An RLT Approach for Solving Binary-Constrained Mixed Linear Complementarity Problems

Transcription

1 An RLT Approach for Solving Binary-Constrained Mixed Linear Complementarity Problems Miguel F. Anjos Professor and Canada Research Chair Director, Trottier Institute for Energy TAI 2015 Washington, DC, USA 30 October 2015

2 Acknowledgement and Reference This is joint work with Franklin Djeumou Fomeni and Steven A. Gabriel. F. Djeumou Fomeni, S.A. Gabriel, and M.F. Anjos. An RLT approach for solving the binary-constrained mixed linear complementarity problem. Cahier du GERAD G , June 2015.

3 Linear Complementarity Problems The linear complementarity problem (LCP) is a well-known problem in optimization. It is characterized by the presence of complementarity constraints: b i a T i x 0, b j a T j x 0, and (b i a T i x)(b j a T j x) = 0 The mixed linear complementarity problem (MLCP) is an LCP in which some of the terms involved in the complementarity constraints are not required to be non-negative. The Binary-Constrained MLCPs (BC-MLCP) is an MLCP in which some variables are restricted to be binary. These classes of problems are NP-hard (cf. LCP) and have numerous engineering and economic applications.

4 Linear Complementarity Problems The linear complementarity problem (LCP) is a well-known problem in optimization. It is characterized by the presence of complementarity constraints: b i a T i x 0, b i a T i x 0, and (b i a T i x)(b j a T j x) = 0 The mixed linear complementarity problem (MLCP) is an LCP in which some of the terms involved in the complementarity constraints are not required to be non-negative. The Binary-Constrained MLCPs (BC-MLCP) is an MLCP in which some variables are restricted to be binary. These classes of problems are NP-hard (cf. LCP) and have numerous engineering and economic applications.

5 BC-MLCP: Applications and Past Research The BC-MLCP has not been studied widely in the literature. Gabriel et al. (2013) proposed a mixed integer linear programming (MILP) approach that relaxes the complementarity constraints as well as the integrality. They showed that their approach was suitable to computing equilibria in energy markets. That approach was also success in solving Nash-Cournot games with application to power markets, and the electricity pool pricing problem (Ruiz et al. (2014)). We motivate the practical interest of BC-MLCPs using a small example from energy economics.

6 Example: An Energy Network Equilibrium Problem Price π 1 Node 1 g 12 Price π 2 Node 2 Demand D 1 (π 1 ) Producer A Producer B Demand D 2 (π 2 ) Producer C Producer D At Node 1: Production levels q A 1 and qb 1 and outflows f A 12 and f B 12. At Node 2: Production levels q C 2 and qd 2.

7 Formulation as an MLCP Formulation for producer A: where max s A 1,qA 1,f A 12 π 1 s A 1 + π 2f A 12 ca 1 (qa 1 ) (τ 12 + τ reg 12 )f A 12 s.t. q A 1 qa (λ A 1 ) s A 1 = qa 1 f A 12 (δ A 1 ) s A 1, qa 1, f A 12 0, c1 A(qA 1 ) = γa 1 qa 1 with γa 1 > 0 is the production cost function, τ reg 12 0 is a (fixed) regulated tariff for using the network from node 1 to node 2, and τ 12 is the (variable) congestion tariff for using the network from node 1 to node 2. The formulation for Producer B is similar.

8 Formulation as an MLCP (ctd) Formulation for producer C: max s C 2,qC 2 π 2 s C 2 cc 2 (qc 2 ) s.t. q C 2 qc (λ C 2 ) s C 2 = qc 2 (δ C 2 ) s C 2, qc 2 0 The formulation for Producer D is similar.

9 Formulation as an MLCP (ctd) The transmission system operator (TSO) solves: max g 12 (τ 12 + τ reg 12 )g 12 c TSO (g 12 ) s.t. g 12 ḡ 12 (ɛ 12 ) g 12 0, where c TSO (g 12 ) = γ TSO g 12 with γ TSO > 0 is the network operation cost function.

10 Formulation as an MLCP (ctd) KKT conditions for Producer A (similar for Producer B): 0 π 1 + δ A 1 sa γ A 1 + λa 1 δa 1 qa π 2 + (τ Reg 12 + τ 12 ) + δ A 1 f A q A 1 qa 1 λa = s A 1 qa 1 + f A 12, δa 1 free KKT conditions for Producer C (similar for Producer D): 0 π 2 + δ C 2 sc γ C 2 + λc 2 δc 2 qc q C 2 qc 2 λc = s C 2 qc 2, δc 2 free

11 Formulation as an MLCP (ctd) KKT conditions for the TSO: Market clearing conditions: 0 τ Reg 12 τ 12 + γ TSO + ɛ 12 g ḡ 12 g 12 ɛ 12 0 Equality of supply and demand: 0 = [s A 1 + sb 1 ] (a 1 b 1 π 1 ), π 1 free, 0 = [s C 2 + sd 2 + f A 12 + f B 12 ] (a 2 b 2 π 2 ), π 2 free Congestion tariff: 0 = g 12 [f A 12 + f B 12 ], τ 12 free. The set of all these conditions is an MLCP.

