University of Erlangen-Nürnberg and Academy of Sciences of the Czech Republic. Solving MPECs by Implicit Programming and NLP Methods p.

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1 Solving MPECs by Implicit Programming and NLP Methods Michal Kočvara University of Erlangen-Nürnberg and Academy of Sciences of the Czech Republic Solving MPECs by Implicit Programming and NLP Methods p.1/31

2 Outline 1. What is MPEC, examples from mechanics 2. What is Implicit Programming 3. MPEC and MPCC 4. ImP and NLP face-to-face 5. Examples Solving MPECs by Implicit Programming and NLP Methods p.2/31

3 MPEC 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 Solving MPECs by Implicit Programming and NLP Methods p.3/31

4 MPEC 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) α... shape of elastic body, thickness, load, material properties, boundary conditions,... Solving MPECs by Implicit Programming and NLP Methods p.4/31

5 MPEC MPEC: Mechanical motivation s.t. min α,u F (α, u) 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 Solving MPECs by Implicit Programming and NLP Methods p.5/31

6 MPEC Examples: Obstacle problem State problem (membrane with rigid obstacle): min Π(u) := 1 2 ut Au f T u s.t. u K := {v R n v i ψ i } Solving MPECs by Implicit Programming and NLP Methods p.6/31

7 MPEC Examples: Obstacle problem State problem (membrane with rigid obstacle): min Π(u) := 1 2 ut Au f T u s.t. u K := {v R n v i ψ i } Au f 0, u ψ 0 (Au f) T (u ψ) = 0 Solving MPECs by Implicit Programming and NLP Methods p.6/31

8 MPEC Examples: Obstacle problem Shape (control) variable α: a n ξ 2 Ω 0 a i (α ( ), ) h a i a i a a c c ξ c Solving MPECs by Implicit Programming and NLP Methods p.7/31

9 MPEC Examples: Obstacle problem Shape (control) variable α: a n ξ 2 Ω 0 a i (α ( ), ) h a i a i LCP(α): a a c c ξ c A(α)u f(α) 0, u ψ(α) 0 (A(α)u f(α)) T (u ψ(α)) = 0 Solving MPECs by Implicit Programming and NLP Methods p.7/31

10 MPEC Examples: Obstacle problem Shape (control) variable α: a n ξ 2 Ω 0 a i (α ( ), ) h a i a i a a c c ξ c min α,u F r(α, u) := meas Ω h (α) + r h 2 s.t. α U ad u solves LCP(α) i D 0 u i ψ i Solving MPECs by Implicit Programming and NLP Methods p.7/31

11 MPEC Examples: Designing Masonry Structures Solving MPECs by Implicit Programming and NLP Methods p.9/31

12 MPEC Examples: Designing Masonry Structures Discrete dual elasticity problem: min σ E(α) 1 2 Ψ(α, σ) (:= σt F (α)σ) E(α) := {τ R 3M A(α)τ = f(α)} Masonry piers: vertical stress component nonpositive min σ E(α) M 1 2 σt F (α)σ M := {τ R 3M σ 22,k 0, k = 1, 2,..., M} Solving MPECs by Implicit Programming and NLP Methods p.10/31

13 MPEC Examples: Designing Masonry Structures MPEC: minimize weight so that the pier does not collapse min F (α, σ, λ) := meas Ω(α) + r h 2 i D0 λi s.t. α Uad (σ, λ) solves QP(α) λ is a multiplier to the inequality constraints in QP(α) Ω λ = 0 Ωα 0 Solving MPECs by Implicit Programming and NLP Methods p.11/31

14 MPEC Examples: Designing Masonry Structures Solving MPECs by Implicit Programming and NLP Methods p.12/31

15 Solving MPECs: ImP and NLP Methods... How about solving MPEC? What is Implicit Programming? Solving MPECs by Implicit Programming and NLP Methods p.13/31

