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18
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
"... ..."
Cones Of Matrices And Successive Convex Relaxations Of Nonconvex Sets
, 2000
"... . Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each ..."
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Cited by 49 (20 self)
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. Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding LovaszSchrijver liftandproject procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...
On the MatrixCut Rank of Polyhedra
 Mathematics of Operations Research
, 2001
"... Lov'asz and Schrijver (1991) described a semidefinite operator for generating strong valid inequalities for the 01 vectors in a prescribed polyhedron. Among their results, they showed that n iterations of the operator are sufficient to generate the convex hull of 01 vectors contained in a poly ..."
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Cited by 31 (0 self)
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Lov'asz and Schrijver (1991) described a semidefinite operator for generating strong valid inequalities for the 01 vectors in a prescribed polyhedron. Among their results, they showed that n iterations of the operator are sufficient to generate the convex hull of 01 vectors contained in a polyhedron in nspace. We give a simple example, having Chv'atal rank 1, that meets this worst case bound of n. We describe another example requiring n iterations even when combining the semidefinite and GomoryChv'atal operators. This second example is used to show that the standard linear programming relaxation of a kcity traveling salesman problem requires at least bk=8c iterations of the combined operator; this bound is best possible, up to a constant factor, as k + 1 iterations suffice. Key words. Semidefinite programming, integer hull, rank of polytopes, cutting planes, projection operators. Many structures in combinatorial optimization can be modeled as a set of 01 vectors in...
The probable value of the LovaszSchrijver relaxations for maximum independent set
 SIAM Journal on Computing
, 2003
"... independent set ..."
Complexity of SemiAlgebraic Proofs
, 2001
"... It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver calculi ..."
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Cited by 25 (2 self)
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It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS^d of LS that operate with polynomial inequalities of degree at most d. It turns out
Solving liftandproject relaxations of binary integer programs
 SIAM Journal on Optimization
"... Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constrain ..."
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Cited by 22 (2 self)
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Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient implementation. Computational results illustrate that our algorithm produces tight bounds more quickly than stateoftheart linear and semidefinite solvers.
SheraliAdams relaxations of the matching polytope
 In STOC’2009
, 2009
"... We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rou ..."
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Cited by 12 (0 self)
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We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K2d+1. We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a “phase transition, ” dropping from close to its maximum value 1 + 1 2d to close to 1 around the threshold k = 2d − √ d. We also show that the rank of the matching polytope (i.e., the number of SheraliAdams rounds until the integer polytope is reached) is exactly 2d − 1.
Semidefinite Relaxations for MaxCut
 The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM
, 2001
"... We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the maxcut problem. This re ..."
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Cited by 9 (1 self)
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We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the maxcut problem. This relaxation Q t (G) can be defined as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of [1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I ; J)th entry depends only on the symmetric difference of the sets I and J . The set F 0 (n) is the basic semidefinite relaxation of maxcut consisting of the semidefinite matrices of order n with an all ones diagonal, while Fn\Gamma2 (n) is the 2 n\Gamma1 dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n \Gamma 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively. 1