  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. FRE-ACO algorithm has been compared with some
related works proposed for solving nonlinear optimization problems with respect
to maxmin 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.
