Semidefinite relaxation and Branch-and-Bound Algorithm for LPECs

Transcription

1 Semidefinite reaxation and Branch-and-Bound Agorithm for LPECs Marcia H. C. Fampa Universidade Federa do Rio de Janeiro Instituto de Matemática e COPPE. Caixa Posta Rio de Janeiro RJ Brasi fampa@cos.ufrj.br Neson Macuan Universidade Federa do Rio de Janeiro COPPE. Caixa Posta Rio de Janeiro RJ Brasi macuan@cos.ufrj.br Wende Aexandre de Meo Universidade Federa do Rio de Janeiro Instituto de Matemática. Caixa Posta Rio de Janeiro RJ Brasi wendeaexandre@gmai.com ABSTRACT In this paper we propose branch-and-bound (B&B) agorithm for the goba resoution of inear programs with inear compementarities constraints (LPECs). The procedure was motivated by the B&B agorithm proposed by Bard and Moore for inear/quadratic bieve programs where compementarities are recursivey enforced. Burer and Vandenbusshe recenty proposed a semidefinite (SDP) reaxation for the nonconvex quadratic programming probem. We extend the Burer-Vandenbusshe approach LPECs and propose the use of SDP reaxations to generate bounds at the nodes of the B&B tree. Computationa resuts compare the quaity of the bounds given by the SDP reaxation with the ones used by Bard and Moore. KEYWORDS. Semidefinite reaxation. LPEC. Bieve program. PM - Programação Matemática 2153

2 1 Introduction Linear programs with compementarity constraints (LPECs) are disjunctive inear optimization probems that contain a set of compementarity conditions. LPECs form a subcass of mathematica programs with equiibrium constraints (MPECs) and incude bieve inear/quadratic programs as a particuar case. The main difficuty on the soution of LPECs is associated to the compementarity constraints which introduce nonconvexities to the probem and are aso responsibe for the ack of reguarity of feasibe points. Since an LPEC is a noninear programming probem it woud be natura to appy NLP agorithms for soving it. However as discussed by Andreani and Martínez in [Andreani (2001)] due to the ack of reguarity these agorithms may have a poor performance on the soution of MPECs or may even be inappicabe. Moreover as shown by Fampa et. a. in [Fampa (2008)] with an appication of bieve programming in energy markets NLP agorithms may converge to a oca optimum of the probem. In this paper we propose a Branch and Bound (B&B) agorithm to sove an LPEC which considers a semidefinite reaxation to the probem in order to compute an upper bound at each node of the enumeration tree. The procedure was motivated by the B&B agorithm proposed by Bard and Moore for the inear/quadratic bieve probem (LQBP) in 1990 in [Bard (1990)] where the authors reformuate the LQBP as an LPEC and use a B&B scheme to recursivey enforce the compementarity constraints. It is we known that one of the key eements in the construction of a B&B agorithm is the method used to obtain a bound for the subprobem at each node of the B&B tree. These bounds have been obtained by the soution of inear and Lagrangean reaxations and more recenty by the soution of semidefinite programming (SDP) reaxations. The deveopment of interior point methods for SDP in the ate 1980s [Nesterov (1994)] made it possibe to efficienty sove these reaxations and motivated the research in this area. The basic idea on the deveopment of SDP reaxations was introduced by Lovasz e Schirijver in [Lovász (1991)] where they present a reformuation for 0-1 inear integer program using SDP. Extensions of the Lovasz-Schirijver approach to combinatoria optimization probems are considered in the surveys pubished by Goemans [Goemans (1997)] and by Hemberg [Hemberg (2002)] and references therein. In [Kojima (2000)] Kojima and Tunçe shows how to extend the approach to any optimization probem that can be expressed by a quadratic objective and quadratic constraints. Burer e Vandenbusshe [Burer (2005)] proposed a B&B agorithm for the nonconvex quadratic programming probem which is based on the the soution SDP reaxations at each node of the enumeration tree. The authors present SDP reaxations of the Karush- Kuhn-Tucker (KKT) conditions of the quadratic program based on the SDP reaxations of the integraity constraints of binary variabes proposed in [Lovász (1991)]. The numerica resuts presented in [Burer (2005)] show the strength of the use of SDP reaxations on the soution of goba optimization probems. We extend the Burer-Vandenbusshe approach to the LPEC and propose the use of SDP reaxations to generate upper bounds for the subprobems considered at the nodes of the enumeration tree of the B&B agorithm presented by Bard and Moore. Computationa resuts compare the quaity of the bounds given by the SDP reaxation with the ones given by the inear reaxation used by Bard and Moore. This inear reaxation is obtained when the compementarity constraints on the reformuation of the LQBP as an LPEC are omitted. 2154

