Second order modified objective function method for twice differentiable vector optimization problems over cone constraints

,Anurag Jayswala,Tadeusz Antczakb,Shalini Jha

doi:10.3934/naco.2019010

Abstract

In the paper, a vector optimization problem with twice differentiable functions and cone constraints is considered. The second order modified objective function method is used for solving such a multiobjective programming problem. In this method, for the considered twice differentiable multi-criteria optimization problem, its associated second order vector optimization problem with the modified objective function is constructed at the given arbitrary feasible solution. Then, the equivalence between the sets of (weakly) efficient solutions in the original twice differentiable vector optimization problem with cone constraints and its associated modified vector optimization problem is established. Further, the relationship between an (weakly) efficient solution in the original vector optimization problem and a saddle-point of the second order Lagrange function defined for the modified vector optimization problem is also analyzed.

Highlights

The extremum problem, in which more than one objective functions are optimized simultaneously, is known in the optimization literature as a vector optimization problem or multiobjective programming problem
We prove the equivalence between an efficient solution in the original multiobjective programming problem (VOP) and an efficient solution in its associated vector optimization problem (VOP)2(x) with the second order modified objective function
The equivalence between an efficient solution in the original vector optimization problem with cone constraints and an efficient solution in its associated vector optimization problem with the second order modified objective function has been established under second order cone convexity hypotheses

Summary

Introduction

The extremum problem, in which more than one objective functions are optimized simultaneously, is known in the optimization literature as a vector optimization problem or multiobjective programming problem. If the Lagrange multiplier λsatisfies λ ∈ intK∗, xis an efficient solution in the associated vector optimization problem (VOP)2(x) with the second order modified objective function.

Results

Conclusion