Nondifferentiable and Two-Level Mathematical Programming

1. Front Matter

   Pages i-xii

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2. Introduction

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 1-12

3. Mathematical Preliminaries

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 13-58

4. Differentiable Nonlinear Programming

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 59-112

5. Nondifferentiable Nonlinear Programming

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 113-127

6. Linear Programming

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 128-187

7. Optimal-Value Functions

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 188-228

8. Two-Level Mathematical Programming Problem

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 229-258

9. Large-Scale Nonlinear Programming: Decomposition Methods

   • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

   Pages 259-270

10. Min-Max Problem

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 271-279

11. Satisfaction Optimization Problem

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 280-291

12. Two-Level Design Problem (Mathematical Programming with Optimal-Value Functions)

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 292-311

13. General Resource Allocation Problem for Decentralized Systems

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 312-317

14. Min-Max Type Multi-Objective Programming Problem

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 318-333

15. Best Approximation Problem by the Chebyshev Norm

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 334-346

16. The Stackelberg Problem: General Case

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 347-390

17. The Stackelberg Problem: Linear and Convex Case

    • Kiyotaka Shimizu, Yo Ishizuka, Jonathan F. Bard

    Pages 391-449

18. Back Matter

    Pages 450-470

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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.

• algorithm
• algorithms
• complex system
• complex systems
• linear optimization
• mathematical programming
• nonlinear optimization
• operations research
• optimization
• programming
• system

• Keio University, Yokohama, Japan

  Kiyotaka Shimizu

• Sophia University, Tokyo, Japan

  Yo Ishizuka

• The University of Texas, Austin, USA

  Jonathan F. Bard

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