Nondifferentiable and Two-Level Mathematical Programming

✍ Scribed by Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard (auth.)

• Publisher: Springer US
• Year: 1997
• Tongue: English
• Leaves: 481
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xii
Introduction....Pages 1-12
Mathematical Preliminaries....Pages 13-58
Differentiable Nonlinear Programming....Pages 59-112
Nondifferentiable Nonlinear Programming....Pages 113-127
Linear Programming....Pages 128-187
Optimal-Value Functions....Pages 188-228
Two-Level Mathematical Programming Problem....Pages 229-258
Large-Scale Nonlinear Programming: Decomposition Methods....Pages 259-270
Min-Max Problem....Pages 271-279
Satisfaction Optimization Problem....Pages 280-291
Two-Level Design Problem (Mathematical Programming with Optimal-Value Functions)....Pages 292-311
General Resource Allocation Problem for Decentralized Systems....Pages 312-317
Min-Max Type Multi-Objective Programming Problem....Pages 318-333
Best Approximation Problem by the Chebyshev Norm....Pages 334-346
The Stackelberg Problem: General Case....Pages 347-390
The Stackelberg Problem: Linear and Convex Case....Pages 391-449
Back Matter....Pages 450-470

✦ Subjects

Operation Research/Decision Theory; Systems Theory, Control; Mathematical Modeling and Industrial Mathematics