Linear & Conic Programming Reformulations of Two-Stage Robust Linear Programs

Transcription

1 1 / 34 Linear & Conic Programming Reformulations of Two-Stage Robust Linear Programs Erick Delage CRC in decision making under uncertainty Department of Decision Sciences HEC Montreal (joint work with Amir Ardestani-Jaafari) (special thanks to Samuel Burer) July 5th, 2017

2 2 / 34 A CLASSICAL DISTRIBUTION PROBLEM A multinational retailing corporation wishes to construct new warehouses Warehouse sites Retailers

3 3 / 34 A CLASSICAL DISTRIBUTION PROBLEM 1. Choose where to build the new warehouses Warehouse sites Retailers

4 4 / 34 A CLASSICAL DISTRIBUTION PROBLEM 2. Observe amount of weekly demand Warehouse sites Retailers

5 5 / 34 A CLASSICAL DISTRIBUTION PROBLEM 3. Transport goods to retailers to maximize profits Warehouse sites Retailers

6 6 / 34 A CLASSICAL DISTRIBUTION PROBLEM Facility location-transportation model maximize I {0,1} n,x,y 0 s. t. sold product location cost { }}{{}}{ η Y ij c T x + K T I + i j Y ij x i, i, j transportation&production cost { }}{ (p i + t ij )Y ij i (Capacity constraint) Y ij d j, j, (Demand constraint) i x i MI i, i, (Facility Size constraint) How can one account for demand uncertainty? j

7 7 / 34 ROBUST OPTIMIZATION IS NOW A WELL ESTABLISHED METHODOLOGY Ben-Tal & Nemirovski s «Robust Convex Optimization»

8 8 / 34 A CLASSICAL ROBUST DISTRIBUTION PROBLEM Robust Facility location-transportation model: max Y 0 maximize I {0,1} n,x min d D h(i, x, d) s. t. x i MI i, i, (Facility Size constraint) where h(i, x, d) is the optimal value of sold product location cost { }}{{}}{ { }}{ η Y ij c T x + K T I + (p i + t ij )Y ij s. t. i j Y ij x i, i, j transportation&production cost i (Capacity constraint) Y ij d j, j, (Demand constraint) i j

9 9 / 34 OUTLINE INTRODUCTION BACKGROUND ON TWO-STAGE ROBUST LINEAR PROGRAMS COPOSITIVE PROGRAMMING REFORMULATIONS APPLICATIONS CONCLUSION

10 10 / 34 OUTLINE INTRODUCTION BACKGROUND ON TWO-STAGE ROBUST LINEAR PROGRAMS COPOSITIVE PROGRAMMING REFORMULATIONS APPLICATIONS CONCLUSION

11 STATIC ROBUST LINEAR PROGRAM [BEN-TAL & NEMIROVSKI (2000), 1296 CITATIONS!] Consider the following static problem: maximize x X,y c T x + f T y (1a) s. t. Ax + By D(x)z, z Z (1b) where we assume n x + n y decision variables, J constraints, and m uncertain parameters. If Z := {z R m z 0, Pz = q} is a non-empty polyhedral set defined by K constraints, then Problem (1) maximize x X,y,Λ c T x + f T y s. t. Ax + By + Λq 0 D(x) + ΛP 0, where Λ R J K. 11 / 34

12 12 / 34 TWO-STAGE ROBUST LINEAR PROGRAMS [BEN-TAL ET AL. (2004), 824 CITATIONS!] Consider the following two-stage problem: (TSRLP) maximize x X,y( ) min z Z ct x + f T y(z) s. t. Ax + By(z) D(x)z z Z where y : R m R ny This problem can also be represented as where (TSRLP) maximize x X min z Z h(x, z) h(x, z) := max y c T x + f T y s. t. Ax + By D(x)z.

13 COMPLEXITY OF TWO-STAGE ROBUST LINEAR PROGRAMS Unfortunately, the two-stage robust linear program is known to be intractable in general [Ben-Tal et al. (2004)]. Conservative approximation obtained by using affine adjustment functions : y(z) := y + Yz The two-stage robust problem reduces to (AARC) maximize x X,y,Y min z Z ct x + f T (y + Yz) s. t. Ax + B(y + Yz) D(x)z z Z Some exact methods have been proposed but without polynomial time convergence guarantees [Zeng & Zhao (2013)] 13 / 34

14 14 / 34 OUTLINE INTRODUCTION BACKGROUND ON TWO-STAGE ROBUST LINEAR PROGRAMS COPOSITIVE PROGRAMMING REFORMULATIONS APPLICATIONS CONCLUSION

15 COPOSITIVE PROGRAMMING REFORMULATION I Assumptions 1. Z is a non-empty and bounded polyhedral set 2. The TSRLP problem is bounded above, i.e. x X, z Z, h(x, z) <. Let our robust optimization problem take the form where ψ(x) := min z Z maximize x X max y ψ(x), c T x + f T y (2a) s. t. Ax + By D(x)z (2b) Since (2) is bounded, strong LP duality applies ψ(x) = min z Z,λ 0 c T x + z T D(x) T λ (Ax) T λ B T λ = f 15 / 34

