L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... b T y and H(y) P m k=1 y k A k Gamma C. Example 2. SDP Relaxations of Binary Combinatorial Optimization Problems In SDP relaxations of binary combinatorial optimization problems, the binary constraints x 2 i = 1; i = 1; 2; n; are relaxed into X ii = 1; i = 1; 2; n; see [19,13,5,16] for the evolution of such relaxations) resulting in primal linear SDP problems in the form of (4) with I = D and B I = I, the identity matrix. The dual form of these SDP relaxations are special instances of (5) In particular, when m = 0, we obtain the MAXCUT SDP relaxation that forms the ....

L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....variable is replaced by a matrix valued continuous variable, resulting in a convex optimization problem called a semidefinite program (SDP) that can be solved to a prescribed accuracy in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [10, 23, 24, 26, 27]. Based on solving the SDP relaxation, Goemans and Williamson [18] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time ....

L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....Houston, Texas 77005, USA. This author was supported in part by DOE Grant DE FG03 97ER25331, DOE LANL Contract 03891 99 23 and NSF Grant DMS 9973339. Email: zhang caam.rice.edu) 1 in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [9, 17, 18, 20, 21]. Based on the SDP relaxation, Goemans and Williamson [14] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time approximation ....

L. Lovasz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166-190, 1991. 15

....continuous variable, resulting in a convex optimization problem called a SDP problem. Since a SDP problem is solvable in polynomial time, one can obtain a bound to the original problem in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [9, 18, 19, 21, 22]. Based on the SDP relaxation, Goemans and Williamson [14] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time approximation ....

L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....integral convex hull by applying a sequence of linear cuts on the fractional convex hull. Chv atal [45] shows that the Gomory cut procedure always terminates in a finite number of steps and obtains the integral convex hull. The number of steps, however, could be very large. Lova sz and Schrijver [108] show that nonlinear cuts do perform better in this aspect. They introduce a nonlinear cut, whose nonlinearity comes from semidefinite constraints, by lifting the fractional convex hull to a higher dimensional space, applying some linear cuts in the higher dimensional space, and then projecting ....

....applied to reduce the time or space complexity of the only known polynomialtime algorithm for solving MAXCLIQUE and MAX STABLE SET on a perfect graph if its chromatic number is a constant. 8.2 Future Work We propose the following future work. ffl There are plenty of other semidefinite programs [106, 108, 63, 55, 51, 52, 150, 113, 32]. It would be nice if our algorithms could be generalized to work on some of them. ffl Our compact algorithms are still far from practical due to their time complexity. It would be nice if one can come up with compact algorithms with better running times. We propose to perform the conjugate ....

L. Lov'asz and A. Schrijver. Cones of Matrices and Setfunctions, and 0-1 Optimization. SIAM Journal on Optimization, 1(2):166--190, 1991.

....have been used by Goldberg et al. [GPST91] to derive sublinear time parallel algorithms for the bounded weight assignment problem. We show that maximum stable sets for perfect graphs can be computed in randomized sublinear parallel time. Furthermore, based on the work of Lov asz and Schrijver [LS91], we argue that in a branch and bound scheme for 2 0 1 programs interior point SDP algorithms may efficiently yield much sharper bounds than possible from linear programming relaxations of such problems. In section 2 we review the so called cone duality theory as specialized to semidefinite ....

....little work has been done in generating nonlinear but convex cuts in the feasible region of the LP relaxation. Generally such cuts may produce far better approximations than planar cuts. An ingenious approach for creating a class of nonlinear cuts has been proposed by Lov asz and Schrijver in [LS91]. The idea is to lift the space from vectors in n to n Theta n symmetric matrices 5 . It is convenient to homogenize integer program by introducing a new variable x 0 as a multiple of b and then imposing the constraint x 0 = 1. After this transformation the homogenized integer programming ....

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L. Lov'asz and A. Schrijver. Cones of Matrices and Setfunctions, and 0-1 Optimization. SIAM J. Optimization, 1(2), 1991.

....problems defined over a cone of positive semi definite matrices. These are models that generalize linear programs and are specializations of convex programming models. There are theoretical and practical algorithms for solving semi definite programs in polynomial time [3] Lov asz and Schrijver [91] suggest a general relaxation strategy for 0 Gamma 1 integer programming problems that obtains semi definite relaxations. The first step is to consider a homogenized version of a 0 Gamma 1 integer program (solvability version) F I = fx 2 n 1 : Ax 0; x 0 = 1; x i 2 f0; 1g for i = 1; 2; ....

