Samuel Burer 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. Citation Manuscript, Department of Mangagement Sciences, University of … Read more
We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lov\’asz and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss … Read more
The low-rank semidefinite programming problem (LRSDP_r) is a restriction of the semidefinite programming problem (SDP) in which a bound r is imposed on the rank of X, and it is well known that LRSDP_r is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDP_r and … Read more