Samuel Burer The convex cone of $n \times n$ completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for $n \le 4$ only, every DNN matrix is CPP. … Read more
Given a 0-1 integer programming problem, several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — that generate the convex hull of integer points in at most $n$ steps. In this paper, we introduce a sequential relaxation technique, which is based on $p$-order cone programming ($1 \le p \le \infty$). … Read more
We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first- and second-order necessary optimality conditions. We compare these relaxations with a basic semidefinite relaxation due to Shor, particularly in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger. We … Read more
Let C be the convex hull of points {(1;x)(1,x’)| x \in F\subset R^n}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. If n
We establish that any nonconvex quadratic program having a mix of binary and continuous variables over a bounded feasible set can be represented as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of … Read more
We revisit a regularization technique of Meszaros for handling free variables within interior-point methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites and recent applications, we demonstrate that the modern code SDPT3 modified to incorporate the proposed … Read more
This paper presents a branch-and-bound algorithm for nonconvex quadratic programming, which is based on solving semidefinite relaxations at each node of the enumeration tree. The method is motivated by a recent branch-and-cut approach for the box-constrained case that employs linear relaxations of the first-order KKT conditions. We discuss certain limitations of linear relaxations when handling … Read more
We present a practical approach to Anstreicher and Lee’s masked spectral bound for maximum-entropy sampling, and we describe favorable results that we have obtained with a Branch-&-Bound algorithm based on our approach. By representing masks in factored form, we are able to easily satisfy a semidefiniteness constraint. Moreover, this representation allows us to restrict the … Read more
Given a finite set of outlets with joint normally distributed demands and identical holding and penalty costs, inventory centralization induces a cooperative cost allocation game with nonempty core. It is well known that for this newsvendor inventory setting the expected cost of centralization can be expressed as a constant multiple of the standard deviation of … Read more
We discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs, an exact linesearch procedure, and a dynamic rank reduction scheme. A truncated Newton method is also introduced, and several preconditioning strategies are proposed. Numerical experiments illustrating these enhancements are provided. CitationManuscript, Department of Mangagement Sciences, University of Iowa, … Read more