....program is given by: fx 2 n 1 : Ax 0; x 0 = 1; 0 x i x 0 for i = 1; 2; Delta Delta Delta ; ng Next, we define two polyhedral cones. K = fx 2 n 1 : Ax 0; 0 x i x 0 for i = 0; 1; Delta Delta Delta ; ng K I = Cone generated by 0 Gamma 1 vectors in P I Lov asz and Schrijver [91] show how we might construct a family of convex cones fCg such that K I C K for each C. i) Partition the cone constraints of K into T 1 = fA 1 x 0g and T 2 = fA 2 x 0g, with the constraints f0 x i x 0 for i = 0; 1; Delta Delta Delta ; ng repeated in both (the overlap can be larger) ....

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L.Lov'asz and A.Schrijver, Cones of matrices and set functions, SIAM Journal on Optimization 1, (1991), pp. 166-190.

....and Schrijver [22,24] to develop the only known (and non combinatorial) polynomial time algorithm to solve the maximum stable set problem for perfect graphs. More recently, the development of efficient interior point algorithms for semidefinite programming, the results of Lov asz and Schrijver [44,45] on stronger formulations using semidefinite programming, improved approximation algorithms for the maximum cut and related problems, and striking hardness of approximation results have spawned much focus on the power (and limitation) of semidefinite programming for combinatorial optimization ....

....the number of constraints is thus only l. Furthermore, the resulting LP after adding the inequalities x j 0 for j = 2 M takes precisely the same form as Delsarte s LP (7) once generalized to association schemes) See [15,59,49] for details. 4 Deriving Valid Inequalities Lov asz and Schrijver [44,45] have proposed a technique for automatically generating stronger and stronger formulations for integer programs. Because of space limitation, we can only briefly describe their approach. We also refer the reader to Sherali and Adams [61] Balas et al. 7] Lov asz [43] for additional results and ....

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L. Lov'asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....y k A k Gamma C. Example 2. 2 SDP Relaxations of Binary Combinatorial Optimization Problems 4 In SDP relaxations of binary combinatorial optimization problems, the binary constraints x i = Sigma1; i = 1; 2; n; are relaxed into X ii = 1; i = 1; 2; n; see Lov asz and Schrijver [10]) resulting in primal linear SDPs in the form of (4) with I = D and B I = I , the identity matrix. The dual form of these SDP relaxations are special instances of (5) In particular, when m = 0, we obtain the MAXCUT SDP relaxation introduced by Goemans and Williamson in [4] Another example ....

L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....problems, the MAXCUT problem can be formulated as a quadratic programming (QP) problem in binary (or Sigma1) variables. The idea that these problems can be naturally relaxed to SDP problems was first observed in Lov asz [16] and Shor [22] and has been used by several authors (e.g. see [1, 6, 14, 17, 18, 20, 23]) Goemans and Williamson [9] developed a randomized algorithm for the MAXCUT problem, based on solving its SDP relaxation, which provides an approximate solution guaranteed to be within a factor of 0:87856 of its optimal value. In practice, their algorithm has been observed to find solutions ....

L. Lov'asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....P m k=1 z k A k C. Example 2. 2 SDP Relaxations of Binary Combinatorial Optimization Problems In SDP relaxations of binary combinatorial optimization problems, the binary constraints x i = Sigma1; i = 1; 2; n; are relaxed into X ii = 1; i = 1; 2; n; see Lov asz and Schrijver [10]) resulting in primal linear SDPs in the form of (4) with I = D and B I = I , the identity matrix. The dual form of these SDP relaxations are special instances of (5) In particular, when m = 0, we obtain the MAXCUT SDP relaxation introduced by Goemans and Williamson in [4] Another example ....

L. Lov'asz and A. Schrijver. Cones of matrices and set functions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1991.

....as an important technique. Lov asz [15] showed semidefinite programming could be used to compute the Shannon capacity of a graph, often referred to as the theta function; this is a number that lies between the size of the maximum clique and the minimumnumber of colors. Lov asz and Schrijver [16] described a way to use semidefinite programming to estimate the value of integer programs. In an important recent breakthrough, Goemans and Williamson [4] discovered an approximation algorithm for graph MAX CUT whose accuracy is significantly better than that of the previously known algorithms. ....