12 Binary-Constrained Version of the Example Suppose that the formulation for Producer A is of the form: max s1 A,qA 1,f 12 A,v 1 A s.t. π 1 s A 1 + π 2f A 12 ca 1 (qa 1 ) (τ 12 + τ reg 12 )f A 12 v A 1 qa min qa 1 v A 1 qa max s A 1 = qa 1 f A 12 s A 1, qa 1, f A 12 0 v A 1 {0, 1}. Then the resulting model will be a BC-MILP.

13 General Form of a BC-MLCP Let N denote the set of the indices of variables s.t. N = N 1 N 2, and z 1 denotes the n 1 complementarity variables with indices in N 1, and z 2 denotes the n 2 free variables with indices in N 2. ( ) ( ) q1 Given q = R q n A11 A and A = 12 R 2 A 21 A n n, ( ) 22 z1 find R n 1 R n 2 with n 1 + n 2 = n such that z 2 where some or all of 0 q 1 + ( ) ( ) z A 11 A 1 12 z z = q 2 + ( ) ( ) z A 21 A 1 22, z 2 free, ( z1 z 2 z 2 ) are required to be binary.

14 The BC-MLCP: A Challenging Problem! Some of the challenges in solving BC-MLCPs are: The feasible region is in general not convex nor connected. The complementarity and binary constraints may lead to an exponential number of combinations to check. There is only one earlier solution method in the literature: Gabriel et al. (2012) propose to relax both complementarity and integrality, and to trade-off between the two. Our objective is a solution method that does NOT relax either of them.

15 Boundedness Assumption We make the following assumption. There exists a finite solution to the BC-MLCP, i.e., all the continuous variables satisfy < l i x i u i < +. Under this assumption, we can write the set of all solutions of the BC-MLCP (that satisfy the assumption) as: F = x R n q i + A i x 0 for i N 1 (q i + A i x)x i = 0 for i N 1 q i + A i x = 0 for i N 2 0 x i 1 for i N \ B x i {0, 1} for i B

16 An RLT relaxation of the BC-MLCP We first relax F using the well-known RLT approach. Our use of RLT differs from the method in Sherali et al. (1998) to solve the LCP (the special case of MLCP with n 2 = 0) because no new binary variables are introduced. The RLT relaxation is constructed following three simple steps: 1 Relax x i {0, 1} to 0 x i 1 for i B. 2 Replace the terms x i x j with new variable y ij for all i, j N. 3 Relax the relationship y ij = x i x j to y ij 0, x i y ij 0, x j y ij 0, and y ij x i x j

17 An RLT relaxation of the BC-MLCP (ctd) The RLT relaxation of the set F is: q i + A i x 0 for i N 1 q i x i + n A ij y ij = 0 for i N 1 j=1 q F = x R n i + A i x = 0 for i N 2 0 x i 1 for i N y ij 0 for i N y ij x i for i N y ij x j for i N y ij + 1 x i + x j for i, j N

18 Bound Refinement The relaxation F can be improved by tightening the bounds on each variable x i. Given the nonconvex nature of F, finding such bounds is non-trivial. Following Sherali and Tunçbilek (1997), we improve the upper bounds on each complementarity variable as follows: 1 Solve x + i 0 = max x i0. (x,y) F 2 The new upper bound is given by min{x + i 0, y + i 0,i 0 }.

19 Reduce the Number of Complementarity Constraints Theorem For i 0 N 1, let xi 0 be an optimal solution value of min x i0. (x,y) F We have the following: If x i 0 0, then the constraint (q i0 + A i0 x)x i0 = 0 can be replaced by q i0 + A i0 x = 0. If xi 0 = 0 and the problem min x i0 is infeasible, then the (x,y) F + i 0 constraint (q i0 + A i0 x)x i0 = 0 can be replaced by x i0 = 0. Here F + i 0 q i x i + n j=1 is obtained from F by replacing the i th 0 A ij y ij = 0 with q i0 + A i0 x = 0. equation of the form

20 Reduce the Number of Complementarity Const. (ctd) Theorem Suppose that F is non-empty. For i 0 N 1, let xi 0 be the optimal objective value of min x i0. (x,y) F Suppose that xi 0 = 0. For i i 0, if xi 0 and ρ i 1, then we can replace the complementarity constraint (q i + A i x)x i = 0 with the equation q i + A i x = 0. Here ρ i is the allowable increase associated at optimality with the variable x i, i = 1,..., n.