16 Solving MPECs: ImP and NLP Methods... How about solving MPEC? What is Implicit Programming? min α,u s.t. F (α, u) α U ad u solves E(α, u) Define solution map S : α u of E(α, u). Assume: (A1) F continuously differentiable on Ã R k, U ad Ã (A2) S single-valued on Ã (A3) E strongly regular at all points (α, u) with α Ã, u = S(α) Solving MPECs by Implicit Programming and NLP Methods p.13/31

17 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 α s.t. Θ(α) := F (α, S(α)) α U ad Standard (but nonsmooth) optimization problem Solve by any nonsmooth algorithm, e.g. BT (Bundle-Trust region). Solving MPECs by Implicit Programming and NLP Methods p.14/31

18 Solving MPEC by ImP and BT min α s.t. Θ(α) := F (α, S(α)) α U ad To use BT, one needs, at each iterate α k the function value Θ(α k ) main task: compute S(α) (solve E) Solving MPECs by Implicit Programming and NLP Methods p.15/31

19 Solving MPEC by ImP and BT min α s.t. Θ(α) := F (α, S(α)) α U ad To use BT, one needs, at each iterate α k and the function value Θ(α k ) 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 Solving MPECs by Implicit Programming and NLP Methods p.15/31

20 Solving MPEC by ImP and BT min α s.t. Θ(α) := F (α, S(α)) α U ad + BT particularly efficient for few variables BT particularly efficient for difficult nonsmoothness BT particularly efficient for only one subgradient available variables separated, E solved by special solvers (high dimension) Solving MPECs by Implicit Programming and NLP Methods p.16/31

21 Solving MPEC by ImP and BT min α s.t. Θ(α) := F (α, S(α)) α U ad + BT particularly efficient for few variables BT particularly efficient for difficult nonsmoothness BT particularly efficient for only one subgradient available variables separated, E solved by special solvers (high dimension) single-valuedness of S (sometimes) nonsmooth codes not efficient and robust Solving MPECs by Implicit Programming and NLP Methods p.16/31

22 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 = (z 0, z 1, z 2 ), z = (z 0, z 1, z 2 ), z 0... control variable (α) z 1... state variable of E z 2... multipler of E Solving MPECs by Implicit Programming and NLP Methods p.17/31

23 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 = (z 0, z 1, z 2 ), z = (z 0, z 1, z 2 ), z 0... control variable (α) z 1... state variable of E z 2... multipler of E So far, our MPEC examples were also MPCC Solving MPECs by Implicit Programming and NLP Methods p.17/31

24 MPCC and MPEC MPCC is almost a subset of MPEC MPEC MPCC: optimum desing with given friction MPEC MPCC: optimum desing with Coulomb friction MPEC MPCC: optimum desing with hemivariational inequalities Solving MPECs by Implicit Programming and NLP Methods p.18/31

25 MPCC and MPEC MPCC is almost a subset of MPEC MPEC MPCC: optimum desing with given friction MPEC MPCC: optimum desing with Coulomb friction MPEC MPCC: optimum desing with hemivariational inequalities MPEC MPCC: z = (z 1, z 2 ) (no control variable) MPEC MPCC: may look as formal reason but it MPEC MPCC: excludes ImP technique Solving MPECs by Implicit Programming and NLP Methods p.18/31

26 MPCC and MPEC MPCC is almost a subset of MPEC MPEC MPCC: optimum desing with given friction MPEC MPCC: optimum desing with Coulomb friction MPEC MPCC: optimum desing with hemivariational inequalities MPEC MPCC: z = (z 1, z 2 ) (no control variable) MPEC MPCC: may look as formal reason but it MPEC MPCC: excludes ImP technique MPEC MPCC min u s.t. F (u) u solves E(u) Solving MPECs by Implicit Programming and NLP Methods p.18/31