3 2 The LPEC We consider in this paper the foowing LPEC c 1 x + d 1 y A 1 x b 1 0 (a) x 0 (b) d 2T + Q 1T x + Q 2T y B 2T µ + λ = 0 (c) A 2 x + B 2 y b 2 0 (d) (A 2 x + B 2 y b 2 ) µ = 0 (e) y λ = 0 (f) y λ µ 0 (g) where A 1 IR m 1 n 1 A 2 IR m 2 n 1 B 2 IR m 2 n 2 Q 1 IR n 1 n 2 Q 2 S n 2 i.e. Q 2 IR n 2 n 2 and is symmetric negative semidefinite and b 1 b 2 c 1 d 1 and d 2 are vectors of conforma dimension. The constraints (1a) (1b) are caed ordinary and the constraints (1c) (1g) are caed equiibrium constraints. Considering that S(x y) = d 2T + Q 1T x + Q 2T y is the gradient with respect to y of the quadratic function s(x y) = d 2 y + x T Q 1 y yt Q 2 y then the equiibrium constraints can be identified as the KKT conditions of y d 2 y + x T Q 1 y yt Q 2 y A 2 x + B 2 y b 2 0 y 0. If we substitute in (1) the equiibrium constraints (1c) (1g) by (2) we obtain the inear/quadratic bieve program (LQBP) x c 1 x + d 1 y A 1 x b 1 x 0 y d 2 y + x T Q 1 y yt Q 2 y A 2 x + B 2 y b 2 y 0 where (2) is known as the foower probem. Let us consider P := {(x y) 0 : A 1 x b 1 0 A 2 x + B 2 y b 2 0} as the constraint set region corresponding to the bieve program (3) and M(x) := {y : y = argmax{d 2 y +x T Q 1 y yt Q 2 y : A 2 x + B 2 y b 2 0 y 0}} as the reaction set which contains the reactions of the foower probem in (3) for a given x. We make the foowing assumptions: P is nonempty and compact and M(x) is a point-to-point map. These assumptions guarantee the existence of a goba optima soution for probem (1). 3 A Semidefinite Programming Reaxation for the LPEC In this section we extend the SDP reaxation presented by Burer and Vandenbusshe for the nonconvex quadratic probem to the LPEC (1). Let s first consider G xy := {(λ µ) 0 : d 2T + Q 1T x + Q 2T y B 2T µ + λ = 0} C xy := {(λ µ) 0 : (A 2 x + B 2 y b 2 ) µ = 0 y λ = 0}. Then probem (1) may be expressed as (1) (2) (3) 2155

4 Now et Z = 1 x y λ µ 1 x y λ µ T = c 1 x + d 1 y (x y) P (λ µ) G xy C xy. 1 x T y T λ T µ T x xx T xy T xλ T xµ T y yx T yy T yλ T yµ T λ λx T λy T λλ T λµ T µ µx T µy T µλ T µµ T (4). (5) The matrix Z is symmetric and positive semidefinite i.e. Z S+ 1+n where n := n 1 + 2n 2 + m 2. If we mutipy the constraints A 1 x b 1 0 and A 2 x + B 2 y b 2 0 of P and the constraints d 2T + Q 1T x + Q 2T y B 2T µ + λ = 0 of G xy by some ν 0 we obtain the quadratic inequaities A 1 xν b 1 ν 0 and A 2 xν + B 2 yν b 2 ν 0 that are vaid for P and d 2T ν + Q 1T xν + Q 2T yν B 2T µν + λν = 0 that are vaid for G xy. Therefore if we define K := (x 0 x y λ µ) IR 1+n + : A 1 x x 0 b 1 0 A 2 x + B 2 y x 0 b 2 0 x 0 d 2T + Q 1T x + Q 2T y B 2T µ + λ = 0 then the foowing set represents a set of vaid inequaities for (1): M := {Z 0 : Ze i K i = n}. where e i IR 1+n is the vector with a components equa to zero except the i-th component which is equa to one. If we consider Z xµ Z yµ and Z yλ as the submatrices of Z given by xµ T yµ T and yµ T respectivey then the compementarity constraints (A 2 x + B 2 y b 2 ) µ = 0 and y λ = 0 in C xy may be written in terms of the matrix Z as diag(a 2 Z xµ + B 2 Z yµ ) = b 2 µ and diag(z yλ ) = 0 respectivey. Finay considering 0 c 1 d Q := 1 c 1 T d 1 T then the objective function of (1) given by c 1 x + d 1 y may be written as Q Z. Therefore we have the foowing SDP reformuation of (1): Q Z Z = 1 x T y T λ T µ T x xx T xy T xλ T xµ T y yx T yy T yλ T yµ T λ λx T λy T λλ T λµ T µ µx T µy T µλ T µµ T (x y) P (λ µ) G xy diag(a 2 Z xµ + B 2 Z yµ ) = b 2 µ diag(z yλ ) = 0. M (6) 2156

