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86
Maximum stable set formulations and heuristics based on continuous optimization
 MATH. PROGRAM., SER. A 94: 137–166 (2002)
, 2002
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Generalized Chebyshev bounds via semidefinite programming
, 2007
"... A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev’s inequality for scalar random variables. Two semidefinite programming formul ..."
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Cited by 20 (1 self)
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A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev’s inequality for scalar random variables. Two semidefinite programming formulations are presented, with a constructive proof based on convex optimization duality and elementary linear algebra.
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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Cited by 18 (1 self)
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This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
A Unified Framework for Obtaining Improved Approximation Algorithms for Maximum Graph Bisection Problems
, 2002
"... We obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX n/ BISECTION  partition the vertices of the graph into two sets of equal size such that the total weight of edges connecting ..."
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Cited by 16 (0 self)
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We obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX n/ BISECTION  partition the vertices of the graph into two sets of equal size such that the total weight of edges connecting vertices from different sides is maximized; MAX n/2VERTEXCOVER  find a set containing half of the vertices such that the total weight of edges touching this set is maximized; MAX n/2DENSESUBGRAPH  find a set containing half of the vertices such that the total weight of edges connecting two vertices from this set is maximized; and MAX n/2UnCUT  partition the vertices into two sets of equal size such that the total weight of edges that do not cross the cut is maximized. We also consider the directed versions of these problems, such as MAX n/2DIRECTEDBISECTION and MAX n/2DIRECTEDUnCUT. These results can be used to obtain improved approximation algorithms for the unbalanced versions of the partition problems mentioned above, where we want to partition the graph into two sets of size k and n  k, where k is not necessarily n/2 . Our results improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others. All these results may be viewed as extensions of the MAX CUT algorithm of Goemans and Williamson, and the MAX 2SAT and MAX DICUT algorithms of Feige and Goemans.
Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 13 (2 self)
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The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.
An Improved Semidefinite Programming Relaxation for the Satisfiability Problem
, 2002
"... The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there ..."
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Cited by 13 (3 self)
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NPcomplete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems.
Global minimization of rational functions and the nearest GCDs
 J. of Global Optimization
"... This paper discusses the global minimization of rational functions with or without constraints. The sum of squares (SOS) relaxations are proposed to find the global minimum and minimizers. Some special features of the SOS relaxations are studied. As an application, we show how to find the nearest co ..."
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Cited by 12 (0 self)
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This paper discusses the global minimization of rational functions with or without constraints. The sum of squares (SOS) relaxations are proposed to find the global minimum and minimizers. Some special features of the SOS relaxations are studied. As an application, we show how to find the nearest common divisors of polynomials via global minimization of rational functions.
Solving Problems with Semidefinite and Related Constraints Using InteriorPoint Methods for Nonlinear Programming
, 2001
"... In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMAC ..."
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Cited by 12 (1 self)
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In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided.