Ebook: Nonlinear Programming and Variational Inequality Problems: A Unified Approach

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.

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The framework of algorithms presented in this book is called Cost Approximation. 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 updating of the iteration points, and a merit function which guides and monitors the convergence of the algorithm. One purpose of the book is to offer this framework as an intuitively appealing tool for describing an algorithm. Another purpose is to provide a convergence analysis of the algorithms in the framework.
Audience: The book will be of interest to all researchers in the field (it includes over 800 references) and can also be used for advanced courses in non-linear optimization with the possibility of being oriented either to algorithm theory or to the numerical aspects of large-scale nonlinear optimization.

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The framework of algorithms presented in this book is called Cost Approximation. 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 updating of the iteration points, and a merit function which guides and monitors the convergence of the algorithm. One purpose of the book is to offer this framework as an intuitively appealing tool for describing an algorithm. Another purpose is to provide a convergence analysis of the algorithms in the framework.
Audience: The book will be of interest to all researchers in the field (it includes over 800 references) and can also be used for advanced courses in non-linear optimization with the possibility of being oriented either to algorithm theory or to the numerical aspects of large-scale nonlinear optimization.
Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-38
Technical preliminaries....Pages 39-56
Instances of the cost approximation algorithm....Pages 57-93
Merit functions for variational inequality problems....Pages 95-133
Convergence of the CA algorithm for nonlinear programs....Pages 135-168
Convergence of the CA algorithm for variational inequality problems....Pages 169-189
Finite identification of active constraints and of solutions....Pages 191-209
Parallel and sequential decomposition CA algorithms....Pages 211-251
A column generation / simplicial decomposition algorithm....Pages 253-276
Back Matter....Pages 277-336

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The framework of algorithms presented in this book is called Cost Approximation. 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 updating of the iteration points, and a merit function which guides and monitors the convergence of the algorithm. One purpose of the book is to offer this framework as an intuitively appealing tool for describing an algorithm. Another purpose is to provide a convergence analysis of the algorithms in the framework.
Audience: The book will be of interest to all researchers in the field (it includes over 800 references) and can also be used for advanced courses in non-linear optimization with the possibility of being oriented either to algorithm theory or to the numerical aspects of large-scale nonlinear optimization.
Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-38
Technical preliminaries....Pages 39-56
Instances of the cost approximation algorithm....Pages 57-93
Merit functions for variational inequality problems....Pages 95-133
Convergence of the CA algorithm for nonlinear programs....Pages 135-168
Convergence of the CA algorithm for variational inequality problems....Pages 169-189
Finite identification of active constraints and of solutions....Pages 191-209
Parallel and sequential decomposition CA algorithms....Pages 211-251
A column generation / simplicial decomposition algorithm....Pages 253-276
Back Matter....Pages 277-336
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