Approaches to mathematical optimization and its applications

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

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.