L. Lov'asz and A. Schrijver. Cones of Matrices and Setfunctions, and 0-1 Optimization. SIAM Journal on Optimization, 1(2):166--190, 1991.

....that the primal is feasible and the dual has an interior so that there is no duality gap between the primal and dual. Denote by X(Q) and ( y 0 (Q) y(Q) an optimal solution pair for the primal (1) and dual (2) The positive semi definite relaxation was first proposed by Lov asz and Shrijver [9], also see recent papers by Alizadeh [1] Anstreicher and Wolkowicz[2] Fujie and Kojima [6] and Polijak, Rendl and Wolkowicz [16] This relaxation problem can be solved in polynomial time, e.g. see Nesterov and Nemirovski [14] We have the following relations between (QP) and (SDP) Proposition ....

L. Lov'asz and A. Shrijver, Cones of matrices and setfunctions, and 0 \Gamma 1 optimization, SIAM Journal on Optimization 1 (1990) 166-190.

....over S n , the cone of positive semidefinite symmetric matrices, and its intersection with a subspace W of E = Sn , the Hilbert space of n Theta n real symmetric matrices. Semidefinite programming finds applications in combinatorial optimization, as well as eigenvalue problems (see e.g. [26], 33] 1] 40] and is a special case of the general class of self concordant programming, defined and studied in [31] It is natural to ask if the scaling dualities proved for K = n extend to the HP formulation of semidefinite programming, or more generally to the HP formulation of ....

L. Lov'asz and A. Schrijver, Cones of matrices and set-functions, and 0-1 optimization, SIAM J. of Optim., 1 (1991), pp. 166-190.

.... solvability of semidefinite programs, this leads to the only known polynomial time algorithm for finding the largest stable set in a perfect graph (Grotschel et al. 25] More recently, there has been increased interest in semidefinite programming from a combinatorial point of view [46, 47, 1, 58, 17, 45]. This started with the work of Lov asz and Schrijver [46, 47] who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programming. Their papers demonstrated the wide applicability and the power of semidefinite programming for ....

.... polynomial time algorithm for finding the largest stable set in a perfect graph (Grotschel et al. 25] More recently, there has been increased interest in semidefinite programming from a combinatorial point of view [46, 47, 1, 58, 17, 45] This started with the work of Lov asz and Schrijver [46, 47], who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programming. Their papers demonstrated the wide applicability and the power of semidefinite programming for combinatorial optimization problems. Our use of semidefinite ....

[Article contains additional citation context not shown here]

L. Lov'asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1990.

....methods have been used by Goldberg et al. [26] to derive sublinear time parallel algorithms for the bounded weight assignment problem. We show that maximum stable sets for perfect graphs can be computed in randomized sublinear parallel time. Furthermore, based on the work of Lov asz and Schrijver [42], we argue that in a branch and bound scheme for 0 1 programs interior point SDP algorithms may efficiently yield much sharper bounds than possible from 4 F. Alizadeh linear programming relaxations of such problems. In section 2 we review the so called cone duality theory as specialized to ....

....formulation for various eigenvalue optimization problems. We also state complementary slackness results for these problems. Finally, in section 5 we study some applications of SDP interior point methods to various combinatorial optimization problems. These include 0 1 integer programs of [42], maximum clique and maximum stable set problems in graphs, and various partitioning and cut problems in graphs. Notation and terminology. Unless otherwise stated the following convention and terminology is used throughout this article. The semidefinite programming problem (SDP) refers to any ....

[Article contains additional citation context not shown here]

L. Lov' asz and A. Schrijver, Cones of Matrices and Setfunctions, and 0-1 Optimization, SIAM J. Optim., 1 (1991), pp. 166--190.

....that the primal is feasible and the dual has an interior so that there is no duality gap between the primal and dual. Denote by X(Q) and ( y(Q) z(Q) an optimal solution pair for the primal (1) and dual (2) The positive semi definite relaxation was first proposed by Lov asz and Shrijver [8], also see recent papers by Alizadeh [1] Fujie and Kojima [5] and Polijak, Rendl and Wolkowicz [12] This relaxation problem can be solved in polynomial time, e.g. see Nesterov and Nemirovskii [10] We have the following relations between (QP) and (SDP) Proposition 1 Let q = q(Q) q = ....

