Locally and globally optimal solutions of global optimisation for max‐plus linear systems

Abstract

Abstract This paper considers the locally and globally optimal solutions of the global optimisation problem whose objective function is a max‐plus vector‐valued function and constraint function is a real affine function. The formulas about global optimisation in the existing work are further explained and simplified. The necessary and sufficient conditions for the existence and uniqueness of locally optimal solutions of a single objective are established, which are then used to deduce these of globally optimal solutions. Furthermore, the general formulas of locally and globally optimal solutions are presented, respectively. The local optimisation is then applied to solve the load distribution problem of distributed systems with no globally optimal solution, and the optimal allocation scheme is proposed to complete the overall task at the earliest time. The proposed method is constructive, and the obtained results are illustrated by the numerical examples.