27 Solution Techniques for MPCC Fukushima-Pang 99, Scholtes 01, Hu-Ralph 01, Fukushima-Tseng Solving MPECs by Implicit Programming and NLP Methods p.19/31

28 Solution Techniques for MPCC Fukushima-Pang 99, Scholtes 01, Hu-Ralph 01, Fukushima-Tseng Sven Leyffer 1999 (sooner?): (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, z T 1 z 2 0 This NLP does not satisfy MFCQ, but why not trying... Experience: many NLP solvers do not work but some do! Solving MPECs by Implicit Programming and NLP Methods p.19/31

29 Solution Techniques for MPCC Fukushima-Pang 99, Scholtes 01, Hu-Ralph 01, Fukushima-Tseng Sven Leyffer 1999 (sooner?): (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, z T 1 z 2 0 This NLP does not satisfy MFCQ, but why not trying... Experience: many NLP solvers do not work but some do!... more in Sven s and Mihai s talks... Solving MPECs by Implicit Programming and NLP Methods p.19/31

30 ImP-MPEC vs. NLP-MPCC ImP-MPEC separated variables well-structured for BT NLP-MPCC works on Cartesian product design var. = state var. Solving MPECs by Implicit Programming and NLP Methods p.20/31

31 ImP-MPEC vs. NLP-MPCC ImP-MPEC separated variables well-structured for BT uniqueness of E nonsmooth solvers not robust NLP-MPCC works on Cartesian product design var. = state var. no uniqueness of E needed robust NLP solvers Solving MPECs by Implicit Programming and NLP Methods p.20/31

32 ImP-MPEC vs. NLP-MPCC ImP-MPEC separated variables well-structured for BT uniqueness of E nonsmooth solvers not robust no state constraints NLP-MPCC works on Cartesian product design var. = state var. no uniqueness of E needed robust NLP solvers can handle state constraints Solving MPECs by Implicit Programming and NLP Methods p.20/31

33 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 Solving MPECs by Implicit Programming and NLP Methods p.20/31

34 ImP-MPEC and NLP-MPCC face-to-face MacMPEC collection of Sven Leyffer: obstacle problem pack-comp1, pack-comp2 problem var constr FilterMPEC LOQO-MPEC p-c p-c p-c p-c p-c p-c Solving MPECs by Implicit Programming and NLP Methods p.21/31

35 ImP-MPEC and NLP-MPCC face-to-face MacMPEC collection of Sven Leyffer: obstacle problems pack-comp1 problem var constr LOQO-MPEC ImP-BT p-c /9 180/ p-c /17 736/ p-c / / Solving MPECs by Implicit Programming and NLP Methods p.22/31

36 ImP-MPEC and NLP-MPCC face-to-face MacMPEC collection of Sven Leyffer: obstacle problems pack-comp1 problem var constr LOQO-MPEC ImP-BT p-c /9 180/ p-c /17 736/ p-c / / p-c / / p-c / / Solving MPECs by Implicit Programming and NLP Methods p.22/31

37 Effective reformulation of truss design problem Truss design problem min f T u t R m,u R n s.t. A(t)u = f, m t i = V, t i 0, i = 1, 2,..., m i=1 A(t) = m i=1 t i A i, t i... bar volumes u i... displacements A i = E i l 2 i γ i γ T i Solving MPECs by Implicit Programming and NLP Methods p.23/31

38 Effective reformulation of truss design problem Truss design problem min f T u t R m,u R n s.t. A(t)u = f, m t i = V, t i 0, i = 1, 2,..., m i=1 A(t) = m i=1 t i A i, t i... bar volumes u i... displacements A i = E i l 2 i γ i γ T i Solving MPECs by Implicit Programming and NLP Methods p.23/31

39 Effective reformulation of truss design problem Truss design problem min f T u t R m,u R n s.t. A(t)u = f, m t i = V, t i 0, i = 1, 2,..., m i=1 Difficult NLP problem, only SNOPT can solve medium-size examples, many codes fail for smallest problems Solving MPECs by Implicit Programming and NLP Methods p.23/31

