Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems

T Antczak

doi:10.1007/s10957-017-1204-2

Abstract

In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.

Highlights

In optimization theory, the coefficients of the extremum problems are usually considered as deterministic values and, the corresponding solutions are precise
The results established in the paper are illustrated by suitable examples of optimization problems with the interval-valued objective function solved by using the exact l1 penalty function method
An associated penalized optimization problem with the interval-valued exact l1 penalty function is constructed for the considered intervalvalued minimization problem

Summary

Introduction

The coefficients of the extremum problems are usually considered as deterministic values and, the corresponding solutions are precise. We first prove that a Karush-Kuhn-Tucker point of the considered nondifferentiable interval-valued constrained optimization problem is a LU-minimizer of its associated penalized optimization problem with the interval-valued exact l1 penalty function. This result is established under convexity hypotheses for all penalty parameters c exceeding the threshold value, which is expressed as the function of Lagrange multipliers. The results established in the paper are illustrated by suitable examples of optimization problems with the interval-valued objective function solved by using the exact l1 penalty function method. We extend and improve the results established by Jayswal and Banerjee [37] for the exact l1 penalty function method, who used it for solving differentiable interval-valued optimization problems with inequality constraints only

Preliminaries and Interval-Valued Optimization

Conclusions