Chanchal Singh

A survey of recent [1985--1995] advances in generalized convexity with applications to duality theory and optimality conditions

Convexity plays a very important role in various branches of mathematical, natural and social sciences. In an effort to extend existing results depending on convexity there has been a steady interest over the years towards its generalizations. Many excellent books, monographs and numerous articles have been written pursuing generalizations of the basic concepts, their characterizations. and studying properties under different conditions. The purpose of this contribution is to gather information on certain generalizations of convexity and their applications to duality theory and optimality conditions

Optimality and duality with generalized convexity

Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.

A class of multiple-criteria fractional programming problems

Abstract Multiple-criteria nonlinear programming results are extended to a class of nonlinear fractional programming problems. Some duality and optimality theorems of standard nonlinear programming are established for the multiple criteria fractional programming problems being considered. They are the converse duality theorems and the Fritz-John-type sufficient conditions.

Optimality conditions for fractional minmax programming

where Y is a compact subset of R”,f(a, .): R” X R” + R is C’ on R” X Rm and g(.): R” + RP is C’ on R”; B is an n X n positive semidefinite matrix; h(.,.):R”xR”+R is C’ on R”XR”. Throughout the paper we assume that h(x, y) > 0 for each (x,~) in XX Y, where X is the set of feasible solutions of Problem (l), i.e., X= {x E R”: g(x) < 0). The necessary and sufficient conditions to be established are based on the optimality conditions developed by Schmitendorf [6] for a static minmax problem. This application is possible when the functions involved are differentiable at an optimal point or a candidate for optimal point. However, differentiability conditions may not hold due to the presence of (x’Bx)“~ in the objective function. In this situation, the constraint qualification used by Aggarwal and Saxena and introduced by Mond [4] is extended for our case. 409 0022-247X/84

Saddlepoint Theory for Nondifferentiable Multiobjective Fractional Programming

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A sufficient optimality criterion in continuous time programming for generalized convex functions

Abstract Saddlepoint theory is developed for the nondifferentiable multiobjective fractional programming problem. Necessary and sufficient conditions of Fritz John and Kuhn-Tucker type are established. As expected, only necessary conditions require convexity restrictions. All functions are assumed to be nondifferentiable.

Generalized convexity and generalized monotonocity

The continuous time programming problem, originating from Bellman’s bottleneck problems [l], has been considered by number of authors. A partial list of references can be found in Farr and Hanson [2, 31. More recently, Singh and Farr [A established the optimality criteria of Kuhn-Tucker and Fritz John type without assuming differentiability of functions involved. However, all the work done so far in the nonlinear case is based on the assumption the functions involved are either convex or concave. In this paper we weaken the convexity/concavity restrictions by considering quasi-convex/quasi-concave and pseudoconvex functions. More specifically we will assume that the objective function involve pseudoconvexity “almost everywhere” and the constraints involve quasi-convex and quasi-concave functions. Having this set up we establish a sufficient optimality criterion of the classical type as presented in Mangasarian [5]. Functions are assumed to be FrCchet differentiable. For definition and properties of FrCchet differential the reader is referred to Luenberger [4] and Rall [6].

Duality in nonlinear multiobjective programming using augmented Lagrangian functions

Abstract General definitions of monotonocity, quasimonotonocity, pseudomonotonocity and strong monotonocity of a vector valued function defined on a subset of an n-dimensional Euclidean space are given. Some properties and characterizations are studied in the genera] setting. A relationship between generalized monotonocity and (Φ1, Φ2)– convexity introduced earlier by the authors, is described under certain conditions. Applications of the general definitions are studied in terms of invexity and B-vexity.

(Ф1,Ф2)convexity

A vector-valued generalized Lagrangian is constructed for a nonlinear multiobjective programming problem. Using the Lagrangian, a multiobjective dual is considered. Without assuming differentiability, weak and strong duality theorems are established using Pareto efficiency.

Further Properties of b-Vex Functions

Convexity of a function and of a set are generalized. The new class being introduced, include many well known classes as its subclasses. The defining functions involved are required to satisfy certain regularity conditions. Some properties are studied with or without differentiability; in the differentiable case, first and second order conditions are stated