Approaches to mathematical optimization and its applications

Abstract or Introduction

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.