L. Lov'asz and A. Shrijver, Cones of matrices and setfunctions, and 0 \Gamma 1 optimization, SIAM Journal on Optimization 1 (1990) 166-190.

....it can be applied to any 0 1 optimization problem. Interesting applications of a related method to the max cut problem were given by Delorme and Poljak (1990) see also Mohar and Poljak (1990) These methods can be extended from quadratic to higher order inequalities. For these extensions, see Lov asz and Schrijver (1990) and Sherali and Adams (1990) ....

L. Lov'asz and A. Schrijver (1990): Cones of matrices and setfunctions, and 0-1 optimization, SIAM J. Optim. 1, 166--190.

.... or equivalently, U x x T 1 # 0: The relaxation thus obtained is a semidefinite program, and is called the Image Convexification Relaxation (ICR) as one can show that ( Ram93] RG94b] Conv(f( n ) fF (U; x)jU Gamma xx T 0g: The ICR is closely related to the N operator defined in [LS91]. In particular, the semidefinite relaxation of the stable set problem as considered by (see [GLS88] and [LS91] as well as that of the Max Cut problem as in [GW94] are precisely the ICRs of the corresponding MQPs: maxfe T xjx i x j = 0 8 (i; j) 2 E; x 2 i = x i 8 ig in the case of the former, ....

.... the Image Convexification Relaxation (ICR) as one can show that ( Ram93] RG94b] Conv(f( n ) fF (U; x)jU Gamma xx T 0g: The ICR is closely related to the N operator defined in [LS91] In particular, the semidefinite relaxation of the stable set problem as considered by (see [GLS88] and [LS91]) as well as that of the Max Cut problem as in [GW94] are precisely the ICRs of the corresponding MQPs: maxfe T xjx i x j = 0 8 (i; j) 2 E; x 2 i = x i 8 ig in the case of the former, and maxfx T Qxjx 2 i = 1 8 ig for the latter. It is NP Hard to check whether the ICR of a given MQP is ....

L. Lov'asz and A. Schrijver, Cones of Matrices and Setfunctions, and 0 \Gamma 1 Optimization, SIAM J. Optimization, 1(1991).

....In this paper, we describe a few applications of semidefinite programming in combinatorial optimization. Because of space limitations, we restrict our attention to the Lov asz theta function, the maximum cut problem [8] and the automatic generation of valid inequalities a la Lov asz Schrijver [17, 18]. This survey is much inspired by another (longer) survey written by the author [7] However, new results on the power and limitations of the Lov asz Schrijver procedure are presented as well as a study of the maximum cut relaxation for graphs arising from association schemes. 1 Supported in ....

.... P i z i C i L(G) As an illustration, the triangle inequalities can be aggregated in order to be of the required form, and thus the semidefinite program with triangle inequalities can be solved as a linear program for association schemes. 4 Deriving Valid Inequalities Lov asz and Schrijver [17, 18] have proposed a technique for automatically generating stronger and stronger formulations for integer programs. We briefly describe their approach here and discuss its power and its limitations. Let P = fx 2 R n : Ax b; 0 x 1g, and let P 0 = conv(P f0; 1g n ) denote the convex hull of 0 ....

[Article contains additional citation context not shown here]

L. Lov'asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization. SIAM J. Opt., 1:166--190, 1991.

.... with the polynomial time solvability of semidefinite programs, this leads to the only known polynomial time algorithm for finding the largest stable set (or the largest clique) in a perfect graph (Grotschel et al. 17] More recently, there has been increased interest in semidefinite programming [33, 34, 1, 42, 45, 12, 32]. This started with the work of Lov asz and Schrijver [33, 34] who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programming. Their paper demonstrated the wide applicability and the power of semidefinite programming for ....

.... only known polynomial time algorithm for finding the largest stable set (or the largest clique) in a perfect graph (Grotschel et al. 17] More recently, there has been increased interest in semidefinite programming [33, 34, 1, 42, 45, 12, 32] This started with the work of Lov asz and Schrijver [33, 34], who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programming. Their paper demonstrated the wide applicability and the power of semidefinite programming for combinatorial optimization problems. A consequence of our ....

[Article contains additional citation context not shown here]

L. Lov'asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization. SIAM Journal on Optimization, 1:166--190, 1990.

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