40 Effective reformulation of truss design problem Truss design problem min f T u t R m,u R n s.t. A(t)u = f, m t i = V, t i 0, i = 1, 2,..., m i=1 Ben-Tal/Bendsøe: min α f T u α R,u R n subject to 1 2 ut A i u α 0, i = 1,..., m Solving MPECs by Implicit Programming and NLP Methods p.23/31

41 Effective reformulation of truss design problem Nonuniqueness in displacements Solving MPECs by Implicit Programming and NLP Methods p.24/31

42 Effective reformulation of truss design problem Nonuniqueness in displacements Nonuniqueness in bar volumes Solving MPECs by Implicit Programming and NLP Methods p.24/31

43 Effective reformulation of truss design problem min α f T u α R,u R n subject to 1 2 ut A i u α 0, i = 1,..., m solution u, KKT vector t, both nonunique Solving MPECs by Implicit Programming and NLP Methods p.25/31

44 Effective reformulation of truss design problem min α f T u α R,u R n subject to 1 2 ut A i u α 0, i = 1,..., m solution u, KKT vector t, both nonunique Identify a physical solution by solving MPCC min t,u G(t, u) s.t. (t, u) solves QQP MPEC MPCC Solving MPECs by Implicit Programming and NLP Methods p.25/31

45 Effective reformulation of truss design problem min t R m,u R n,α R G(t, u) ( m ) s.t. t i A i u + f = 0 i=1 m t i 1 i=1 0 (α 1 2 ut A i u) t i 0, i = 1,..., m. Solving MPECs by Implicit Programming and NLP Methods p.26/31

46 Effective reformulation of truss design problem min t R m,u R n,α R G(t, u) ( m ) s.t. t i A i u + f = 0 i=1 m t i 1 i=1 0 (α 1 2 ut A i u) t i 0, i = 1,..., m. Resembles min f T u t R m,u R n s.t. A(t)u = f, m t i 1, t i 0, i = 1, 2,..., m i=1 Solving MPECs by Implicit Programming and NLP Methods p.26/31

47 Effective reformulation of truss design problem problem var constr SNOPT KNITRO MINOS tro_3x3f tro_4x4f F 1588 tro_5x5f F F Solving MPECs by Implicit Programming and NLP Methods p.27/31

48 Effective reformulation of truss design problem Identify a physical solution by solving MPCC min t,u G(t, u) s.t. (t, u) solves QQP min α R,u R n α f T u subject to 1 2 ut A i u α 0, i = 1,..., m Solving MPECs by Implicit Programming and NLP Methods p.28/31

49 Effective reformulation of truss design problem min α f T u α R,u R n (QQP) s.t. 1 2 ut A i u α 0, i = 1,..., m Solving MPECs by Implicit Programming and NLP Methods p.29/31

50 Effective reformulation of truss design problem min α f T u α R,u R n (QQP) s.t. 1 2 ut A i u α 0, i = 1,..., m Proposition: Assume that G only depends on u. Let (α, u ) be a solution of (QQP). Then ûis the solution of the "unique "truss design problem if and only if it solves the following problem min G(u) u R n subject to 1 2 ut A i u α 0, i = 1,..., m α α = 0 f T (u u ) = 0. Solving MPECs by Implicit Programming and NLP Methods p.29/31

51 Effective reformulation of truss design problem problem var constr LOQO Filter SNOPT KNITRO PENNON tro_3x tro_4x F tro_5x tro_6x tro_11x tro_21x tro_41x F F 78 tro_21x tro_41x n.a n.a. 8.8 Solving MPECs by Implicit Programming and NLP Methods p.30/31

52 THE END Solving MPECs by Implicit Programming and NLP Methods p.31/31

Solving MPECs Implicit Programming and NLP Methods

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