Vector Exponential Penalty Function Method for Nondifferentiable Multiobjective Programming Problems

Tadeusz Antczak

doi:10.1007/s40840-016-0340-4

Abstract

In this paper, a new vector exponential penalty function method for nondifferentiable multiobjective programming problems with inequality constraints is introduced. First, the case when a sequence of vector penalized optimization problems with vector exponential penalty function constructed for the original multiobjective programming problem is considered, and the convergence of this method is established. Further, the exactness property of a vector exact penalty function method is defined and analyzed in the context of the introduced vector exponential penalty function method. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto solutions of the considered nondifferentiable multiobjective programming problem and the associated vector penalized optimization problem with the vector exact exponential penalty function. This equivalence is established for nondifferentiable vector optimization problems with inequality constraints in which involving functions are r-invex.

Highlights

The field of multiobjective programming, known as vector programming, has attracted a lot of attention since many real-world problems in decision theory, economics, engineering problems, game theory, management sciences, physics, optimal control can be modeled as nonlinear vector optimization problems
In order to avoid the need of a sequence of unbounded penalty parameters, in other words, a sequence of unbounded penalized optimization problems, we prove that the introduced vector exponential penalty function method is exact in the sense that a Pareto solution in the original multiobjective programming problem is equivalent to a Pareto solution in the associated vector penalized problem, for a finite value of the penalty parameter
The exact vector exponential penalty function method has been used for solving nonconvex nondifferentiable multiobjective programming problems with inequality constraints

Summary

Introduction

The field of multiobjective programming, known as vector programming, has attracted a lot of attention since many real-world problems in decision theory, economics, engineering problems, game theory, management sciences, physics, optimal control can be modeled as nonlinear vector optimization problems. We introduce a new vector exponential penalty function method, and we use it for solving a class of nondifferentiable multiobjective programming problems involving r -invex functions (with respect to the same function η). This method is based on such a construction of an exact absolute value penalty function, which is minimized in the exponential penalized optimization problem constructed in this method. We prove that there exists a finite lower bound of the penalty parameter c such that, for every penalty parameter c exceeding this threshold, a (weak) Pareto solution in the considered nonconvex nondifferentiable multiobjective programming problem coincides with an unconstrained (weak) Pareto solution in its associated vector penalized optimization problem with the vector exact exponential penalty function.

Preliminaries

Exactness of the Introduced Vector Exponential Penalty Function Method

Conclusion