Solving MPECs Implicit Programming and NLP Methods Michal Kočvara Academy of Sciences of the Czech Republic September 2005 1
Mathematical Programs with Equilibrium Constraints Mechanical motivation Mechanical equilibrium (static problems): min Π( u) s.t. u K E( u) Π( u)... (quadratic) potential energy u... displacements K... set (cone) of admissible dispalcements When K space Π( u) = 0, A u f = 0 September 2005 2
Mathematical Programs with Equilibrium Constraints Mechanical motivation Mechanical equilibrium (static problems): min Π(α, u) s.t. u K(α) E(α, u) Π(α, u)... (quadratic) potential energy u... displacements K(α)... set (cone) of admissible dispalcements When K space Π(α, u) = 0, A(α)u f(α) = 0 α... shape of elastic body, thickness, load, material properties, boundary conditions,... September 2005 3
Mathematical Programs with Equilibrium Constraints MPEC: Mechanical motivation min α,u F(α, u) s.t. α U ad u solves E(α, u) F(α, u)... cost functional α... design variable u... state variable U ad... admissible designs natural MPEC September 2005 4
Solving MPECs: ImP and NLP Methods... What is Implicit Programming? min α,u F(α, u) s.t. α U ad u solves E(α, u) Define solution map S : α u of E(α, u). Assume: (A1) F continuously differentiable on à R k, U ad à (A2) S single-valued on à (A3) E strongly regular at all points (α, u) with α Ã, u = S(α) September 2005 5
Implicit Programming (ImP) Technique Using S, write as min α,u s.t. F(α, u) α U ad u solves E(α, u) min Θ(α) := F(α, S(α)) α α U ad s.t. Standard (but nonsmooth) optimization problem Solve by any nonsmooth algorithm, e.g. BT (Bundle-Trust region). September 2005 6
Solving MPEC by ImP and BT min Θ(α) := F(α, S(α)) α α U ad s.t. To use BT, one needs, at each iterate α k the function value Θ(α k ) and main task: compute S(α) (solve E) one element (subgradient) of the generalized Jacobian Θ(α k ) implicit programming technique developed in 90s Outrata-MK-Zowe, Kluwer 1998 September 2005 7
Solving MPEC by ImP and BT (example) Example: convex quadratic equilibrium problem min u, C(α)u b(α), u 2 Au = c s.t. u 1 Bu d Denote λ... Lagrangian multiplier for inequality constraints Adjoint problem: 1 min p 2 p, C(α)p uf(α, u), p s.t. Ap = 0 B j p = 0, j I + (α, u) M i (α, u) September 2005 8
Solving MPEC by ImP and BT (example cont.) Adjoint problem: 1 min p 2 p, C(α)p uf(α, u), p s.t. Ap = 0 B j p = 0, j I + (α, u) M i (α, u) where I(α, u) = I + (α, u) = {i {1, 2,..., m} B i, u = d i} { } i I(α, u) λ i > 0 I 0 (α, u) = I(α, u) \ I + (x, u). September 2005 9
Solving MPEC by ImP and BT (example cont.) Adjoint problem: Then min u, C(α)u b(α), u 2 Au = c s.t. u 1 Bu d 1 min p 2 p, C(α)p uf(α, u), p s.t. Ap = 0 B j p = 0, j I + (α, u) M i (α, u) α f(α, u) [ α (C(α)u b(α))] T p Θ(α) September 2005 10
Solving MPEC by ImP and BT + min Θ(α) := F(α, S(α)) α α U ad s.t. BT particularly efficient for few variables difficult nonsmoothness only one subgradient available variables separated, E solved by special solvers (high dimension) single-valuedness of S (sometimes) nonsmooth codes not efficient and robust September 2005 11
MP with Complementarity Constraints (MPCC) min F(z) s.t. g j (z) 0, j J eq g j (z) = 0, j J in 0 z 1 z 2 0 z = (z 0, z 1, z 2 ), z 0... control variable (α) z 1... state variable of E z 2... multipler of E September 2005 12
MPCC and MPEC MPCC is almost a subset of MPEC MPEC MPCC: optimum desing with given friction Coulomb friction hemivariational inequalities MPEC MPCC: z = (z 1, z 2 ) (no control variable) may look as formal reason but it excludes ImP technique MPEC MPCC min F(u) u u solves E(u) s.t. September 2005 13
