Nonsmooth Vector Optimization Problem Involving Second-Order Semipseudo, Semiquasi Cone-Convex Functions

Sunila Sharma,Priyanka Yadav

doi:10.19139/soic-2310-5070-839

Abstract

Recently, Suneja et al. [26] introduced new classes of second-order cone-(η; ξ)-convex functions along with theirgeneralizations and used them to prove second-order Karush–Kuhn–Tucker (KKT) type optimality conditions and duality results for the vector optimization problem involving first-order differentiable and second-order directionally differentiable functions. In this paper, we move one step ahead and study a nonsmooth vector optimization problem wherein the functions involved are first and second-order directionally differentiable. We introduce new classes of nonsmooth second-order cone-semipseudoconvex and nonsmooth second-order cone-semiquasiconvex functions in terms of second-order directional derivatives. Second-order KKT type sufficient optimality conditions and duality results for the same problem are proved using these functions.

Highlights

Second-order optimality conditions have been widely studied for past many years because they refine first-order by second-order information which is very useful for recognizing efficient solutions
Various types of second-order convex functions like second-order (F, ρ) convex [2], second-order (F, α, ρ, d) convex [3], second-order cone-convex [25] and recently many others like second-order univexities, second-order hybrid (Φ, ρ, η, ζ, θ)-invexity [29, 30, 31] along with their weaker notions have been defined for twice differentiable functions and used to study second-order duality results for multiobjective and vector optimization problems
Using the idea of cones, Suneja et al [26] extended the functions introduced by Ivanov [17] to second-order cone-(η, ξ)-convex and its weaker notions and used them to derive second-order KKT type optimality and duality results for vector optimization problem over cones involving first-order differentiable and second-order directionally differentiable vector valued functions

Summary

Introduction

Second-order optimality conditions have been widely studied for past many years because they refine first-order by second-order information which is very useful for recognizing efficient solutions. In the absence of second-order derivatives, Ivanov [17] defined second-order (type I) invexity for firstorder differentiable and second-order directionally differentiable functions He used them to prove necessary and sufficient optimality conditions for nonlinear programming problem. Using the idea of cones, Suneja et al [26] extended the functions introduced by Ivanov [17] to second-order cone-(η, ξ)-convex and its weaker notions and used them to derive second-order KKT type optimality and duality results for vector optimization problem over cones involving first-order differentiable and second-order directionally differentiable vector valued functions.

Notations and Definitions

Second-Order Optimality Conditions

Second-Order Duality

Conclusion