Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is … Read more
Global optimization algorithms, e.g., spatial branch-and-bound approaches like those implemented in codes such as BARON and COUENNE, have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer nonlinear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances … Read more
The best known method to find exact or at least epsilon-approximate solutions to polynomial programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the … Read more
Global optimization seeks a minimum or maximum of a multimodal function over a discrete or continuous domain. In this paper, we propose a hybrid heuristic – based on the CGRASP and GENCAN methods – for finding approximate solutions for continuous global optimization problems subject to box constraints. Experimental results illustrate the relative effectiveness of CGRASP-GENCAN … Read more
We consider a constrained optimization problem with mixed integer and real variables. It models optimal placement of communications relay nodes in the presence of obstacles. This problem is widely encountered, for instance, in robotics, where it is required to survey some target located in one point and convey the gathered information back to a base … Read more
The facility layout problem is a global optimization problem that seeks to arrange a given number of rectangular facilities so as to minimize the total cost associated with the (known or projected) interactions between them. This paper is concerned with the single-row facility layout problem (SRFLP), the one-dimensional version of facility layout that is also … Read more
This paper addresses a constrained two-dimensional (2D), non-guillotine restricted, packing problem, where a fixed set of small rectangles has to be packed into a larger stock rectangle so as to maximize the value of the rectangles packed. The algorithm we propose hybridizes a novel placement procedure with a genetic algorithm based on random keys. We … Read more
It has recently been shown (Burer, 2006) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs (CPPs), i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are necessarily NP-hard. A basic tractable relaxation … Read more
Given a subset of all the pair-wise distances between a set of points in a fixed dimension, and possibly the positions of few of the points (called anchors), can we estimate the (relative) positions of all the unknown points (in the given dimension) accurately? This problem is known as the Euclidean Distance Geometry or Graph … Read more
If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via Branch-and-Bound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for automatically finding the formulation group of any given Mixed-Integer Nonlinear Program, and reformulating the problem so … Read more