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Optimality & Duality in Non-linear Programming Involving Convex Function
Summary
Submitted to
M.J.P. Rohilkhand University, Bareilly
for the award of the Degree
of
Doctor of Philosophy
in
Mathematics
2007
Supervisor : Researcher :
Dr. Sanjeev Rajan Vishnu Dev Pandey
Reader & Head
Dept. of Mathematics,
Hindu (P.G.) College,
Moradabad
Summary
The present thesis entitled "optimality & Duality in non-linear programming involving convex function" has been completed under the inspiring guidance of Dr. Sanjeev Rajan,Reader & Head of the Department of Mathematics Hindu (P.G.) College, Moradabad.
In the present work different aspects of optimality & Duality in non-linear programming have been studied to use keeping in mind the demand of such literature in the field of optimizing theory.
We discussed a branch of non-linear optimizing widely applied to the modelization of problems in operation research and management sciences is fractional programming in many occasions optimal performances of economic system is formulated as ratios in the thesis. Obviously there are certain advantages of these application.
The thesis is divided into five chapters along with two research papers, The chapters are :-
1.Introduction
2.Generalization of Convex function with respect to non-linear Programming.
3.Non-Linear fractional Programming, Involving semi locally convex.
4.Pre-Invex & Cone-Pre-Invex Functions.
5.Generalized non smooth invexity.
The research papers are :-
1.Generalizations of convexity & non smoothness functions.
2.Error Bounds for convex & non convex Programmes.
These two research papers have been accepted for publication in reputed scientific Journal (Ref. given in the thesis).
Chapter I entitled "Introduction" consists a very clear view on the optimizing & Duality theory, it's variation along with its components optimization.
Optimization theory is one of the most exciting branches of modern mathematics. It has been provided in the beginning so as to present a brief survey of the literature. A brief review of the optimization techniques along with a statement of the problem studied has also been included with definition.
Chapter II entitled "Generalization of convex funtion with non linear programming" In this chapter deals with the study of various extensions of convex function. The first section deals overview of convex functions and their important properties, The section deals with different generalizations of convex function. Some shown properties of convex function are studied stress is given to establish certain interrelations among these functions, these result are also extended for vector valued functions, on the lines of work done by Jeya Kumar and Mond second order extensions, of some of these functions are given for single valued and vector valued functions.
In this section we are going to provide an overview of convex function and their properties.
Chapter III entitled "Non linear fractional programming Involving semi locally Convex" In this chapter Suneja and Gupta established the necessary optimality conditions without assuming the semi local convexity of the objective and constraints functions but taking their right differentials at the optimal point to be convex. Derive kuhn - tucker type necessary optimality criteria for the problem (FP) as in suneja and Gupta by assuming the generalized slater constraint qualification & sufficient optimally criteria were obtained by Kaul & Kaur for a non-linear programming problem involving semilocally convex and related functions.
Chapter IV entitled "Pre-Invex & cone-pre-Invex functions" In this chapter there is a discussion a class of functions having the property that their exists an n-dimensional vector function such that for all , for weir & Jeya Kumar called such functions Pre-invex weir & Jeya Kumar considered the class of cone pre invex functions and obtained optimality conditions and duality theorems for both scalar and vector valued programs on a real normed space in the cases where the objective and the constraint functions are directionally differentiable colads, Li and Wang obtained certain necessary and sufficient conditions for a feasible solution to be weak minimum (a minimum, a strong minimum or a proper minimum) with respect to cones of a nonsmooth vector optimization problem in abstract spaces, using clarke's generalized gradient and properties of cone subconvex like functions and cone pre invex functions.
Chapter V entitled "Generalized non smooth invexity". In this chapter there is the theory of non smooth optimization using locally lipschitz functions was put forward by Francis clarke in the early 1970's locally lipschitz real valued function defined on a finite dimensional space is differentiable almost everywhere. This property is quiet similar to real valued convex functions defined on an open convex subset of a finite dimensional space.
In the section we introduce certain classes of generalized convex function using the clarke's generalized gradient. Among them are the classes of quasiconvex function,generalized invex function, - convex function etc. Further the clarke's generalized gradient is used to characterize quasiconvex functions.
Finally some of the remaining problems in this area of which are pointed out.
Together with this research two research papers are accepted to for publication in a reputed scientific Journal.
The first paper is entitled " Generalizations of Convexity & Non smoothness functions". According to this paper we introduce clases of generalized convex function using the clarke's generalized gradient we now define the class of -pseudo convex, quasiconvex functions which can through of as non smooth generalizations of the usual notions of pseudoconvex function and quasi convex function for differentiable functions.
The second paper is entitled "Error Bounds for convex and non-convex programs" In this paper given a single feasible solution and a single infeasible solution of a mathematical program, we provide an upper bounds to the optimal dual value. we assume that satisfies a weakness form of the slater condition.
We apply the bound to convex programs and we discuss it's relation to Hoffman like bounds as a special case. We recover a bound due to mangassarian on the distance of a point to a convex set specified by Inequalities.
Vishnu Dev Pandey
(Research Scholar)