Optimization Problems with Equilibrium Constraints · Primal-dual MPEC system with x 1;x 2 in...

Optimization Problems with EquilibriumConstraints

GIAN Short Course on Optimization:Applications, Algorithms, and Computation

Sven Leyffer

Argonne National Laboratory

September 12-24, 2016

Outline

1 Solving MPECs as NLPs

2 Convergence for Sequential Quadratic Programming Methods

3 Convergence for Interior-Point Methods

4 An SLPEC-EQP ApproachCounter Example for SQPECSLPEC MethodAccelerating Local Convergence

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Solving MPECs as NLPs

Mathematical Program with Equilibrium Constraints (MPEC)minimize

x ,yf (x , y)

subject to c(x , y) ≥ 00 ≤ y ⊥ F (x , y) ≥ 0

Equivalent smooth (lazy) nonlinear program (NLP):minimize

x ,yf (x , y)

subject to c(x , y) ≥ 0F (x , y) = s, s ≥ 0, y ≥ 0 and yT s ≤ 0

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Switching Notation

To understand convergence analysis, we switch notation:x = (x0, x1, x2):

minimizex

f (x)

subject to c(x) ≥ 00 ≤ x1 ⊥ x2 ≥ 0

Equivalent smooth nonlinear program (NLP):minimize

xf (x)

subject to c(x) ≥ 0x1 ≥ 0, x2 ≥ 0, xT

1 x2 ≤ 0

Now examine convergence properties of NLP solvers ...

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A Nonlinear Programming Approach

Replace equilibrium 0 ≤ x1 ⊥ x2 ≥ 0 by X1x2 ≤ 0 or xT1 x2 ≤ 0

⇒ standard nonlinear program (NLP)

(NLP)

minimize

xf (x)

subject to c(x) ≥ 0x1, x2 ≥ 0

X1x2 ≤ 0x

x

1

2

Advantage: standard (?) NLP; use large-scale solvers ...Snag: nonlinear program (NLP) violates standard assumptions!

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Strong Stationarity & Unbounded MultipliersExample x∗ = (0, 1):{

minx

12(x1 − 1)2 + (x2 − 1)2

s.t. x1, x2 ≥ 0, x1x2 ≤ 0

first order conditions:(−1

0

)=

(ν10

)−(ξ0

)ν1 multiplier of x1 ≥ 0; ξ multiplier of x1x2 ≤ 0.

Equivalent NLP (x1x2 ≤ 0) violates MFCQ ⇒ unboundedmultipliers

x

x

1

2

x*

ν

ξ

−1= −ν ξ

multipliers form a ray ⇒ ∃ bounded multipliers

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Outline





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The Relaxed NLP

Define index sets

X1 := {i : x∗1i = 0} & X2 := {i : x∗2i = 0} ,

complements X⊥j := {1, . . . , p} − Xj

⇒ relaxed NLP given by

minimizex

f (x)

subject to c(x) ≥ 0x1j = 0 ∀j ∈ X⊥2x2j = 0 ∀j ∈ X⊥1x1, x2 ≥ 0

... i.e. µi multiplier of “equality” constraints

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Equivalence to KKT Conditions

KKT conditions of equivalent NLP: ∃λ∗, ν∗1 , ν∗2 , ξ∗ ≥ 0

∇f (x∗)−∇c(x∗)Tλ∗ −

0ν∗1 − X ∗2 ξ

∗

ν∗2 − X ∗1 ξ∗

= 0 1st order

c(x∗) ≥ 0, x∗1 ≥ 0, x∗2 ≥ 0 and X ∗1 x∗2 ≤ 0 primal feas.

c(x∗)Tλ = x∗T

1 ν∗1 = x∗T

2 ν∗2 = 0 compl. slack.

... ξ > 0 allows µ1 < 0

... multipliers of relaxed NLP µ1 = ν1 − X ∗2 ξ, and µ2 = ν2 − X ∗1 ξ⇒ KKT multipliers bounded if ‖ξ∗‖ <∞

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Convergence of SQP for MPECs

Sequential Quadratic Programming (SQP) ... compute step d

minx

f (x)

s.t. c(x) ≥ 0x1 ≥ 0x2 ≥ 0X1x2 ≤ 0

→

mind∇f T

k d + 12dTHkd

s.t. ck +∇cTk d ≥ 0

xk1 + d1 ≥ 0xk2 + d2 ≥ 0Xk1xk2 + Xk1d2 + Xk2d1 ≤ 0

where Hk ' ∇2fk −∑λi∇2ck Hessian of the Lagrangian.

