Abstract
One of the relevant research topics to which Chris Floudas contributed was quadratically constrained quadratic programming (QCQP). This paper considers one of the simplest hard cases of QCQP, the two trust region subproblem (TTRS). In this case, one needs to minimize a quadratic function constrained by the intersection of two ellipsoids. The Lagrangian dual of the TTRS is a semidefinite program (SDP) and this result has been extensively used to solve the problem efficiently. We focus on numerical aspects of branch-and-bound solvers with three goals in mind. We provide (i) a detailed analysis of the ability of state-of-the-art solvers to complete the global search for a solution, (ii) a quantitative approach for measuring the cluster effect on each solver and (iii) a comparison between the branch-and-bound and the SDP approaches. We perform the numerical experiments on a set of 212 challenging problems provided by Kurt Anstreicher. Our findings indicate that SDP relaxations and branch-and-bound have orthogonal difficulties, thus pointing to a possible benefit of a combined method. The following solvers were selected for the experiments: Antigone 1.1, Baron 16.12.7, Lindo Global 10.0, Couenne 0.5 and SCIP 3.2.
Highlights
Constrained quadratic programming (QCQP) is the task of finding the global minimum of a linear or quadratic function in a domain defined by finitely many linear and quadratic equations or inequalities
Quadratically constrained quadratic programming (QCQP) attracted the attention of the optimization community due to its importance in science and engineering
Couenne and SCIP could not solve any instance with n = 20 within the time limit of 900 s, while Baron and Antigone were competitive in time with the semidefinite program (SDP) approach
Summary
Constrained quadratic programming (QCQP) is the task of finding the global minimum of a linear or quadratic function in a domain defined by finitely many linear and quadratic equations or inequalities. The efforts of Chris Floudas advanced the field of QCQPs (and more general global optimization problems) in different directions. There are algorithms to solve a relaxation of the problem in polynomial time It is a well-known result that the Lagrangian dual of the TTRS is a semidefinite program(SDP). Several authors proposed relaxations to the canonical SDP to find out efficient algorithms for solving the TTRS within a specified tolerance. The first two questions are of interest as a challenge to complete global optimization solvers and to point out possible directions of improvements for them.
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