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### Transcript of An Extended Mathematical Programming Framework (Nonlinear) Nash Equilibrium where each player solves

• An Extended Mathematical Programming Framework

Michael C. Ferris

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

y solves OPF(αi , ᾱ−i )

Note that objective involves multiplier from OPF problem

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

(y , λ) solves KKT(OPF(αi , ᾱ−i ))

This is an example of an MPCC since KKT form complementarity constraints

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

y solves OPF(αi , ᾱ−i )

Note that objective involves multiplier from OPF problem

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

(y , λ) solves KKT(OPF(αi , ᾱ−i ))

This is an example of an MPCC since KKT form complementarity constraints

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

y solves OPF(αi , ᾱ−i )

Note that objective involves multiplier from OPF problem

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

(y , λ) solves KKT(OPF(αi , ᾱ−i ))

This is an example of an MPCC since KKT form complementarity constraints

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

• Multi-player EPEC and security constraints

(ᾱ1, ᾱ2, . . . , ᾱm) is an equilibrium if

(Nonlinear) Nash Equilibrium where each player solves an MPCC

MPCC is hard (lacks a constraint qualification)

In practice, also require contingency (scenario) constraints imposed in the OPF problem

Model involves: complementarity, hierarchy, interacting agents, risk measures

How to convey and exploit such model structure

Ferris (Univ. Wisconsin) EMP IMA, October 2010 3 / 35

• Multi-player EPEC and security constraints

(ᾱ1, ᾱ2, . . . , ᾱm) is an equilibrium if

(Nonlinear) Nash Equilibrium where each player solves an MPCC

MPCC is hard (lacks a constraint qualification)

In practice, also require contingency (scenario) constraints imposed in the OPF problem

Model involves: complementarity, hierarchy, interacting agents, risk measures

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 point algorithms for example yλ = �, 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)

dx dt (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)

dx dt (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)

dx dt (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

Co m pl em en ta ri ty Sy st em s

F er ri s (U n iv . W is co n si n )

E M P

N o n sm o o th S ch o o l, Ju n e 2 0 1 0

2 / 4 2 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 polyhedral sets) 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 and T via convex analysis Alternatively, can split F (z) = A(z) + B(z) (and possibly T also) so we 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 of affine map in cell

Update direction when hit boundary (pivot)

Solves or determines infeasible if M is copositive-plus on rec(C )

Solves 2-person bimatrix games, 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 an AVI

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 PATH Size (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 convey new 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

EM