Set xk+1 = xk + d & update multiplier estimates

Two cases: ∃k : Xk1xk2 = 0 ... or ... Xk1xk2 > 0, ∀k

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Convergence of SQP Part 1: Xk1xk2 = 0

wlog have xk1 = 0 (and for simplicity assume xk2 > 0)

⇒ QP contains constraints

xk1 + d1 ≥ 0xk2 + d2 ≥ 0

Xk1xk2 + Xk2d1 + Xk1d2 ≤ 0

⇒d1 ≥ 0

xk2 + d2 ≥ 0Xk2d1 ≤ 0

⇒ d1 = 0

⇒ xk+11 = xk1 + d1 = 0 ... stay on same axis

⇒ same tangent cone as NLP with x1 = 0 ... relaxed NLP⇒ fast local convergence

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Convergence of SQP Part 2: Xk1xk2 > 0

wlog x∗1 = 0, but Xk1xk2 > 0, i.e. off axis

QP picks nonsingular basis, subset of 0 0∇ck I Xk2

0 Xk1

Assume all QPs consistent ... 2 cases:

��

��

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��

case 1: true subset ⇒ non-singular ⇒ quadratic convergence

case 2: full set ⇒ xk1 > 0 (otherwise singular)⇒ X k+1

1 xk+12 = 0 now see Part (1) as before ...

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Consistency of QP Approximations

Are QPs always consistent for MPECs?

NO! Linearization can be inconsistent arbitrarily close to solution

minimizex

x1 + x2

subject to x22 ≥ 1

x1 ≥ 0x2 ≥ 0x1 x2 ≤ 0

x (k)

1x

2x

x*

generic problem ⇒ solvers take arbitrary steps

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Consistency of QP Approximations

Relax linearization of X1x2 ≤ 0 ...... heuristic for infeasible QPs (0 < δ, κ < 1 constants)

Xk1xk2 + Xk2d1 + Xk1d2 ≤ δ(

xTk1xk2

)1+κe

x (k)

1x

2x

x*

... works in well practice with δ = 0.1, κ = 1

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The Slacks Matter!!!

How important was the introduction of slack variables?

Consider MPEC without slacks ...

(P)

minimize

z−x1 − 1

2x2

subject to x1 + x2 ≤ 20 ≤ x2

1 − x1 ⊥ x2 ≥ 0 .

with solutions (2, 0)T with f ∗ = −2 and (0, 2)T with f ∗ = −1

1

1 2

2

z1

z2

f−

z(0)

zoo

Start (−ε, t)T

Nonstationary limit(0, t)T for any t.

Avoid failure with slacks

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Outline





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Interior Point Penalty Methods for MPECs

Equivalent NLP: minimize

xf (x)

subject to c(x) ≥ 0x1 ≥ 0, x2 ≥ 0,xT1 x2 ≤ 0

Consider `1 penalty of complementarity constraintminimize

xf (x)+πxT

1 x2

subject to c(x) ≥ 0x1 ≥ 0, x2 ≥ 0

... form primal-dual system with a twist ...

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Primal-dual MPEC system with x1, x2 in primal form∇f (x)−∇c(x)Tλ−

0

µX−11 e − X2π

µX−12 e − X1π

= 0

c(x)− s = 0Sλ = µe

Algorithm I: Interior Penalty Method for MPECs

1 Choose barrier parameter µk , and tolerance εk2 Solve PD system to tolerance εk and ensure

‖min{xk1, xk2}‖ ≤√εk by adjusting πk

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Theorem

If Algorithm I generates an infinite sequence, then:

1 xk → x∗ is feasible,

2 LICQ for relaxed NLP ⇒ x∗ is C-stationary,

3 πkxki → 0 ⇒ x∗ strongly stationary,

4 superlinear convergence for suitable barrier updates

Practical implementation

dynamic penalty πk update during inner iteration

non-monotone reduction of complementarity: πj = 10πj if,

x jT

1 x j2 > 0.9 max

{x(j−1)T1 x

(j−1)2 , . . . , x

(j−m+1)T

1 x(j−m+1)2

}avoid trouble with badly scaled MPECs

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Relaxed Interior Point Methods for MPECs

Perturb rhs of complementarity constraint ... X1x2 ≤ Cµe

... where µ > 0 barrier parameter ⇒ primal dual system ...

∇f (x)−∇c(x)Tλ−

0

µX−11 e − X2ξ

µX−12 e − X1ξ

= 0

c(x)− s = 0Sλ = µe

X1x2 + t = CµeT ξ = µe

⇒ central path (x(µ), ν(µ), ξ(µ)) for µ > 0[Raghunathan and Biegler, 2002, Liu and Sun, 2002]

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Relaxed Interior Point Methods for MPECs

Compare relaxation and penalization

ξ = πt = Cµe − X1x2

T ξ = µe

⇒ Penalization ⇔ Relaxation, if

πi =µ

µCi − x1ix2ior Ci =

µ+ πix1ix2iµπi

... convergence proofs carry over!

