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A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems

  • Hanif D. Sherali
  • Warren P. Adams

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 31)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Introduction

    1. Hanif D. Sherali, Warren P. Adams
      Pages 1-20
  3. Discrete Nonconvex Programs

    1. Front Matter
      Pages 21-21
    2. Hanif D. Sherali, Warren P. Adams
      Pages 23-60
    3. Hanif D. Sherali, Warren P. Adams
      Pages 103-129
    4. Hanif D. Sherali, Warren P. Adams
      Pages 131-183
    5. Hanif D. Sherali, Warren P. Adams
      Pages 185-260
  4. Continuous Nonconvex Programs

  5. Special Applications to Discrete and Continuous Nonconvex Programs

    1. Front Matter
      Pages 403-403
    2. Hanif D. Sherali, Warren P. Adams
      Pages 405-439
    3. Hanif D. Sherali, Warren P. Adams
      Pages 441-491
  6. Back Matter
    Pages 493-516

About this book

Introduction

This book deals with the theory and applications of the Reformulation- Linearization/Convexification Technique (RL T) for solving nonconvex optimization problems. A unified treatment of discrete and continuous nonconvex programming problems is presented using this approach. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. For example, the binariness on a 0-1 variable x . can be equivalently J expressed as the polynomial constraint x . (1-x . ) = 0. The motivation for this book is J J the role of tight linear/convex programming representations or relaxations in solving such discrete and continuous nonconvex programming problems. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through automatic reformulation and constraint generation techniques. As mentioned above, the focal point of this book is the development and application of RL T for use as an automatic reformulation procedure, and also, to generate strong valid inequalities. The RLT operates in two phases. In the Reformulation Phase, certain types of additional implied polynomial constraints, that include the aforementioned constraints in the case of binary variables, are appended to the problem. The resulting problem is subsequently linearized, except that certain convex constraints are sometimes retained in XV particular special cases, in the Linearization/Convexijication Phase. This is done via the definition of suitable new variables to replace each distinct variable-product term. The higher dimensional representation yields a linear (or convex) programming relaxation.

Keywords

algorithms optimization optimization algorithm programming scheduling

Authors and affiliations

  • Hanif D. Sherali
    • 1
  • Warren P. Adams
    • 2
  1. 1.Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-4388-3
  • Copyright Information Springer-Verlag US 1999
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-4808-3
  • Online ISBN 978-1-4757-4388-3
  • Series Print ISSN 1571-568X
  • Buy this book on publisher's site