This book comprises various optimality criteria, duality and mixed duality in a variety of mathematical programming, that includes nondifferentiable nonlinear programming problems, nondifferentiable nonlinear fractional programming problems, nondifferentiable minimax fractional programming problems etc.
Mathematical Programming is concerned with the determination of a minimum or maximum of a function of several variables, which are required to satisfy a number of constraints. Such solutions are sought are sought in diverse fields, including Engineering, Operations Research, Management Science and Economics. Often these situations are mathematical representations of certain real world problems, and hence are turned as mathematical programming problems.
Optimality criteria and duality have played an important role in the development of mathematical programming. Optimality conditions were first investigated by Fritz John and later on, independently by Karush and Kuhn – Tucker. The inception of duality theory in linear programming may be traced to the classical minmax theorem of Von Neumann, which was subsequently formulated in a precise form by Gale, Kuhn and Tucker. Since then optimality criteria and duality have remained as one of the most widely investigated area in mathematical programming. Karush-Kuhn-Tucker conditions not only laid down the foundations for many computational techniques in mathematical programming, but also are a great deal responsible for the development of the duality theory.
An extensive use of duality in mathematical programming has been made for many theoretical and computational developments in mathematical programming itself, economics, control theory, business problems and many other diverse fields. It is well known that duality principle connects two programs, one of which, called the Primal problem, is a constrained maximization (or minimization) problem, and the other one called the Dual, is a constrained minimization (or maximization) problem, in such a way that the existence of an optimal solution to one of them guarantees an optimal solution to the other and optimal values of the two problems are equal. A pair of dual problems is called symmetric if the dual of the dual is the primal itself.
Inhaltsverzeichnis (Table of Contents)
- Preface
- 1. Introduction
- 1.1 General Introduction
- 1.2 Preliminaries
- 1.2.1 Notations
- 1.2.2 Definitions
- 1.3 Review of related work
- 1.3.1 Duality in differentiable mathematical programming
- 1.3.2 Duality in nondifferentiable mathematical programming
- 1.3.3 Symmetric Duality in differentiable mathematical programming
- 2. Variational Problems involving higher order derivatives
- 2.1 Introductory remarks
- 2.2 Invexity
- 2.3 Variational problems and optimality conditions
- 2.4 Wolfe type duality
- 2.5 Mond-Wier Type duality
- 2.6 Special cases
- 2.7 Related problems
- 3. Continuous programming containing support function
- 3.1 Introductory remarks
- 3.2 Preliminaries
- 3.3 Continuous programming problems and optimality
- 3.4 Duality
- 3.5 Converse duality
- 3.6 Mixed duality
- 3.7 Special cases
- 3.8 Related nonlinear programming problems
- 4. Continuous time fractional minmax programming
- 4.1 Introduction
- 4.2 Primal problem and pre-requisites
- 4.3 Optimality conditions
- 4.4 Duality
- 5. Nondifferentiable fractional minmax programming
- 5.1 Introduction
- 5.2 Preliminaries
- 5.3 Necessary optimality conditions
- 5.4 Sufficient optimality conditions
- 5.5 Mond-Weir type duality
- 5.6 Schaible type duality
- 6. Mixed Type symmetric and self duality for variational problems
- 6.1 Introduction
- 6.2 Notations and Preliminaries
- 6.3 Statement of the problem
- 6.4 Mixed type symmetric duality
- 6.5 Self duality
- 6.6 Natural boundry values
- 6.7 Mathematical Programming
- 6.8 Remarks
- 7. Symmetric duality for nondifferentialbe maxmin variational problems
- 7.1 Introduction
- 7.2 Preliminaries
- 7.3 Wolfe type symmetric and self duality
- 7.4 Mond-Weir type symmetric and self duality
- 7.5 Related problems
- 8. Sufficient Fritz John optimality criteria and duality for continuous programming problems
- 8.1 Introduction
- 8.2 Preliminaries
- 8.3 Fritz John type dulaity
- 8.4 Related problems
- 9. Fritz John second order duality for programming problems
- 9.1 Introduction remarks and Preliminaries
- 9.2 Second order dulaity
- 9.3 Special Cases
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This book aims to explore various optimization criteria and duality concepts within the field of mathematical programming, with a particular focus on nondifferentiable problems. The authors delve into optimization techniques for diverse problems including nondifferentiable nonlinear programming, fractional programming, and minmax programming.
- Duality in mathematical programming
- Nondifferentiable optimization problems
- Applications of optimization techniques in various fields
- Variational problems and their solutions
- Continuous programming and its related problems
Zusammenfassung der Kapitel (Chapter Summaries)
- Chapter 1: Introduction This chapter provides an overview of mathematical programming, including its applications and the concept of duality. It introduces key definitions and notations used throughout the book, along with a review of previous work on duality in differentiable and nondifferentiable mathematical programming.
- Chapter 2: Variational Problems involving higher order derivatives This chapter focuses on the application of optimization techniques to problems involving higher-order derivatives. The concepts of invexity and optimality conditions are explored, along with different duality approaches, including Wolfe and Mond-Weir types.
- Chapter 3: Continuous programming containing support function This chapter investigates continuous programming problems involving support functions. It examines optimality conditions and different duality concepts, including converse duality and mixed duality. The chapter also considers special cases and related nonlinear programming problems.
- Chapter 4: Continuous time fractional minmax programming This chapter delves into the optimization of fractional minmax programming problems in a continuous time setting. It outlines the primal problem, optimality conditions, and duality concepts for this type of problem.
- Chapter 5: Nondifferentiable fractional minmax programming This chapter focuses on optimizing nondifferentiable fractional minmax programming problems. It covers necessary and sufficient optimality conditions, along with different duality approaches such as Mond-Weir and Schaible types.
- Chapter 6: Mixed Type symmetric and self duality for variational problems This chapter explores mixed-type symmetric and self-duality for variational problems. It presents the problem statement, analyzes mixed-type symmetric duality, and discusses self-duality concepts. The chapter also addresses natural boundary values and related mathematical programming problems.
- Chapter 7: Symmetric duality for nondifferentialbe maxmin variational problems This chapter examines symmetric duality for nondifferentiable maxmin variational problems. It includes discussions on Wolfe and Mond-Weir type symmetric and self-duality, as well as related problems.
- Chapter 8: Sufficient Fritz John optimality criteria and duality for continuous programming problems This chapter explores sufficient Fritz John optimality criteria and duality for continuous programming problems. It introduces Fritz John type duality and related problems.
- Chapter 9: Fritz John second order duality for programming problems This chapter focuses on Fritz John second-order duality for programming problems. It covers second-order duality concepts and discusses special cases.
Schlüsselwörter (Keywords)
This book focuses on optimization, duality, and mathematical programming, primarily exploring nondifferentiable problems. The key concepts include duality theory, optimality conditions, fractional programming, minmax programming, variational problems, continuous programming, and Fritz John conditions. The book addresses diverse types of programming problems, including nonlinear, fractional, and continuous.
- Quote paper
- Dr. Zamrooda Jabeen (Author), Dr. Farida Khursheed (Author), 2015, Approaches to mathematical optimization and its applications, Munich, GRIN Verlag, https://www.grin.com/document/457232