16 16 / 34 COPOSITIVE PROGRAMMING REFORMULATION I The function ψ(x) minimizes a non-convex quadratic function over a polyhedron in the non-negative orthant ψ(x) = min z 0 c T x + z T Q(x) z c(x)t z à z = b, where z := [λ T z T ] R J+m and where [ ] [ 0 (1/2)D(x) (1/2)Ax Q(x) := (1/2)D(x) T c(x) := 0 0 [ ] [ ] B T 0 d à := b := 0 P q ]

17 17 / 34 COPOSITIVE PROGRAMMING REFORMULATION I The function ψ(x) minimizes a non-convex quadratic function over a polyhedron in the non-negative orthant ψ(x) = min z 0 c T x + trace( Q(x) T z z T ) c(x) T z à z = b à z z T = b z T, where z := [λ T z T ] R J+m and where [ ] [ 0 (1/2)D(x) (1/2)Ax Q(x) := (1/2)D(x) T c(x) := 0 0 [ ] [ ] B T 0 d à := b := 0 P q ]

18 18 / 34 COPOSITIVE PROGRAMMING REFORMULATION I The function ψ(x) has an equivalent convex optimization reformulation ( Z := z z T ) [Burer (2009)] ψ(x) = min Z, z c T x + trace( Q(x) T Z) c(x)t z à z = b [ à Z = b z T ] Z z z T K 1 CP ([ Z z & rank z T 1 ]) = 1 where K CP is the cone of completely positive matrices, i.e. { K CP := M M = } z k z T k for some { z k} k K R J+m+1 +. k K

19 19 / 34 COPOSITIVE PROGRAMMING REFORMULATION I The function ψ(x) has an equivalent convex optimization reformulation ( Z := z z T ) [Burer (2009)] ψ(x) = min Z, z c T x + trace( Q(x) T Z) c(x)t z à z = b [ à Z = b z T ] Z z z T K 1 CP ([ Z z & rank z T 1 ]) = 1 where K CP is the cone of completely positive matrices, i.e. { K CP := M M = } z k z T k for some { z k} k K R J+m+1 +. k K

20 20 / 34 COPOSITIVE PROGRAMMING REFORMULATION I By conic duality we get ψ(x) max W, w,ṽ,t c(x) T x + b T w t s. t. ṽ = c(x) (1/2)(ÃT w W T b) [ Q(x) (1/2)( WT Ã + ÃT W) ṽ ṽ T t ] K Cop, where K Cop is the cone of copositive matrices, i.e. { } K Cop := M M = M T, z T Mz 0, z R J+m+1. +

21 21 / 34 COPOSITIVE PROGRAMMING REFORMULATION I Theorem 1 [Xu & Burer (2016), Hanasusanto & Kuhn (2016)] If the TSRLP problem has complete recourse, i.e. then the copositive program y R ny, By < 0, (Copos 1 ) maximize c T x + b T w t x X, W, w,ṽ,t s. t. ṽ = c(x) (1/2)(ÃT w W T b) [ Q(x) (1/2)( WT Ã + ÃT W) ṽ ṽ T t ] K Cop, provides an exact reformulation of the TSRLP problem. Otherwise, Copos 1 only provides a conservative approximation.

22 22 / 34 RELATION TO AARC Theorem 2 [Xu & Burer (2016)] When K Cop is replaced with N := R J+m+1 J+m+1 + K Cop the copositive programming reformulation is equivalent to AARC. Hence, for any cone K such that N K K Cop, Copos 1 with K provides a tighter approximation than AARC There exists a hierarchy of semidefinite and polyhedral cones {K i } i=1, with N K 1 K 2 K Cop, such that for all M K Cop, there is a i, M K i [Parrilo (2000), Bomze & de Klerk (2002)] This is valuable for complete recourse problems but what about relatively complete recourse problems?

23 HOW TO FIX RELATIVELY COMPLETE RECOURSE Assumption : The TSRLP problem has relatively complete recourse, i.e. This ensures that : x X z Z, y, Ax + By D(x)z h(x, z) = min λ c T x + z T D(x) T λ (Ax) T λ R s. t. λ P := {λ λ 0, B T λ = f } Hence, always an optimal solution λ (x, z) at a vertex of P Since number of vertices is finite, there exists u R J + : ψ(x) = min h(x, z) = min z Z z Z,λ P s. t. λ u c T x + z T D(x) T λ (Ax) T λ 23 / 34

24 24 / 34 COPOSITIVE PROGRAMMING REFORMULATION II The function ψ(x) minimizes a non-convex quadratic function over a polyhedron in the non-negative orthant ψ(x) = min ȳ 0 c T x + ȳ T Q(x)ȳ c(x)tȳ Āȳ = b, where ȳ := [λ T z T s T ] R 2J+m and where Q(x) := Ā := 0 (1/2)D(x) 0 (1/2)D(x) T B T P 0 I 0 I c(x) := b := (1/2)Ax 0 0 d q u.

