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.

翻译：本文研究约束条件为具有最大-最小复合运算的模糊关系方程(FRE)的非线性优化问题。由于FRE的可行解集通常是非凸集合，且求解FRE属于NP难问题，传统非线性方法往往涉及较高的计算复杂度。基于问题的理论特性，本文提出一种算法（称为FRE-ACO算法），该算法融合了FRE的结构特性、离散蚁群优化算法（记为ACO）处理组合问题的能力，以及连续蚁群优化算法（记为ACOR）求解连续优化问题的优势。在本方法中，ACO与ACOR的基本思想相结合，形成了一种有效求解此类非凸区域约束下非线性优化问题的方法。此外，FRE-ACO算法能够保持新生成解的可行性，无需预先计算可行域的最小解或在生成新解后进行可行性验证。通过与现有针对最大-最小FRE约束非线性优化问题的相关研究进行比较，结果表明所提算法具有更高的收敛速度，且相较于其他对比算法所需的函数评估次数更少。