Nondifferentiable and Two-Level Mathematical Programming

The analysis and design of engineering and industrial systems has come to rely heavily on the use of optimization techniques. The theory developed over the last 40 years, coupled with an increasing number of powerful computational procedures, has made it possible to routinely solve problems arising in such diverse fields as aircraft design, material flow, curve fitting, capital expansion, and oil refining just to name a few. Mathematical programming plays a central role in each of these areas and can be considered the primary tool for systems optimization. Limits have been placed on the types of problems that can be solved, though, by the difficulty of handling functions that are not everywhere differentiable. To deal with real applications, it is often necessary to be able to optimize functions that while continuous are not differentiable in the classical sense. As the title of the book indicates, our chief concern is with (i) nondifferentiable mathematical programs, and (ii) two-level optimization problems. In the first half of the book, we study basic theory for general smooth and nonsmooth functions of many variables. After providing some background, we extend traditional (differentiable) nonlinear programming to the nondifferentiable case. The term used for the resultant problem is nondifferentiable mathematical programming. The major focus is on the derivation of optimality conditions for general nondifferentiable nonlinear programs. We introduce the concept of the generalized gradient and derive Kuhn-Tucker-type optimality conditions for the corresponding formulations.

Klappentext

The first objective of this book is to provide an up-to-date exposition of the techniques that are available for optimizing complex systems and to do so in a way that unifies the theory of nondifferentiable and two-level mathematical programming. The second objective is to highlight the most effective algorithms developed for solving a particular instance of the two-level problem known as a static Stackelberg game. In approaching these objectives, close attention is paid to two ideas: (i) the integration of material on differentiable and nondifferentiable mathematical programming, and (ii) the treatment of various two-level mathematical programming problems in a unified manner. This book is intended for the use of researchers, graduate students and practitioners specializing in systems optimization and its applications. In particular, operations researchers, system designers, management scientists, control engineers and mathematicians who work on either applied or theoretical aspects of optimization will find the book useful and beneficial.

Inhalt
1 Introduction.- 2 Mathematical Preliminaries.- 3 Differentiable Nonlinear Programming.- 4 Nondifferentiable Nonlinear Programming.- 5 Linear Programming.- 6 Optimal-Value Functions.- 7 Two-Level Mathematical Programming Problem.- 8 Large-Scale Nonlinear Programming: Decomposition Methods.- 9 Min-Max Problem.- 10 Satisfaction Optimization Problem.- 11 Two-Level Design Problem (Mathematical Programming with Optimal-Value Functions).- 12 General Resource Allocation Problem for Decentralized Systems.- 13 Min-Max Type Multi-Objective Programming Problem.- 14 Best Approximation Problem by Chebyshev Norm.- 15 The Stackelberg Problem: General Case.- 16 The Stackelberg Problem: Linear and Convex Case.- References.