Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming

by Ivo Nowak

Overview

Nonlinear optimization problems containing both continuous and discrete variables are called mixed integer nonlinear programs (MINLP). Such problems arise in many fields, such as process industry, engineering design, communications, and—nance. There is currently a huge gap between MINLP and mixed integer linear programming(MIP) solvertechnology.With a modernstate-of-the-artMIP solver it is possible to solve models with millions of variables and constraints, whereas the dimension of solvable MINLPs is often limited by a number that is smaller by three or four orders of magnitude. It is theoretically possible to approximate a general MINLP by a MIP with arbitrary precision. However, good MIP approximations are usually much larger than the original problem. Moreover, the approximation of nonlinear functions by piecewise linear functions can be difficult and ti- consuming. In this book relaxation and decomposition methods for solving nonconvex structured MINLPs are proposed. In particular, a generic branch-cut-and-price (BCP) framework for MINLP is presented. BCP is the underlying concept in almost all modern MIP solvers. Providing a powerful decomposition framework for both sequential and parallel solvers, it made the success of the current MIP technology possible. So far generic BCP frameworks have been developed only for MIP, for example,COIN/BCP (IBM, 2003) andABACUS (OREAS GmbH, 1999). In order to generalize MIP-BCP to MINLP-BCP, the following points have to be taken into account: • A given (sparse) MINLP is reformulated as a block-separable program with linear coupling constraints.The block structure makes it possible to generate Lagrangian cuts and to apply Lagrangian heuristics. • In order to facilitate the generation of polyhedral relaxations, nonlinear c- vex relaxations are constructed.• The MINLP separation and pricing subproblems for generating cuts and columns are solved with specialized MINLP solvers.

Product Details

ISBN-13:	9783764372385
Publisher:	Birkh�user Basel
Publication date:	09/27/2005
Series:	International Series of Numerical Mathematics , #152
Edition description:	2005
Pages:	213
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

Basic Concepts.- Problem Formulations.- Convex and Lagrangian Relaxations.- Decomposition Methods.- Semidefinite Relaxations.- Convex Underestimators.- Cuts, Lower Bounds and Box Reduction.- Local and Global Optimality Criteria.- Adaptive Discretization of Infinite Dimensional MINLPs.- Algorithms.- Overview of Global Optimization Methods.- Deformation Heuristics.- Rounding, Partitioning and Lagrangian Heuristics.- Branch-Cut-and-Price Algorithms.- LaGO — An Object-Oriented Library for Solving MINLPs.

Editorial Reviews

From the reviews:

“In his monograph, the author treats mixed integer nonlinear programs (MINLPs), that is nonlinear optimization problems containing both continuous and discrete variables. … This self-contained monograph is rich in content, provides the reader with a wealth of information, and motivates his or her further interest in the subject. The book offers fairly comprehensive description of the MINLP theory and algorithms.” (Jan Chleboun, Applications of Mathematics, Issue 3, 2012)

From the Publisher