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Copositivity detection by differenceofconvex decomposition and ωsubdivision
 MATHEMATICAL PROGRAMMING
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Vacuum Stability Conditions From Copositivity Criteria, Eur.Phys.J. C72
, 2012
"... A scalar potential of the form λabϕ 2 aϕ ..."
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NEW APPROXIMATIONS FOR THE CONE OF COPOSITIVE MATRICES AND ITS DUAL
"... Abstract. We provide convergent hierarchies for the convex cone C of copositive matrices and its dual C ∗ , the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual C ∗ ..."
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Abstract. We provide convergent hierarchies for the convex cone C of copositive matrices and its dual C ∗ , the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual C ∗), thus complementing previous inner (resp. outer) approximations for C (for C ∗). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to Kcopositivity and Kcomplete positivity for a closed convex cone K, is straightforward. hal00545755, version 2 19 Jan 2012 1.
Global quadratic minimization over bivalent constraints: Necessary and sufficient global optimality condition
 J. Optim. Theory Appl
"... In this paper, we establish global optimality conditions for quadratic optimization problems with quadratic equality and bivalent constraints. We first present a necessary and sufficient condition for a global minimizer of quadratic optimization problems with quadratic equality and bivalent constr ..."
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In this paper, we establish global optimality conditions for quadratic optimization problems with quadratic equality and bivalent constraints. We first present a necessary and sufficient condition for a global minimizer of quadratic optimization problems with quadratic equality and bivalent constraints. Then, we examine situations where this optimality condition is equivalent to checking the positive semidefiniteness of a related matrix, and so, can be verified in polynomial time by using elementary eigenvalues decomposition techniques. As a consequence, we also present simple sufficient global optimality conditions, which can be verified by solving a linear matrix inequality problem, extending several known sufficient optimality conditions in the existing literature.
A Continuous Quadratic Programming Formulation of the Vertex Separator Problem
, 2013
"... The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi ..."
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The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi: 10.1007/s1010701306441), the authors announced a new continuous bilinear quadratic programming formulation of the VSP, and they used this quadratic programming problem to illustrate the new optimality conditions. The current paper develops conditions for the equivalence between this continuous quadratic program and the vertex separator problem, and it examines the relationship between the continuous formulation of the VSP and continuous quadratic programming formulations for both the edge separator problem and maximum clique problem.
Copositivity and Constrained Fractional Quadratic Problems
, 2011
"... ... formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branchandbound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and ..."
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... formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branchandbound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed.
Copositivity tests based on the Linear Complementarity Problem
, 2014
"... Copositivity tests are presented based on new necessary and sufficient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke’s method, an enumerative algorithm and a linear mixedinteger programming formulation are proposed to solve the required L ..."
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Copositivity tests are presented based on new necessary and sufficient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke’s method, an enumerative algorithm and a linear mixedinteger programming formulation are proposed to solve the required LCPs. A new necessary condition for (strict) copositivity based on solving a Linear Program (LP) is also discussed, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to maxclique instances with matrices up to dimension 496 × 496. We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere.
Continuous Optimization Continuous quadratic programming formulations of optimization problems on graphs
"... a b s t r a c t Four NPhard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program ..."
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a b s t r a c t Four NPhard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.