Branch-and-bound is the workhorse of all state-of-the-art mixed integer linear programming (MILP) solvers. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained by fixing some variable to an integer value v in one node and to v + 1 in the other node. Even though modern MILP solvers are … Read more
A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of n∆ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and ∆ denotes the maximum of the absolute values of the subdeterminants of the … Read more
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
Time transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal con trol problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. … Read more
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. CitationDept. of Business Analytics, … Read more
We consider quadratic optimization in variables (x,y), 0
While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k times k principal submatrices — we call this the sparse SDP relaxation. Surprisingly, … Read more
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such … Read more
We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is … Read more
The problem of finding global minima of nonlinear discrete functions arises in many fields of practical matters. In recent years, methods based on discrete filled functions become popular as ways of solving these sort of problems. However, they rely on the steepest descent method for local searches. Here we present an approach that does not … Read more