25 25 / 34 COPOSITIVE PROGRAMMING REFORMULATION II Theorem 3 [AJ&D (2016b)] If the TSRLP problem has relatively complete recourse, then the copositive program (Copos 2 ) max x X, W, w, v,t c T x + b T w t s. t. v = c(x) (1/2)(ĀT w W T b) [ Q(x) (1/2)( WT Ā + ĀT W) v v T t ] K Cop, provides an exact reformulation of the TSRLP problem.

26 THE PENALIZED AARC MODEL Theorem 4 [AJ&D (2016b)] When K Cop is replaced with N the Copos 2 reformulation is equivalent to applying affine adjustments to: (TSRLP ) maximize x X,y( ),θ( ) min z Z ct x + f T y(z) u T θ(z) s. t. Ax + By(z) D(x)z + θ(z) z Z. Moreover, affine (and static) adjustments are always feasible in TSRLP. u can be interpreted as a marginal penalty for violating constraints TSRLP TSRLP since u is such that there always exists an optimal solution triplet with θ(z) := 0. Method for converting a relatively complete recourse multi-stage linear program into a complete recourse one 26 / 34

27 27 / 34 OUTLINE INTRODUCTION BACKGROUND ON TWO-STAGE ROBUST LINEAR PROGRAMS COPOSITIVE PROGRAMMING REFORMULATIONS APPLICATIONS CONCLUSION

28 ROBUST FACILITY LOCATION-TRANSPORTATION PROBLEM In AJ&D (2016b), we identify an instance for which AARC Penalized AARC Exact model (a.k.a. Copos 2 (N )) model Bound on w.-c. profit W.-c. profit of x We recently randomly generated problem instances, 5 facilities & 10 customer locations. Optimality Proportion of instances gap AARC Penalized AARC = 0% 20.6% 23.8% 0.1% 20.9% 27.4% 1% 28.4% 56.3% Avg. Gap 10.5% 1.6% Max Gap 50.0% 13.3% 28 / 34

29 WHAT SIZE PROBLEMS CAN WE SOLVE? [AJ&D (2017)] (T,L,N) (1,50,100) (20,15,30) Γ Penalized AARC Exact Full form Row generation C&CG sec sec sec sec sec sec sec 7 sec < 1 sec 2 sec Avg sec sec 184 sec sec sec sec sec sec <1 sec Avg sec - ( stands for more than two days of computation) 29 / 34

30 30 / 34 ROBUST MULTI-ITEM NEWSVENDOR In AJ&D (2016a): the robust multi-item newsvendor problem with uncorrelated demand can be solved optimally by AARC/Copos(N ) when using budgeted uncertainty set with integer Γ. In AJ&D (2016b): if demand is correlated than solution improves using Copos with K N : AARC Copos(K 4 LP ) Copos(K1 SDP ) Exact W.-c. profit bound Actual w.-c. profit

31 31 / 34 OUTLINE INTRODUCTION BACKGROUND ON TWO-STAGE ROBUST LINEAR PROGRAMS COPOSITIVE PROGRAMMING REFORMULATIONS APPLICATIONS CONCLUSION

32 32 / 34 CONCLUSION & OPEN QUESTIONS 1. Copositive programming is a useful tool for generating conservative approximations for TSRLP Copos(K) with K N always improves on AARC Although hierarchy of polyhedral cones N KLP d K Cop provide LP reformulations, preliminary results indicate that classical ones perform poorly Can Copos(K) provide intuition on approximate policies? Do Copos(K) reformulations exist for multi-stage problems? 2. Penalized violations transform any two-stage LP with relatively complete recourse in one with complete recourse A useful preprocessing step for AARC when feasibility is a challenge Is it possible to generalize this approach to robust multi-stage non-linear problems?

33 33 / 34 BIBLIOGRAPHY A. Ardestani-Jaafari and E. Delage. Robust Optimization of Sums of Piecewise Linear Functions with Application to Inventory Problems. Operations Research, 64(2): , 2016a. A. Ardestani-Jaafari and E. Delage. Linearized Robust Counterparts of Two-stage Robust Optimization Problem with Applications in Operations Management. Working draft, 2016b. A. Ardestani-Jaafari and E. Delage. The Value of Flexibility in Robust LocationTransportation Problems. Transportation Science, A. Ben-Tal and A. Nemirovski. Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, Series A, 88(3): , A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical Programming A, 99(2): , I. M. Bomze and E. de Klerk. Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Journal of Global Optimization, 24(2): , S. Burer. On the copositive representation of binary and continuous nonconvex quadratic programs. Mathematical Programming A, 120(2): , G. A. Hanasusanto and D. Kuhn. Conic programming reformulations of two-stage distributionally robust linear programs over Wasserstein balls. Working draft, P. A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, G. Xu and S. Burer. A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides. Working draft, B. Zeng and L. Zhao. Solving two-stage robust optimization problems using a column-and-constraint generation method. Operations Research Letters, 41(5): , 2013.

34 34 / 34 Questions & Comments Thank you!