21 Final MILP Reformulation of the BC-MLCP Let and N + 1 = {i N 1 : (q i + A i x)x i = 0 is replaced by q i + A i x = 0} N 0 1 = {i N 1 : (q i + A i x)x i = 0 is replaced by x i = 0}. Clearly either N + 1 N0 1 = N 1 or N + 1 N0 1 N 1. In the first case, all the complementarity constraints have been replaced and therefore the set F is equivalent to F 1 = x R n q i + A i x = 0 for i N + 1 x i = 0 for i N1 0 q i + A i x = 0 for i N \ N 1 0 x i 1 for i N \ B x i {0, 1} for i B. We find a solution to the BC-MLCP by solving an MILP over F 1.

22 Final MILP Reformulation of the BC-MLCP (ctd) In the second case, only some of the complementarity constraints have been replaced and the set F is equivalent to F 2 = x R n q i + A i x 0 for i N 1 \ (N + 1 N0 1 ) (q i + A i x)x i = 0 for i N 1 \ (N + 1 N0 1 ) q i + A i x = 0 for i N + 1 x i = 0 for i N1 0 q i + A i x = 0 for i N \ N 1 0 x i 1 for i N \ B x i {0, 1} for i B. In this case, we follow the work of Sherali et al. (1998) and define an MILP equivalent to finding a solution in F 2.

23 Final MILP Reformulation of the BC-MLCP (ctd) Introduce new binary variables for each complementarity variables such that: { 0 if xi = 0 v i = 1 if x i > 0 Introduce the variables w ij = v i x j Relax the quadratic equation w ij = v i x j using the McCormick inequalities.

24 Final MILP Problem min x,v,w s.t. q T v + i N 1 \(N + 1 N0 1 ) j N A ijw ij n j=1 A kjw ij + q k v i 0 i, k N 1 \ (N + 1 N 0 1 ) n j=1 A kjx j + q k n j=1 A kjw ij + q k v i for i, k N 1 \ (N + 1 N 0 1 ) q i + A i x = 0 for i N + 1 x i = 0 for i N1 0 q i + A i x = 0 for i N \ N 1 x j {0, 1} for j B 0 w ij 1 for i N 1 \ (N + 1 N1 0 ), j N w jj = x j for i N 1 \ (N + 1 N1 0 ) v i {0, 1} for i N 1 \ (N + 1 N1 0 ) w ij 0 for i N 1 \ (N + 1 N1 0 ), j N w ij x j for i N 1 \ (N + 1 N1 0 ), j N w ij v i for i N 1 \ (N + 1 N1 0 ), j N w ij + 1 x j + v i for i N 1 \ (N + 1 N1 0 ), j N.

25 Computational Results

26 LCP Benchmark Instances Instance n # iter. time (sec) compl. B&B Yes No LCP Yes No Yes No Yes No LCP Yes No Yes No Yes No LCP Yes No Yes No Yes No LCP Yes No Yes No Yes No LCP Yes No Yes No Fathi (1979), Murty (1988), and Geiger & Kanzow (1996).

27 MLCP from Two-Node Example Prod. A Prod. B Prod. C Prod. D TSO s A 1 5 s B 1 3 s C 2 5 s D 2 0 g 12 5 # iter. 11 q A 1 10 q B 1 3 q C 2 5 q D 2 0 ɛ 12 2 time f A 12 5 f B 12 0 λ C 2 0 λ D 2 0 π 1 12 compl. Yes λ A 1 2 λ B 1 0 δ C 2 15 δ D 2 15 π 2 15 B&B No δ A 1 12 δ B 1 12 τ Solution different from those reported in Gabriel et al. (2013)

28 Randomly Generated BC-MLCP Instances 200 variables; 100 compl. variables; matrix A with 90% non-zeros

29 Randomly Generated BC-MLCP Instances 200 variables; 100 compl. variables; matrix A with 90% non-zeros

30 A BC-MLCP Model of a Market-Clearing Problem Example from Gabriel et al. (2012) 6 nodes, 8 producers, 4 demands DC power line model Start-up and shut-down costs modelled using binary variables Our algorithm replaced 99 complementarities (out of 248) by linear equations

31 Summary We proposed a novel solution approach for the BC-MLCP. It starts with an RLT relaxation of the BC-MLCP. It fixes complementarity constraints to linear equations as much as possible. If no complementarity constraints remain, it suffices to solve a single MILP. Otherwise, it solves the MILP reformulation of the reduced (binary) complementarity problem. Neither the complementarity nor the integrality are relaxed. It is successful on instances of LCPs, MLCPs and BC-MLCPs, including models of electric energy markets. Thank you for your attention.

32 Summary We proposed a novel solution approach for the BC-MLCP. It starts with an RLT relaxation of the BC-MLCP. It fixes complementarity constraints to linear equations as much as possible. If no complementarity constraints remain, it suffices to solve a single MILP. Otherwise, it solves the MILP reformulation of the reduced (binary) complementarity problem. Neither the complementarity nor the integrality are relaxed. It is successful on instances of LCPs, MLCPs and BC-MLCPs, including models of electric energy markets. Thank you for your attention.