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.
Table of Contents
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
Objective and Research Topics
The work focuses on the determination of optimal solutions (minima or maxima) for various mathematical programming problems, specifically exploring optimality criteria and duality theory across diverse fields such as engineering, operations research, and economics. The core research question addresses the formulation and conceptualization of mixed duality, its application in constructing dual models, and the investigation of optimality criteria for scalar and multiobjective programming problems, particularly under weakened convexity requirements like invexity.
- Mathematical Programming and Duality Theory
- Variational Problems and Higher-Order Derivatives
- Nondifferentiable Minmax Programming
- Symmetric and Self-Duality in Variational and Programming Problems
- Optimality Criteria and Generalized Invexity
Excerpt from the Book
1.1 General Introduction
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 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 condition were first investigated by Fritz John [43] and later on, independently by Karush [44] and Kuhn – Tucker [49]. The inception of duality theory in linear programming may be traced to the classical minmax theorem of Von Neumann [82], which was subsequently formulated in a precise form by Gale, Kuhn and Tucker [34]. Since then optimality criteria and duality have remained as one of the most widely investigated area in mathematical programming. Karush-Kuhn-Tucker conditions [49] 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.
Summary of Chapters
1. Introduction: Provides foundational definitions and an overview of optimality criteria and duality theory in mathematical programming.
2. Variational Problems involving higher order derivatives: Studies constrained variational problems with higher-order derivatives using Fritz John theorems and duality models.
3. Continuous programming containing support function: Examines necessary and sufficient optimality conditions for continuous programming problems using support functions.
4. Continuous time fractional minmax programming: Derives optimality conditions and duality results for fractional minmax problems in a continuous-time setting.
5. Nondifferentiable fractional minmax programming: Formulates dual models for nondifferentiable fractional minmax problems.
6. Mixed Type symmetric and self duality for variational problems: Explores mixed type symmetric and self-duality for variational problems by adapting results from nonlinear programming.
7. Symmetric duality for nondifferentialbe maxmin variational problems: Extends symmetric duality to nondifferentiable maxmin variational problems under generalized convexity.
8. Sufficient Fritz John optimality criteria and duality for continuous programming problems: Investigates sufficiency of Fritz John optimality conditions without requiring constraint qualifications.
9. Fritz John second order duality for programming problems: Formulates second-order dual models using Fritz John optimality conditions for nonlinear programming.
Keywords
Mathematical Programming, Duality Theory, Optimality Criteria, Variational Problems, Higher Order Derivatives, Continuous Programming, Support Functions, Minmax Programming, Invexity, Pseudoconvexity, Symmetric Duality, Self-Duality, Fritz John Conditions, Fractional Programming, Nonlinear Programming
Frequently Asked Questions
What is the core focus of this research?
The work primarily deals with the development and analysis of duality theory and optimality criteria for various classes of mathematical programming and variational problems.
What are the primary fields of application?
The concepts are applied to problems in Engineering, Operations Research, Management Science, and Economics.
What is the main objective or research question?
The objective is to explore mixed duality, construct dual programs, and establish optimality criteria, especially when standard convexity assumptions are weakened.
Which scientific methods are primarily utilized?
The research utilizes mathematical modeling, variational calculus, Fritz John and Karush-Kuhn-Tucker optimality conditions, and various forms of duality (Wolfe, Mond-Weir, Symmetric).
What is covered in the main body of the work?
The work systematically covers differentiable and nondifferentiable programming, variational problems, fractional minmax programming, and provides sufficient optimality conditions and duality theorems for each.
Which terms characterize this research?
The research is characterized by terms such as duality, invexity, optimality conditions, minmax programming, and variational problems.
How does this work handle non-smooth functions?
The research addresses non-smoothness in programming problems by employing support functions and generalized gradients (Clarke generalized gradients).
What is the significance of the "mixed duality" concept here?
Mixed duality unifies existing symmetric dual models and allows for a more flexible construction of dual problems in scalar and multiobjective settings.
- 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
