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(α): miny energy dispatch cost (y , α)s.t. conservation of power flow at nodes

Kirchoff’s voltage law, and simple bound constraints

α are (given) price bids, parametric optimization

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (αi , y , λ)s.t. 0 ≤ αi ≤ αi

y solves OPF(αi , α−i )

Note that objective involves multiplier from OPF problem

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (y , λ, α)s.t. 0 ≤ αi ≤ αi

(y , λ) solves KKT(OPF(αi , α−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(α): miny energy dispatch cost (y , α)s.t. conservation of power flow at nodes

Kirchoff’s voltage law, and simple bound constraints

α are (given) price bids, parametric optimization

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (αi , y , λ)s.t. 0 ≤ αi ≤ αi

y solves OPF(αi , α−i )

Note that objective involves multiplier from OPF problem

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (y , λ, α)s.t. 0 ≤ αi ≤ αi

(y , λ) solves KKT(OPF(αi , α−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(α): miny energy dispatch cost (y , α)s.t. conservation of power flow at nodes

Kirchoff’s voltage law, and simple bound constraints

α are (given) price bids, parametric optimization

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (αi , y , λ)s.t. 0 ≤ αi ≤ αi

y solves OPF(αi , α−i )

Note that objective involves multiplier from OPF problem

Leader(α−i ): maxαi ,y ,λ firm i ’s profit (y , λ, α)s.t. 0 ≤ αi ≤ αi

(y , λ) solves KKT(OPF(αi , α−i ))

This is an example of an MPCC since KKT form complementarityconstraints

Ferris (Univ. Wisconsin) EMP IMA, October 2010 2 / 35

Multi-player EPEC and security constraints

(α1, α2, . . . , αm) is an equilibrium if

αi solves Leader(α−i ), ∀i

(Nonlinear) Nash Equilibrium where each player solves an MPCC

MPCC is hard (lacks a constraint qualification)

Nash Equilibrium is PPAD-complete (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

Multi-player EPEC and security constraints

(α1, α2, . . . , αm) is an equilibrium if

αi solves Leader(α−i ), ∀i

(Nonlinear) Nash Equilibrium where each player solves an MPCC

MPCC is hard (lacks a constraint qualification)

Nash Equilibrium is PPAD-complete (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 (λ) ⇐⇒ (λ,−y) ∈ T ⇐⇒ 0 ≤ λ ⊥ y ≥ 0

By approximating (smoothing) graph can generate interior pointalgorithms for example yλ = ε, y , λ > 0

−F (x) ∈ NRn+

(x) ⇐⇒ (x ,−F (x)) ∈ T n ⇐⇒ 0 ≤ x ⊥ F (x) ≥ 0

Ferris (Univ. Wisconsin) EMP IMA, October 2010 4 / 35

Complementarity Systems (DVI)

dxdt (t) = f (x(t), λ(t))

y(t) = h(x(t), λ(t))

0 ≤ y(t) ⊥ λ(t) ≥ 0

Ferris (Univ. Wisconsin) EMP IMA, October 2010 5 / 35

Complementarity Systems (DVI)

dxdt (t) = f (x(t), λ(t))

y(t) = h(x(t), λ(t))

0 ≤ y(t) ⊥ λ(t) ≥ 0

Ferris (Univ. Wisconsin) EMP IMA, October 2010 5 / 35

Complementarity Systems (DVI)

dxdt (t) = f (x(t), λ(t))

y(t) = h(x(t), λ(t))

(λ(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 (λ) 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, single-valued,non-expansive)

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 analysis

Alternatively, 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”

Forward-Backward 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 so-called 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 iscopositive-plus on rec(C )

Solves 2-person bimatrixgames, 3-person 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 copositive-plus on rec(C)

•  Nails 2-person 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 Mn×n symmetric indefinite

PathAVI PATHSize (m,n) Resid Iter Resid Iter

(180, 60) 3× 10−14 193 0.9 10176(360, 120) 3× 10−14 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 aren’t 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 aren’t 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, disjunctions

Complementarity “⊥” constraints available in AIMMS, AMPL andGAMS

NLPEC: use the convert tool to automatically reformulate as aparameteric sequence of NLP’s

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, disjunctions

Complementarity “⊥” constraints available in AIMMS, AMPL andGAMS

NLPEC: use the convert tool to automatically reformulate as aparameteric sequence of NLP’s

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∗,λ

f (x∗, y∗)

s.t. g(x∗, y∗) ≤ 0,0 ≤ ∇v(x∗, y∗) + λT∇h(x∗, y∗) ⊥ y∗ ≥ 00 ≤ −h(x∗, y∗) ⊥ λ ≥ 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 constraint-based 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 steady-state 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? What’s 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? What’s 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 (⊥ λ ≥ 0)

Agent v: H(x , y , λ) = 0 (⊥ y free)

Difficult to implement correctly (multiple optimization models)Can do automatically - simply annotate equationsempinfo: equilibriummin f x defgvifunc H y dualvar λ defgEMP tool automatically creates an MCP

∇x f (x , y) + λT∇g(x , y) = 0

0 ≤ −g(x , y) ⊥ λ ≥ 0

H(x , y , λ) = 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

x∗i ∈ arg minxi∈Xi

`i (xi , x∗−i , q),∀i ∈ I

x−i 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: Discrete-Time Finite-State 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 non-zero 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 ) : maxxk∈Xk

Uk(xk) s.t. pT xk ≤ ik(y , p)

(I ) :ik(y , p) = pTωk +∑

j

αkjpTgj(yj)

(P) : maxyj∈Yj

pTgj(yj)

(M) : maxp≥0

pT

∑k

xk −∑k

ωk −∑

j

gj(yj)

s.t.∑

l

pl = 1

Can reformulate as embedded problem (Ermoliev et al):

maxx∈X ,y∈Y

∑k

tkβk

log Uk(xk)

s.t.∑k

xk ≤∑k

ωk +∑

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 ) : maxxk∈Xk

Uk(xk) s.t. pT xk ≤ ik(y , p)

(I ) :ik(y , p) = pTωk +∑

j

αkjpTgj(yj)

(P) : maxyj∈Yj

pTgj(yj)

(M) : maxp≥0

pT

∑k

xk −∑k

ωk −∑

j

gj(yj)

s.t.∑

l

pl = 1

Can reformulate as embedded problem (Ermoliev et al):

maxx∈X ,y∈Y

∑k

tkβk

log Uk(xk)

s.t.∑k

xk ≤∑k

ωk +∑

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

πs

(κ−

∏l

cαh,l

h,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 (ω)x + W (ω)y(ω) ≥ h(ω), for all ω ∈ Ω,

x ≥ 0, y(ω) ≥ 0, for all ω ∈ Ω.

Annotations are slightly more involved but straightforward:

Need to describe probability distribution

Define (multi-stage) 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 (ω)x + W (ω)y(ω) ≥ h(ω), for all ω ∈ Ω,

x ≥ 0, y(ω) ≥ 0, for all ω ∈ Ω.

Annotations are slightly more involved but straightforward:

Need to describe probability distribution

Define (multi-stage) 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

minx∈X

f0(x)+θ(f1(x), . . . , fm(x))

Examples of different θ

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 θ functions

In general any piecewise linear penalty function can be used: (differentupside/downside costs).General form:

θ(u) = supy∈YyTu − k(y)

θ nonsmooth due to the max term; θ separable in example.θ 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 = DJ−1D ′, F (x) = (f1(x), . . . , fm(x))

min f0(x) + s ′z + 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