We present a new heuristic for the global solution of box constrained quadratic problems, based on the classical results which hold for the minimization of quadratic problems with ellipsoidal constraints. The approach...
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We present a new heuristic for the global solution of box constrained quadratic problems, based on the classical results which hold for the minimization of quadratic problems with ellipsoidal constraints. The approach is tested on several problems randomly generated and on graph instances from the DIMACS challenge, medium size instances of the Maximum Clique Problem. The numerical results seem to suggest some effectiveness of the proposed approach.
We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper...
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We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvexquadratic programs.
Let be a quadratically constrained, possibly nonconvex, bounded set, and let denote ellipsoids contained in with non-intersecting interiors. We prove that minimizing an arbitrary quadratic over is no more difficult th...
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Let be a quadratically constrained, possibly nonconvex, bounded set, and let denote ellipsoids contained in with non-intersecting interiors. We prove that minimizing an arbitrary quadratic over is no more difficult than minimizing over in the following sense: if a given semidefinite-programming (SDP) relaxation for is tight, then the addition of l linear constraints derived from yields a tight SDP relaxation for . We also prove that the convex hull of equals the intersection of the convex hull of with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.
In this paper, we propose a branch-and-bound algorithm for solving nonconvex quadratic programming problems with box constraints (BoxQP). Our approach combines existing tools, such as semidefinite programming (SDP) bo...
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In this paper, we propose a branch-and-bound algorithm for solving nonconvex quadratic programming problems with box constraints (BoxQP). Our approach combines existing tools, such as semidefinite programming (SDP) bounds strengthened through valid inequalities, with a new class of optimality-based linear cuts which leads to variable fixing. The most important effect of fixing the value of some variables is the size reduction along the branch-and-bound tree, allowing to compute bounds by solving SDPs of smaller dimension. Extensive computational experiments over large dimensional (up to n=200\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=200$$\end{document}) test instances show that our method is the state-of-the-art solver on large-scale BoxQPs. Furthermore, we test the proposed approach on the class of binary QP problems, where it exhibits competitive performance with state-of-the-art solvers.
The classical trust-region subproblem (TRS) minimizes a nonconvexquadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS,...
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The classical trust-region subproblem (TRS) minimizes a nonconvexquadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone (SOC) constraints. For the case where an additional ellipsoidal constraint is added to TRS, resulting in the "two trust-region subproblem" (TTRS), we provide a new relaxation including SOC constraints that strengthens the usual semidefinite programming (SDP) relaxation.
Semidefinite programming (SDP) problems typically utilize a constraint of the form X >= xxT to obtain a convex relaxation of the condition X = xx(T), where x is an element of R-n. In this paper, we consider a new h...
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Semidefinite programming (SDP) problems typically utilize a constraint of the form X >= xxT to obtain a convex relaxation of the condition X = xx(T), where x is an element of R-n. In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of X - xx(T). This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383-411, 2010, https:// ***/10.1007/s10107-0100371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem.
A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown-by several authors using different techniques-t...
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A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown-by several authors using different techniques-that the convex hull of the intersection of an ellipsoid, , and a split disjunction, with , equals the intersection of with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form and , where is a SOCr cone, is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations and , where is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.
A method is presented for the generation of test problems for global optimization algorithms. Given a bounded polyhedron in R and a vertex, the method constructs nonconvexquadratic functions (concave or indefinite) w...
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A method is presented for the generation of test problems for global optimization algorithms. Given a bounded polyhedron in R and a vertex, the method constructs nonconvexquadratic functions (concave or indefinite) whose global minimum is attained at the selected vertex. The construction requires only the use of linear programming and linear systems of equations.
We introduce a new relaxation framework for nonconvexquadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate feat...
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We introduce a new relaxation framework for nonconvexquadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.
This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with linear inequality constraints. When , or and the linear constraints are parallel, it is known that ...
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This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with linear inequality constraints. When , or and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial time. However, it is also known that, when and at least two of the linear constraints intersect within the ball, i.e., some feasible point of the eTRS satisfies both linear constraints at equality, then the same convex relaxation may admit a gap with eTRS. This paper shows that the convex relaxation has no gap for arbitrary as long as the linear constraints are non-intersecting.
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