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© 2005

Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming

  • 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

Book

Part of the International Series of Numerical Mathematics book series (ISNM, volume 152)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Basic Concepts

    1. Front Matter
      Pages 1-1
    2. Ivo Nowak
      Pages 3-7
    3. Ivo Nowak
      Pages 9-19
    4. Ivo Nowak
      Pages 21-31
    5. Ivo Nowak
      Pages 33-53
    6. Ivo Nowak
      Pages 55-71
    7. Ivo Nowak
      Pages 73-81
    8. Ivo Nowak
      Pages 83-97
    9. Ivo Nowak
      Pages 99-111
  3. Algorithms

    1. Front Matter
      Pages 119-119
    2. Ivo Nowak
      Pages 121-128
    3. Ivo Nowak
      Pages 129-142
    4. Ivo Nowak
      Pages 155-179
  4. Back Matter
    Pages 187-213

About this book

Introduction

This book presents a comprehensive description of theory, algorithms and software for solving nonconvex mixed integer nonlinear programs (MINLP). The main focus is on deterministic global optimization methods, which play a very important role in integer linear programming, and are used only recently in MINLP.

The presented material consists of two parts. The first part describes basic optimization tools, such as block-separable reformulations, convex and Lagrangian relaxations, decomposition methods and global optimality criteria. Some of these results are presented here for the first time.

The second part is devoted to algorithms. Starting with a short overview on existing methods, deformation, rounding, partitioning and Lagrangian heuristics, and a branch-cut-and-price algorithm are presented. The algorithms are implemented as part of an object-oriented library, called LaGO. Numerical results on several mixed integer nonlinear programs are reported to show abilities and limits of the proposed solution methods.

The book contains many illustrations and an up-to-date bibliography. Because of the emphasis on practical methods, as well as the introduction into the basic theory, it is accessible to a wide audience and can be used both as a research as well as a graduate text.

Keywords

Branch-and-bound Branch-cut-and-price Convex relaxation Decomposition Heuristics Lagrangian relaxation Nonconvex programming Nonlinear programming Semidefinite relaxation algorithm algorithms linear optimization nonlinear optimization optimization programming

Authors and affiliations

  1. 1.BerlinGermany

Bibliographic information

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)