This paper considers the global optimization of max-plus linear systems with affine equality constraints. For both the cases that the variables are real and non-negative, the necessary and sufficient conditions for the existence and uniqueness of globally optimal solutions are given, respectively. The proposed approaches are constructive and yield two polynomial algorithms for checking the solvabilities of the global optimization problems and finding all globally optimal solutions, in which the analytic expressions of general solutions are presented. The global optimization is then applied in the load scheduling of distributed systems with different processor capacities. The optimal allocation scheme is designed to minimize the completion time of the overall task. Some illustrative examples are presented to demonstrate the results.

本文考虑具有仿射等式约束的最大加线性系统的全局优化。对于变量是实数和非负数的两种情况，分别给出了全局最优解的存在和唯一性的充要条件。所提出的方法是建设性的，并产生了两个多项式算法来检查全局优化问题的可解性并找到所有全局最优解，其中给出了一般解的解析表达式。然后，将全局优化应用于具有不同处理器容量的分布式系统的负载调度中。最佳分配方案旨在最小化整个任务的完成时间。给出了一些说明性的例子来证明结果。