A clique algorithm for standard quadratic programming

Abstract

A standard Quadratic Programming problem (StQP) consists in minimizing a (nonconvex) quadratic form over the standard simplex. For solving a SLQP we present an exact and a heuristic algorithm, that are based on new theoretical results for quadratic and convex optimization problems. With these results a StQP is reduced to a constrained nonlinear minimum weight clique problern in an associated graph. Such a Clique problem, which does not seem to have been Studied before, is then solved with all exact and a heuristic algorithm. Some computational experience shows that Our algorithms are able to solve StQP problems of at least one order of magnitude larger than those reported in the literature. (c) 2007 Elsevier B.V. All rights reserved

The fundamental theorem of linear programming: extensions and applications

“…We first recall the following two results by Scozzari and Tardella [32] that establish the existence of a sequence of nested faces with interior minimizers for this type of problems. THEOREM 2.10 Let f be a strictly quasi-convex differentiable function that attains its minimum on a polytope P at an interior point x à of P. Then there exists a facet F of P such that the minimum of f on F is attained at a point in its relative interior.…”

“…We point out that the maximum weight clique formulation for the standard quadratic optimization problem has been used by Scozzari and Tardella [32] to efficiently solve the latter problem by means of clique algorithms on the associated graph. This approach has led to the solution of problems of up to two orders of magnitude greater than those solved in the literature.…”

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

“…Using the theoretical results in [224,236] and extending [45,46,47], Cesarone et al [44] have shown that the problem (3.1) with cardinality constraints (3.16) has the same optimal solution of problem (3.1) with equality cardinality constraints (3.17) and reduce this kind of programs to Standard Quadratic Programming Problem (see [23,24]), avoiding to explicitly introduce binary variables and considering an exact tailored solving procedure, called Increasing Set Algorithm. The Standard Quadratic Programming Problem is an NP-hard problem when the Hessian matrix of the objective function is indefinite, i.e., if the Hessian matrix of the objective function is neither positive nor negative semidefinite [23].…”

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

“…QCQP: Box QP; Standard QP; Randomly-Generated QCQP GLOBALLib Example 2.1.9 is a toy problem that all solvers address in seconds. To illustrate the advantages of the GloMIQO 2 branch-and-cut framework, we consider: (1) the BoxConstrained Quadratic Programming (BoxQP) problems of Vandenbussche and Nemhauser [113,114]; (2) the Standard Quadratic Programming (StQP) test cases of Scozzari and Tardella [96]; (3) the 135 randomly-generated QCQP of Bao et al [18]. The BoxQP test set was expanded to 90 problems by Burer and Vandenbussche [25]; we used the problems at http: //dollar.biz.uiowa.edu/˜sburer/pmwiki/pmwiki.php.html.…”