Generalized parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions

G J Zalmai,Ram U Verma

doi:10.1007/s40096-016-0193-x

Abstract

In this paper, we construct six generalized second-order parameter-free duality models, and prove several weak, strong, and strict converse duality theorems for a discrete minmax fractional programming problem using two partitioning schemes and various types of generalized second-order $$(\mathcal {F},\beta ,\phi ,\rho ,\theta ,m)$$ -univexity (more compactly, ’second-order univexity’ is referred to as ’sounivexity’) assumptions. The obtained results are new and generalize most of results on discrete minmax fractional programming involving the second-order invexity as well as on second-order univexity in the literature.

Highlights

Verma and Zalmai [10] dealt with some details on discrete minmax fractional programming, a fairly extensive list of currently available publications dealing with various second-order necessary and sufficient optimality conditions for several types of optimization problems, some modifications of the concepts of second-order invexity, pseudoinvexity, and quasiinvexity originally defined by Hanson [3], a set of second-order necessary optimality conditions, and making use of the new classes of generalized secondorder invex functions, a fairly large number of sets of second-order sufficient optimality criteria
Remark 5.1 Using a direct nonparametric approach, in this paper, we have formulated six generalized secondorder parameter-free duality models for a discrete minmax fractional programming problem and established numerous duality results using a variety of generalized ðF ; b; /; q; h; mÞ-sounivexity assumptions
Each one of the six duality models considered in this paper is, a family of dual problems whose members can be identified by appropriate choices of certain sets and functions

Summary

Introduction

Introduction and preliminariesBased on a close observation on second-order necessary and sufficient optimality conditions for minmax fractional programming problems, which have not received much attention in the literature of mathematical programming, that is in sharp contrast to the case of minmax programming problems, numerous second-order necessary and sufficient optimality conditions for various classes of nonlinear programming problems with single and multiple objective functions have been investigated in the literature, including [1, 8,9,10,11, 13, 15, 19,20,21]. Theorem 2.2 (Weak duality) Let x and S ðy; z; u; v; wÞ be arbitrary feasible solutions of (P) and (DI), respectively, and assume that any one of the following four sets of hypotheses is satisfied: 1.

Results

Conclusion