Overview
- Presents the first branch-cut-and-price algorithm for mixed integer nonlinear programming (MINLP)
- Several new MINLP cuts based on semidefinite programming, interval-gradients and Bezier polynomials are proposed
- A description of the MINLP solver LaGO, including numerical results for a wide range of applications, is provided
Part of the book series: International Series of Numerical Mathematics (ISNM, volume 152)
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About this book
Nonlinearoptimizationproblemscontainingbothcontinuousanddiscretevariables are called mixed integer nonlinear programs (MINLP). Such problems arise in many ?elds, 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 itispossibletosolvemodelswithmillionsofvariablesandconstraints,whereasthe dimensionofsolvableMINLPsisoftenlimitedbyanumberthatissmallerbythree 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 di?cult 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.
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Table of contents (14 chapters)
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Front Matter
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Back Matter
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)
Authors and Affiliations
Bibliographic Information
Book Title: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming
Authors: Ivo Nowak
Series Title: International Series of Numerical Mathematics
DOI: https://doi.org/10.1007/3-7643-7374-1
Publisher: Birkhäuser Basel
eBook Packages: Computer Science, Computer Science (R0)
Copyright Information: Birkhäuser Basel 2005
Hardcover ISBN: 978-3-7643-7238-5Published: 15 August 2005
eBook ISBN: 978-3-7643-7374-0Published: 28 March 2006
Series ISSN: 0373-3149
Series E-ISSN: 2296-6072
Edition Number: 1
Number of Pages: XVI, 213
Topics: Math Applications in Computer Science, Applications of Mathematics, Algorithms, Computational Science and Engineering, Programming Techniques