Nonlinear Programming and Variational Inequality Problems

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   • Michael Patriksson

   Pages 1-38

3. Technical preliminaries

   • Michael Patriksson

   Pages 39-56

4. Instances of the cost approximation algorithm

   • Michael Patriksson

   Pages 57-93

5. Merit functions for variational inequality problems

   • Michael Patriksson

   Pages 95-133

6. Convergence of the CA algorithm for nonlinear programs

   • Michael Patriksson

   Pages 135-168

7. Convergence of the CA algorithm for variational inequality problems

   • Michael Patriksson

   Pages 169-189

8. Finite identification of active constraints and of solutions

   • Michael Patriksson

   Pages 191-209

9. Parallel and sequential decomposition CA algorithms

   • Michael Patriksson

   Pages 211-251

10. A column generation / simplicial decomposition algorithm

    • Michael Patriksson

    Pages 253-276

11. Back Matter

    Pages 277-336

    PDF

Since I started working in the area of nonlinear programming and, later on, variational inequality problems, I have frequently been surprised to find that many algorithms, however scattered in numerous journals, monographs and books, and described rather differently, are closely related to each other. This book is meant to help the reader understand and relate algorithms to each other in some intuitive fashion, and represents, in this respect, a consolidation of the field. The framework of algorithms presented in this book is called Cost Approxi­ mation. (The preface of the Ph.D. thesis [Pat93d] explains the background to the work that lead to the thesis, and ultimately to this book.) It describes, for a given formulation of a variational inequality or nonlinear programming problem, an algorithm by means of approximating mappings and problems, a principle for the update of the iteration points, and a merit function which guides and monitors the convergence of the algorithm. One purpose of this book is to offer this framework as an intuitively appeal­ ing tool for describing an algorithm. One of the advantages of the framework, or any reasonable framework for that matter, is that two algorithms may be easily related and compared through its use. This framework is particular in that it covers a vast number of methods, while still being fairly detailed; the level of abstraction is in fact the same as that of the original problem statement.

• Approximation
• algorithms
• linear optimization
• nonlinear optimization
• optimization
• programming

• Chalmers University of Technology, Gothenburg, Sweden

  Michael Patriksson

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