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55
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 37 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Valid inequalities for mixed integer linear programs
 Mathematical Programming B
, 2006
"... Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mi ..."
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Cited by 31 (0 self)
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Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, liftandproject, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.
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.
LP decoding
 In Proc. 41st Annual Allerton Conference on Communication, Control, and Computing
, 2003
"... Abstract. Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximumlikelihood (ML) decoding. An arbitrary binaryinput memoryless channel is consider ..."
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Cited by 21 (3 self)
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Abstract. Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximumlikelihood (ML) decoding. An arbitrary binaryinput memoryless channel is considered. This treatment of the LP decoding method places our previous work on turbo codes [6] and lowdensity paritycheck (LDPC) codes [8] into a generic framework. We define the notion of a proper relaxation, and show that any LP decoder that uses a proper relaxation exhibits many useful properties. We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels. The fractional distance of an LP decoder is defined, and it is shown that LP decoders correct a number of errors equal to half the fractional distance. We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code. We show that if a relaxation is symmetric, one may assume that the allzeros codeword is transmitted. 1
Towards Sharp Inapproximability For Any 2CSP
, 2008
"... We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good ..."
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Cited by 21 (1 self)
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We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clausebyclause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are “hard” to round in a certain sense, we obtain a Unique Gamesbased inapproximability matching this “hardness ” of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the “hardness ” of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2CSPs. As an application, we show that MAX 2AND is hard to approximate within 0.87435. Thisimproves upon the best previous hardness of αGW + ɛ ≈ 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor αGW.
CSP Gaps and Reductions in the Lasserre Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 104 (2008)
, 2008
"... We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as l ..."
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Cited by 21 (5 self)
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We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as large as 2 even after Ω(n) rounds of the Lasserre hierarchy. We show that for the general MAX kCSP problem over binary domain, the ratio of SDP optimum to the value achieved by the optimal assignment, can be as large as 2 k /2k − ɛ even after Ω(n) rounds of the Lasserre hierarchy. For alphabet size q which is a prime, we give a lower bound of q k /q(q − 1)k − ɛ for Ω(n) rounds. The method of proof also gives optimal integrality gaps for a predicate chosen at random. We also explore how to translate gaps for CSP into integrality gaps for other problems using reductions, and establish SDP gaps for Maximum Independent Set, Approximate Graph Coloring, Chromatic Number and Minimum Vertex Cover. For Independent Set and Chromatic Number, we show integrality gaps of n/2 O( √ log nlog log n) even after 2
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 19 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Subset Algebra Lift Operators for 01 Integer Programming
, 2002
"... We extend the SheraliAdams, LovaszSchrijver, BalasCeriaCornuejols and Lasserre liftandproject methods for 01 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a setcovering problem at least as strong as that given ..."
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Cited by 19 (3 self)
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We extend the SheraliAdams, LovaszSchrijver, BalasCeriaCornuejols and Lasserre liftandproject methods for 01 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a setcovering problem at least as strong as that given by the set of all valid inequalities with small coefficients, and, more generally, all valid inequalities where the righthand side is not very large relative to the positive coefficients in the lefthand side. Applied to generalizations of vertexpacking problems, our methods yield, in polynomial time, relaxations that have unbounded rank using for example the N+ operator.
Conic mixedinteger rounding cuts
 University of CaliforniaBerkeley
, 2006
"... Abstract. A conic integer program is an integer programming problem with conic constraints. Many important problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixedinteger sets defined by secondorder conic constr ..."
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Cited by 17 (5 self)
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Abstract. A conic integer program is an integer programming problem with conic constraints. Many important problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixedinteger sets defined by secondorder conic constraints. We introduce generalpurpose cuts for conic mixedinteger programming based on polyhedral conic substructures of secondorder conic sets. These cuts can be readily incorporated in branchandbound algorithms that solve continuous conic programming or linear programming relaxations of conic integer programs at the nodes of the branchandbound tree. Central to our approach is a reformulation of the secondorder conic constraints with polyhedral secondorder conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to computationally efficient implementation of nonlinear cuts for conic mixedinteger programming. The reformulation also allows the use of polyhedral methods for conic integer programming. Our computational experiments show that conic mixedinteger rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixedinteger programs and, hence, improving their solvability.