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Interior Point Method with Two Sided Relaxation

Clever idea by Friedlander, de Miguel & Scholtes [2003]:

MPECs have no strict interior

Relax X1x2 ≤ τe ⇒ interior → 0⇒ relax X1x2 ≤ τe

and x1 ≥ −δe, x2 ≥ −δe

Adjust τ , δ as µ→ 0

Theorem

In limit τ → 0 or δ → 0 but not both⇒ relaxed problem has non-empty interior in limit⇒ interior point methods faster & more robust

MPEC multiplier µi < 0 ⇒ reduce τi ↘ 0 ...

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Outline





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SQPEC Approach [Scholtes, 2004]

Sequential QPEC approach (similar to piecewise SQP)

minimized

g (k)T d + 12dTH(k)d

subject to c(k) + A(k)T d ≥ 0,

0 ≤ x(k)1 + d1 ⊥ x

(k)2 d2 ≥ 0

where g (k) = ∇f (x (k)) and A(k) = ∇c(x (k), y (k)),

Solve sequence of QPECs, set x (k+1) = x (k) + d

Theorem [Scholtes, 04]: Local B-stationary convergence.

SQPEC has correct tangent cone ⇒ global convergence???

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No! Counter Example for SQPECConsiderminimize (x1 − 1)2 + x3

2 + x22 subject to 0 ≤ x1 ⊥ x2 ≥ 0

SQPEC: x (k+1) =(

0, 3x(k)2

2 /(6x(k)2 + 2)

)→ (0, 0) spurious

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A Sequential LPEC Methodwhile (not optimal) begin

1 Compute step d from LPEC subproblem

minimized

g (k)T d

subject to c(k) + A(k)T d ≥ 0,

0 ≤ x(k)1 + d1 ⊥ x

(k)2 + d2 ≥ 0

‖d‖∞ ≤ ∆k trust-region

2 if x (k) + d acceptable thenx (k+1) = x (k) + d & increase TR ∆(k+1) = 2 ∗∆k

else x (k+1) = x (k) & decrease TR ∆(k+1) = ∆k/2

end

1 Like steepest descend: Can we speed up convergence?

2 When is x (k) + d acceptable?

3 How do we solve the LPEC subproblem?

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Spurious A/M-Stationarity Revisited

Consider min (x1 − 1)2 + x32 + x2

2 subject to 0 ≤ x2 ⊥ x1 ≥ 0

SLPEC pivots through (0, 0) ... get onto x1-axis

SLPEC converges to B-stationary limit (1, 0)

... cannot get stuck in spurious stationary points

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2 subject to 0 ≤ x2 ⊥ x1 ≥ 0

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trust−region




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2 subject to 0 ≤ x2 ⊥ x1 ≥ 0

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trust−region




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Accelerating Local Convergence

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Equality Constrained Quadratic Program (EQP)

Given active set estimate from LPEC step d :

Ac(d) :={

i : c(k)i + a

(k)T

i d = 0}

A1(d) :={

i : x(k)1i + d1i = 0

}A2(d) :=

{i : x

(k)2i + d2i = 0

}solve corresponding equality QP

EQPk(d)

minimize

sg (k)T s + 1

2sTH(k)s

subject to c(k)i + a

(k)T

i s = 0, ∀i ∈ Ac(d)

x(k)1i + s1i = 0, ∀i ∈ A1(d)

x(k)2i + s2i = 0, ∀i ∈ A2(d)

for 2nd order step s.

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A Filter Method for MPECs

MPEC have three competing aims

1 Minimize f (x , y)

2 Minimize h(x , y) := ‖c−(x , y)‖ ... more important

3 Minimize hc(x , y) := ‖min(x1, x2)‖ ... most important

... for plots, let h(x) := h(x , y) + hc(x , y)

c(x)

f(x)

dom

inate

d

h(x) =

(h , f )kk

Borrow concept of domination frommulti-objective optimization

(hk , hck , fk) dominates (hl , h

cl , fl)

iff hk ≤ hl & hck ≤ hc

l & fk ≤ fl

i.e. (x (k), y (k)) at least as good as(x (l), y (l))

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Global Convergence to B-Stationarity

Assumptions:

MPEC-MFCQ (i.e. every piece satisfies MFCQ) weak

x (k) remain in compact set strong

f , c twice continuously differentiable

Theorem

Outcome of SLPEC is one of:

1 restoration phase fails to find feasible point, or

2 d = 0 solves LPEC ⇒ B-stationary, or

3 limit is B-stationary.

Proof: exploit fact that LPEC ≡ disjunctive LPs

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Conclusions

Considered convergence of four classes of methods for MPECs1 SLPEC-EQP Method

Method of choice, but LPEC hard to solveDeveloping active-set type solver for LPECs⇒ base on standard LP solvers ...

2 Sequential Quadratic Programming Methods

Often works very well ... my preferred methodFails to converge or converges slowly for degenerate MPECs

3 Interior-Point Methods

Works mostly well ... not as robust as SQPFails to converge or converges slowly for degenerate MPECs

4 Sequential penalization or regularization methods

Not as effective as SQP or IPM above... solve sequence of NLPs versus a single one!Fails to converge or converges slowly for degenerate MPECs

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