On the nonlinear programming problems subject to a system of generalized bipolar fuzzy relational equalities defined with continuous t-norms

TL;DR

This paper generalizes bipolar fuzzy relational equations using continuous t-norms, analyzes their feasible solutions, and develops algorithms for solving nonlinear programming problems with these constraints.

Contribution

It introduces a comprehensive framework for bipolar fuzzy relational equations with continuous t-norms and proposes methods to find feasible solutions and optimize problems under these constraints.

Findings

Feasible solutions form finite, possibly disconnected, compact sets.

An algorithm is provided to identify the feasible region.

Optimization models with diverse nonlinear objectives are solvable with the proposed approach.

Abstract

As a starting point, this paper develops the system of bipolar fuzzy relational equations (FRE) to the most general case, where bipolar FREs are defined by an arbitrary continuous t-norm. Due to the fact that fuzzy relational equations are special cases of bipolar FREs, the proposed system can also be viewed as a generalization of traditional FREs, in which the fuzzy composition can be defined by a continuous t-norm. In order to determine the feasibility of the proposed system, some necessary and sufficient conditions are presented for studying continuous bipolar FREs. This is followed by a complete analysis of the set of feasible solutions to the problem. Contrary to FREs and bipolar FREs defined by continuous Archimedean t-norms, the feasible solutions set of generalized bipolar FREs consists of a finite number of compact sets that are not necessarily connected. Further, five techniques have been outlined in an attempt to simplify the current problem, and then an algorithm has been presented to find the feasible region of the problem. Next, we present a class of optimization models subject to continuous bipolar FRE constraints, in which the objective function incorporates a wide range of (non)linear functions, such as maximum functions, geometric mean functions, log-sum-exp functions, maximum eigenvalues of symmetric matrices, support functions for sets, etc. Considering that the problem has a finite number of local optimal solutions, the global optimal solution can always be obtained by choosing the point with the minimum objective value among these local optimal solutions. Lastly, as a means to illustrate the definitions, theorems, and algorithms presented in the paper, a step-by-step example is presented in several sections, in which the constraints are a system of bipolar FREs defined by the Dubois-Prade t-norm, which is a continuous non-Archimedean t-norm.