5 If we omit the ast n coumns of equation (5) we arrive at the foowing inear SDP reaxation of (1): Q Z Z M Ze 0 = (1; x; y; λ; µ) (x y) P (λ µ) G xy diag(a 2 Z xµ + B 2 Z yµ ) = b 2 µ diag(z yλ ) = 0. Assumption 1 guarantee that (7) has an optima soution. (7) 4 A Branch-and-Bound Agorithm We propose the use of a B&B agorithm to sove the LPEC (1). The basic idea of the B&B scheme is to recursivey enforce the compementarities through branching. In order to describe the agorithm et s consider ( ) ( ) µ A u := g := 2 x + B 2 y b 2 λ y and T as the set of indices corresponding to the components of u and g i.e. T := {1... m 2 + n 2 }. The compementarities in (1) can then be expressed as u i g i = 0 i T. We associate to a particuar node on the B&B tree two sets T + and T such that T + T T and T + T =. The constraints u j = 0 and g k = 0 are enforced at node for every j T + and k T. If is the root node then T + = T = and if is a eaf node T + T = T. Branching is accompished at node by choosing one compementarity constraint uîgî = 0 to be enforced and creating two chidren and such that T + {î} T T T {î} T + T + T + T The node of the tree is therefore associated to the subprobem obtained when we add to (1) the foowing constraints:. u j = 0 j T + g k = 0 k T. (8) In order to obtain an upper bound to this subprobem we propose the soution of the SDP reaxation derived from (7) where the constraints (8) are added by repacing the sets P G xy and M by P G xy and M defined beow. P := (x y) 0 : A 1 x b 1 0 A 2 x + B 2 y b 2 0 A 2 j x + B2 j y b2 ++ j = 0 j T y k = 0 k T

6 d2t + Q1T x + Q2T y B2T µ + λ = 0 G xy := (λ µ) 0 : µ j = 0 j T + λ k = 0 k T M := {Z 0 : Ze i K i = 1... n} A 1 x x 0 b 1 0 A 2 x + B 2 y x 0 b 2 0 x 0 d 2T + Q 1T x + Q 2T y B 2T µ + λ = 0 K := (x 0 x y λ µ) IR + 1+n : A 2 j x + B2 j y x 0b 2 ++ j = 0 j T µ j = 0 j T + y k = 0 k T + λ k = 0 k T where T ++ T + := {i T + : i m 2 } T + := {i T + : i > m 2 } := {i T : i m 2 } T := {i T : i > m 2 }. The reaxation soved at node is therefore given by 5 Numerica Resuts Q Z Z M Ze 0 = (1; x; y; λ; µ) (x y) P (λ µ) G xy diag(a 2 Z xµ + B 2 Z yµ ) = b 2 µ diag(z yλ ) = 0 (9) In order to test the strength of the SDP reaxation for LPECs we have impemented the semidefinite-based B&B agorithm refereed in the section as SDP agorithm and compared it with the B&B agorithm proposed by Bard and Moore [Bard (1990)] refeered as LP agorithm. The difference between both agorithms is the method used to compute upper bounds to the subprobems considered at the nodes of the enumeration tree. Whie our agorithm appy the SDP reaxation (9) to compute the bound the agorithm presented by Bard and Moore appy the LP reaxation obtained when the compementarity constraints in (1) are omitted and the constraints (8) are added. More specificay the soution of the foowing inear program gives the upper bound at node : c 1 x + d 1 y (x y) P (λ µ) G xy A 2 j x + B2 j y b2 ++ j = 0 j T µ j = 0 j T + y k = 0 k T + λ k = 0 k T (10) 2158

