Quadratic factorization heuristics for copositive programming

Abstract

Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.

Copositive optimization – Recent developments and applications

“…[63, p.81]. To summarize, under appropriate conditions one has the following rough picture: strict copositivity ⇒ strict local solution ⇒ local solution ⇒ copositivity , (23) while for QPs, relation (23) can be sharpened as follows: strict copositivity ⇔ strict local solution ⇒ local solution ⇔ copositivity . (24) Another consequence of (22) is that a local solution satisfying the strict complementarity condition is necessarily strict, if Q is nonsingular (then all four conditions in (24) are equivalent).…”

“…(24) Another consequence of (22) is that a local solution satisfying the strict complementarity condition is necessarily strict, if Q is nonsingular (then all four conditions in (24) are equivalent). Unfortunately, this kind of argument does not carry over to (23), because from nonsingularity of a copositive matrix Q one cannot infer strict copositivity [14]; this reflects the fact that there are non-strict local solutions which do satisfy strict complementarity.…”

“…This is the adaptive aspect which turns a copositivity check into a fullyfledged procedure to solve copositive optimization problems, and apparently the algorithm in [31] is hard to beat up to now, but see [126,128]. Working on the dual cone C, the recent paper [23] suggests a factorization heuristic for solving copositive optimization problems.…”

Copositive Programming – a Survey

“…Optimizing over C * A recent attempt to solve optimization problems over C * is a feasible descent method by Bomze et al [14], who approximate the steepest descent path from a feasible starting point in C * . They study the problem…”

Non-Linear Distance Transformation Algorithm and its Application in Medical Image Processing in Healthcare

“…The types of non-linear systems considered are restricted while focusing towards the intrested problems. The types of non-linear can be classified into two main types: systems with component wise non-linearlities and systems with a single non-linearity [13].…”