Interval global optimization problem in max-plus algebra

Abstract

Consider the global optimization problem of minimizing the max-plus product $A\otimes x$, where A is a given matrix and the constraint set is the set of column vectors x such that the sum of products $k_j{x}_j$ is equal to c and c is a given positive real constant, $k_j$ are non-negative numbers with sum equal to 1. We show that the solvability of the given global optimization problem is independent of the number c if the components of the vector x can also be negative. From a practical point of view, we further consider the solvability of the global optimization problem with non-negative constraints. We propose an algorithm which decides whether a given problem is solvable, extend the problem to interval matrices and provide an algorithm to verify the solvability of interval global optimization problem.