Solution Techniques for MPCC Note that min F(z) s.t. g j (z) 0, j J eq g j (z) = 0, j J in z 1 0, z 2 0, z1 T z 2 = 0 is an NLP as such. BUT: Mangasarian Fromowitz constraint qualification (MFCQ) for this problem is violated at all feasible points expect serious difficulties of standard NLP algorithms. September 2005 14
Solution Techniques for MPCC (cont.) Several techniques proposed: replace z T 1 z 2 = 0 by z T 1 z 2 τ with some τ > 0 solve a sequence of problems with τ 0 Scheel-Scholtes, Ferris-Kanzow. replace 0 z 1 z 2 0 by a smooth equation: smoothened min-function (Facchinei et al.) (z i 1 zi 2 )2 + 4τ z i 1 zi 2 = 0 smoothened Fischer-Burmeister function (Jiang and Ralph) (z i 1 )2 + (z i 2 )2 + τ z i 1 zi 2 = 0 with τ > 0. Solve (inexactly) a sequence of NLPs with τ 0. September 2005 15
Solution Techniques for MPCC (cont.) A direct NLP approach (recently mostly used) Sven Leyffer 1999: (Scheel-Scholtes, Anitescu) Formulate MPCC as NLP, use SQP solvers: min F(z) s.t. g j (z) 0, j J eq g j (z) = 0, j J in z 1 0, z 2 0, z1 T z 2 0 This NLP does not satisfy MFCQ, but why not trying... Experience: many NLP solvers do not work but some do! September 2005 16
Why does this work? MFCQ not satisfied Lagrangian multipliers unbounded. Fletcher et al.: there exists a basic multiplier; the multiplier set is a ray based in this basic multiplier vector. SQP methods converge quadratically to the basic multiplier, provided all QP subproblems remain consistent. MFCQ not satisfied QP subproblem in SQP may be inconsistent. Anitescu: elastic mode, implemented in some SQP codes (SNOPT) helps. Modify the NLP by relaxing the constraints and add a penalty term to the objective function. SQP with elastic mode converges globally. September 2005 17
What about interior-point codes? l penalization (used e.g. by LOQO): replace min F(z) s.t. g j (z) 0, j J eq g j (z) = 0, j J in z 1 0, z 2 0, z T 1 z 2 = 0 by min F(z) + ρζ s.t. g j (z) 0, j J eq g j (z) = 0, j J in z 1 0, z 2 0, ζ 0, z i 1 zi 2 ζ i September 2005 18
Penalization not globally convergent! Anitescu: Any solution to MPCC is a solution to the penalized problem with large enough ρ BUT: the converse is no true. No matter how large the penalty, there will always be solutions of the penalized problem which are not solutions for MPCC. September 2005 19
ImP-MPEC vs. NLP-MPCC ImP-MPEC separated variables well-structured for BT uniqueness of E nonsmooth solvers not robust no state constraints can solve non-mpcc problems NLP-MPCC works on Cartesian product design var. = state var. no uniqueness of E needed robust NLP solvers can handle state constraints can solve non-mpec problems MPEC MPCC September 2005 20
ImP-MPEC and NLP-MPCC face-to-face MacMPEC collection of Sven Leyffer: obstacle problem problem size obj. LOQO PENNON pack-rig1-8 49-29-40 0.787932 22 Nwt 0.2s 149 Nwt 1s pack-rig1-16 209-99-192 0.826013 66 Nwt 3s 309 Nwt 16s pack-rig1-32 865-521-832 0.850895 failure 1129 Nwt 5m44s pack-rig2-8 49-29-40 0.780404 43 Nwt 0.4s 198 Nwt 2s pack-rig2-16 209-99-192 (Infeas) failure pack-rig2-32 865-521-832 (Infeas) failure September 2005 21
ImP-MPEC and NLP-MPCC face-to-face MacMPEC collection of Sven Leyffer: obstacle problem problem size obj. PENNON ImP+BT pack-rig1-8 49-29-40 0.787932 149 Nwt 1s 83 fun 0.2s pack-rig1-16 209-99-192 0.826013 309 Nwt 16s 59 fun 0.8s pack-rig1-32 865-521-832 0.850895 1129 Nwt 5m44s 114 fun 7.4s pack-rig2-8 49-29-40 0.780404 198 Nwt 2s 69 fun 0.2s pack-rig2-16 209-99-192 (Infeas) failure 91 fun 1.3s pack-rig2-32 865-521-832 (Infeas) failure 158 fun 10.5s September 2005 22