Approximate Optimality Conditions in Fractional Semi-Infinite Multiobjective Optimization (Study on Nonlinear Analysis and Convex Analysis)

This paper is based on the manuscript "Approximate necessary optimality in fractional semi-infinite multiobjective optimization" written by T. Shitkovskaya, Z. Hong, D.S. Kim and G.R. Piao, which was accepted to J. Nonlinear Convex Anal.This paper provides some new results on weak approximate solutions in fractional multiobjective optimization problems. Specifically, we establish necessary optimality conditions of Fritz-John type for a local weakly E-efficient solution in fuzzy form and, by using limiting constraint qualification, we provide necessary optimality conditions of Karush-Kuhn-Tucker type for a weakly E-quasi-efficient solution. To this purpose advanced tools of variational analysis and generalized differentiation are used

Necessary Conditions in Multiobjective Optimization With Equilibrium Constraints

In this paper we study multiobjective optimization problems with equilibrium constraints (MOECs) described by generalized equations in the form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models particularly arise from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data

New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem

We study a nondifferentiable fractional programming problem as follows: (P)minx∈Kf(x)/g(x) subject to x∈K⊆X,  hi(x)≤0,  i=1,2,…,m, where K is a semiconnected subset in a locally convex topological vector space X, f:K→ℝ, g:K→ℝ+ and hi:K→ℝ, i=1,2,…, m. If f, -g, and hi, i=1,2,…,m, are arc-directionally differentiable, semipreinvex maps with respect to a continuous map γ:[0,1]→K⊆X satisfying γ(0)=0 and γ′(0+)∈K, then the necessary and sufficient conditions for optimality of (P) are established

Some contributions to optimality criteria and duality in Multiobjective mathematical programming.

This thesis entitled, “some contributions to optimality criteria and duality in multiobjective mathematical programming”, offers an extensive study on optimality, duality and mixed duality in a variety of multiobjective mathematical programming that includes nondifferentiable nonlinear programming, variational problems containing square roots of a certain quadratic forms and support functions which are prominent nondifferentiable convex functions. This thesis also deals with optimality, duality and mixed duality for differentiable and nondifferentiable variational problems involving higher order derivatives, and presents a close relationship between the results of continuous programming problems through the problems with natural boundary conditions between results of their counter parts in nonlinear programming. Finally it formulates a pair of mixed symmetric and self dual differentiable variational problems and gives the validation of various duality results under appropriate invexity and generalized invexity hypotheses. These results are further extended to a nondifferentiable case that involves support functions.Digital copy of Thesis.University of Kashmir

Second-order optimality conditions for interval-valued functions

This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of second-order optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the first-order conditions. We will define new concepts such as second-order gH-derivative for interval-valued functions, 2-critical points, and 2-KKT-critical points. We obtain and present new types of interval-valued functions, such as 2-pseudoinvex, characterized by the property that all their second-order stationary points are global minima. We extend the optimality criteria to the semi-infinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with interval-valued functions.Funding for open access publishing: Universidad de Cádiz/CBUA. The research has been supported by MCIN through grant MCIN/AEI/PID2021-123051NB-I00

Duality in mathematical programming.

In this thesis entitled, “Duality in Mathematical Programming”, the emphasis is given on formulation and conceptualization of the concepts of second-order duality, second-order mixed duality, second-order symmetric duality in a variety of nondifferentiable nonlinear programming under suitable second-order convexity/second-order invexity and generalized second-order convexity / generalized second-order invexity. Throughout the thesis nondifferentiablity occurs due to square root function and support functions. A support function which is more general than square root of a positive definite quadratic form. This thesis also addresses second-order duality in variational problems under suitable second-order invexity/secondorder generalized invexity. The duality results obtained for the variational problems are shown to be a dynamic generalization for thesis of nonlinear programming problem.Digital copy of Thesis.University of Kashmir