## Solving Two-Trust-Region Subproblems using Semidefinite Optimization with Eigenvector Branching

Semidefinite programming (SDP) problems typically utilize the constraint that X-xx’ is PSD to obtain a convex relaxation of the condition X=xx’, where x is an n-vector. In this paper we consider a new hyperplane branching method for SDP based on using an eigenvector of X-xx’. This branching technique is related to previous work of Saxeena, … Read more

## Convex Hull Representations for Bounded Products of Variables

It is well known that the convex hull of {(x,y,xy)}, where (x,y) is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti et al. (2010) and Miller et al. (2011) showed that if there are additional upper and/or lower bounds on the product z=xy, then the convex hull can … Read more

## Testing Copositivity via Mixed-Integer Linear Programming

We describe a simple method to test if a given matrix is copositive by solving a single mixed-integer linear programming (MILP) problem. This methodology requires no special coding to implement and takes advantage of the computational power of modern MILP solvers. Numerical experiments demonstrate that the method is robust and efficient. Citation Dept. of Business … Read more

## Quadratic Optimization with Switching Variables: The Convex Hull for n=2

We consider quadratic optimization in variables (x,y), 0

## Efficient Solution of Maximum-Entropy Sampling Problems

We consider a new approach for the maximum-entropy sampling problem (MESP) that is based on bounds obtained by maximizing a function of the form ldet M(x) over linear constraints, where M(x)is linear in the n-vector x. These bounds can be computed very efficiently and are superior to all previously known bounds for MESP on most … Read more

## Maximum-Entropy Sampling and the Boolean Quadric Polytope

We consider a bound for the maximum-entropy sampling problem (MESP) that is based on solving a max-det problem over a relaxation of the Boolean Quadric Polytope (BQP). This approach to MESP was first suggested by Christoph Helmberg over 15 years ago, but has apparently never been further elaborated or computationally investigated. We find that the … Read more

## Kronecker Product Constraints for Semidefinite Optimization

We consider semidefinite optimization problems that include constraints that G(x) and H(x) are positive semidefinite (PSD), where the components of the symmetric matrices G(x) and H(x) are affine functions of an n-vector x. In such a case we obtain a new constraint that a matrix K(x,X) is PSD, where the components of K(x,X) are affine … Read more

Let $\F$ be a quadratically constrained, possibly nonconvex, bounded set, and let $\E_1, \ldots, \E_l$ denote ellipsoids contained in $\F$ with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $q(\cdot)$ over $\G := \F \setminus \cup_{k=1}^\ell \myint(\E_k)$ is no more difficult than minimizing $q(\cdot)$ over $\F$ in the following sense: if a given semidefinite-programming … Read more