7 We coded both agorithms in MATLAB and used the toobox YALMIP R [Löfberg (2004)] and CSDP [Borchers 99] to sove the LPs and SDPs reaxations. The experiments were run on a desktop computer with 2.8GHz Inte Ceeron and 512 Mb RAM. The test probems are randomy generated and are instances of the inear bieve probem represented by 3 with m 1 Q 1 and Q 2 equa to zero and n 1 + n 2 = 10. The generator of random probems was aso coded in MATLAB and is based on the generator described in [Bard (1990)]. A probems are generated to be bounded which is guaranteed by the incorporation of the constraint A 2 i x + B2 i y b2 i in (3) such that a components of A2 i and Bi 2 are nonnegative and b 2 i is equa to sum of the components. The computationa resuts for the test probems are reported in Figures 1 and 2. In Figure 1 we summarize the resuts obtained by the soution of 900 random probems. A the probems have four constraints on the foower probem (m 2 = 4) and a tota of ten variabes (n 1 + n 2 = 10). The probems were divided into 9 groups each has a different proportion of the number of variabes controed by the eader (n 1 ) and by the foower (n 2 ) of the bieve program. Note that when increasing the number of variabes controed by the foower we increase the difficuty of the probem since the number of compementarities increases which does not happen if n 1 is increased. We give average resuts for the 100 probems with each n 2 that were soved by both the LP and SDP agorithms. The vertica axis refers to the average number of nodes on the B&B tree and the the horizonta axis abes n 2. On these probems the strength of the SDP reaxation reduces the the average number of nodes in the B&B tree by a substantia factor that appears to be growing with n 2 for the LP agorithm and decreasing with n 2 for the SDP agorithm. This resut indicates that the SDP reaxation becomes stronger when the proportion of the number of variabes controed by the foower increases which does not happen with the LP reaxation. In Figure 2 we summarize the resuts obtained by the soution of 1000 random probems. A the probems have five variabes controed by the eader and five controed by the foower (n 1 = n 2 = 5). The probems were divided into 10 groups each has a different number of constraints on the foower probem (m 2 ) which varies from one to ten. We give average resuts for the 100 probems with each m 2 that were soved by both the LP and SDP agorithms. The vertica axis refers to the average gap between the soution of the reation on the root node of the B&B tree and the optima soution of the probem obtained when the B&B agorithms were used. The arger average vaue for the gap when we use the SDP reaxation is 0060 for m 2 = 9 whie the average gaps vary from to when we use the LP reaxation. The resuts again confirm the superiority of the SDP reaxation speciay for probems with smaer m 2. We shoud remark that the SDP reaxation presented in this paper was motivated by a recent work of Burer and Vandenbussche [Burer (2005)] which provides efficient computationa techniques for soving simiar SDP probems. Such SDPs are too arge to be soved with conventiona SDP agorithms as the one used in the current study and therefore we were ony abe to sove sma instances of the probem. Our propose in this paper is to show the strongness of the SDP reaxation for LPECs but to verify its efficiency we woud need to use computationa techniques such as the ones used in [Burer (2005)]. In this paper the authors show that a simiar SDP approach is faster than the LP approach considered to sove arge instances of nonconvex quadratic programs. 2159

8 Figure 1: Average number of nodes on the B&B tree Figure 2: Average gap at the root node of the B&B tree 2160

9 Bibiography Andreani R. and Martínez J. M. (2001) On the soution of mathematica programming probems with equiibrium constraints by means of noninear programming agoritms Mathematica Methods of Operations Research Bard J. F. and Moore J. T. (1990) A branch and bound agorithm for the bieve programming probem SIAM J. Scientific and Statistica Computing 11(2) Borchers B. (1999) CSDP a C ibrary for semidefinite programming Optimization Methods & Software 11(2) Burer S. and Vandenbussche D. Semidefinite-based branch-and-bound for nonconvex quadratic programming University of Iowa ( sburer/papers/017-qpbb.pdf) Fampa M. Barroso L. A. Canda D. and Simonetti L. (2008) Bieve optimization appied to strategic pricing in competitive eectricity markets Computationa Optimization and Appications Goemans M. X. (1997) Semidefinite programming in combinatoria optimization Mathematica Programming Hemberg C. (2002) Semidefinite Programming European Journa of Operationa Research Kojima M. and Tunçe L. (2000) Cones of matrices and sucessive convex reaxations of nonconvex sets SIAM J. on Optimization 10(3) Löfberg J. (2004) YALMIP : A Toobox for Modeing and Optimization in MATLAB Proceedings of the CACSD Conference Taipei Taiwan. Lovász L. and Schirijver A. (1991) Cones of matrices and set-functions and 0-1 optimization SIAM J. on Optimization Nesterov Y. and Nemirovskii A. Interior-point poynomia agorithms in convex programming SIAM Studies in Appied Mathematics Phiadephia Outrata J. Kocvara M. and Zowe J. Nonsmooth Aproach to Optimization Probems with Equiibrium Constraints: Theory Appications and Numerica Resuts Kuwer Academic Pubishers