)
s.t.k
xk k
k +
j
gj(yj)
tk = ik(y , p) where p is multiplier on NLP constraint
Leads to sequential joint maximization algorithm (Rutherford)Ferris (Univ. Wisconsin) EMP IMA, October 2010 26 / 35

Competing agent models (with Wets)
Competing agents (consumers, or generators in energy market)
Each agent maximizes objective independently (utility)
Market prices are function of all agents activities
Additional twist: model must hedge against uncertainty
Facilitated by allowing contracts bought now, for goods delivered later
Conceptually allows to transfer goods from one period to another(provides wealth retention or pricing of ancilliary services in energymarket)
Ferris (Univ. Wisconsin) EMP IMA, October 2010 27 / 35

The model details: c.f. Brown, Demarzo, EavesEach agent maximizes:
uh =
s
s
(
l
ch,lh,s,l
)Time 0:
dh,0,l = ch,0,l eh,0,l ,
l
p0,ldh,0,l +k
qkzh,k 0
Time 1:
dh,s,l = ch,s,l eh,s,l k
Ds,l ,k zh,k ,
l
ps,ldh,s,l 0
Additional constraints (complementarity) outside of control of agents:
0 h
zh,k qk 0
0 h
dh,s,l ps,l 0Ferris (Univ. Wisconsin) EMP IMA, October 2010 28 / 35

EMP(iv): Stochastic programming and risk measures
SP: min c>x + E[d>y ]
s.t. Ax = b
T ()x + W ()y() h(), for all ,
x 0, y() 0, for all .
Annotations are slightly more involved but straightforward:
Need to describe probability distribution
Define (multistage) structure (what variables and constraints belongto each stage)
Define random parameters and process to generate scenarios
Can also define risk measures on variables
Ferris (Univ. Wisconsin) EMP IMA, October 2010 29 / 35

EMP(iv): Stochastic programming and risk measures
SP: min c>x + R[d>y ]
s.t. Ax = b
T ()x + W ()y() h(), for all ,
x 0, y() 0, for all .
Annotations are slightly more involved but straightforward:
Need to describe probability distribution
Define (multistage) structure (what variables and constraints belongto each stage)
Define random parameters and process to generate scenarios
Can also define risk measures on variables
Automatic reformulation (deterministic equivalent), solvers such asDECIS, etc.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 30 / 35

EMP(v): Extended nonlinear programs
minxX
f0(x)+(f1(x), . . . , fm(x))
Examples of different
least squares, absolute value, Huber functionSolution reformulations are very differentHuber function used in robust statistics.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 31 / 35

More general functions
In general any piecewise linear penalty function can be used: (differentupside/downside costs).General form:
(u) = supyY{yTu k(y)}
nonsmooth due to the max term; separable in example. is always convex.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 32 / 35

First order conditions
Solution via reformulation. One way:
0 xL(x , y) + NX (x)0 yL(x , y) + NY (y)
NX (x) is the normal cone to the closed convex set X at x .
Automatically creates an MCP (or a VI)
Already available!
To do: extend X and Y beyond simple bound sets.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 33 / 35

Alternative Reformulations
Convert does symbolic/numeric reformulations. Alternative NLPformulations also possible.
k(y) =1
2y Qy , X = {x : Rx r} , Y =
{y : S y s
}Defining
Q = DJ1D , F (x) = (f1(x), . . . , fm(x))
min f0(x) + sz + 12wJw
s.t. Rx r , z 0,F (x) Sz Dw = 0
Can set up better (solver) specific formulation.
Ferris (Univ. Wisconsin) EMP IMA, October 2010 34 / 35

Conclusions
Modern optimization within applications requires multiple modelformats, computational tools and sophisticated solvers
EMP model type is clear and extensible, additional structure availableto solver
Extended Mathematical Programming available within the GAMSmodeling system
Able to pass additional (structure) information to solvers
Embedded optimization models automatically reformulated forappropriate solution engine
Exploit structure in solvers
Extend application usage further
Ferris (Univ. Wisconsin) EMP IMA, October 2010 35 / 35
MotivationExtended Mathematical ProgrammingMPCCHierarchical ModelsBioreactorVariational InequalitiesEmbedded modelsNash EquilibriaStochastic ProgrammingExtended Nonlinear Programming
Conclusions