A two - phase - ACO algorithm for solving nonlinear optimization problems subjected to fuzzy relational equations

In this paper, we investigate nonlinear optimization problems whose constraints are defined as fuzzy relational equations (FRE) with max-min composition. Since the feasible solution set of the FRE is often a non-convex set and the resolution of the FREs is an NP-hard problem, conventional nonlinear approaches may involve high computational complexity. Based on the theoretical aspects of the problem, an algorithm (called FRE-ACO algorithm) is presented which benefits from the structural properties of the FREs, the ability of discrete ant colony optimization algorithm (denoted by ACO) to tackle combinatorial problems, and that of continuous ant colony optimization algorithm (denoted by ACOR) to solve continuous optimization problems. In the current method, the fundamental ideas underlying ACO and ACOR are combined and form an efficient approach to solve the nonlinear optimization problems constrained with such non-convex regions. Moreover, FRE-ACO algorithm preserves the feasibility of new generated solutions without having to initially find the minimal solutions of the feasible region or check the feasibility after generating the new solutions. The FRE-ACO algorithm has been compared with some related works proposed for solving nonlinearoptimization problems with respect to max-min FREs. The obtained results demonstrate that the proposed algorithm has a higher convergence rate and requires a less number of function evaluations compared to other considered algorithms.