A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

Pooling problem: Alternate formulations and solution methods

Copyright @ 2004 INFORMSThe pooling problem, which is fundamental to the petroleum industry, describes a situation in which products possessing different attribute qualities are mixed in a series of pools in such a way that the attribute qualities of the blended products of the end pools must satisfy given requirements. It is well known that the pooling problem can be modeled through bilinear and nonconvex quadratic programming. In this paper, we investigate how best to apply a new branch-and-cut quadratic programming algorithm to solve the pooling problem. To this effect, we consider two standard models: One is based primarily on flow variables, and the other relies on the proportion. of flows entering pools. A hybrid of these two models is proposed for general pooling problems. Comparison of the computational properties of flow and proportion models is made on several problem instances taken from the literature. Moreover, a simple alternating procedure and a variable neighborhood search heuristic are developed to solve large instances and compared with the well-known method of successive linear programming. Solution of difficult test problems from the literature is substantially accelerated, and larger ones are solved exactly or approximately.This project was funded by Ultramar Canada and Luc Massé. The work of C. Audet was supported by NSERC (Natural Sciences and Engineering Research Council) fellowship PDF-207432-1998 and by CRPC (Center for Research on Parallel Computation). The work of J. Brimberg was supported by NSERC grant #OGP205041. The work of P. Hansen was supported by FCAR(Fonds pour la Formation des Chercheurs et l’Aide à la Recherche) grant #95ER1048, and NSERC grant #GP0105574