Log-sum-exp optimization problem subjected to Lukasiewicz fuzzy relational inequalities

TL;DR

This paper introduces a nonlinear optimization problem with a log-sum-exp objective constrained by Lukasiewicz fuzzy relational inequalities, providing conditions for feasibility and an algorithm for exact solutions.

Contribution

It formulates a new optimization problem involving fuzzy inequalities and develops an algorithm to find exact solutions despite non-convex feasible regions.

Findings

Derived necessary and sufficient feasibility conditions.

Characterized the feasible set as finite convex cells.

Proposed an algorithm that finds the exact optimal solution.

Abstract

In this paper, we introduce a nonlinear optimization problem whose objective function is the convex log-sum-exp function and the feasible region is defined as a system of fuzzy relational inequalities (FRI) defined by the Lukasiewicz t-norm. Some necessary and sufficient conditions are derived to determine the feasibility of the problem. The feasible solution set is characterized in terms of a finite number of closed convex cells. Since the feasible solutions set of FRIs is non-convex, conventional methods may not be directly employed. An algorithm is presented for solving this nonlinear problem. It is proved that the algorithm can find the exact optimal solution and an example is presented to illustrate the proposed algorithm.