Complete solution of an optimization problem in tropical semifield

We consider a multidimensional optimization problem that is formulated in the framework of tropical mathematics to minimize a function defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible multiplication). The function, given by a matrix and calculated through a multiplicative conjugate transposition, is nonlinear in the tropical mathematics sense. We show that all solutions of the problem satisfy a vector inequality, and then use this inequality to establish characteristic properties of the solution set. We examine the problem when the matrix is irreducible. We derive the minimum value in the problem, and find a set of solutions. The results are then extended to the case of arbitrary matrices. Furthermore, we represent all solutions of the problem as a family of subsets, each defined by a matrix that is obtained by using a matrix sparsification technique. We describe a backtracking procedure that offers an economical way to obtain all subsets in the family. Finally, the characteristic properties of the solution set are used to provide a complete solution in a closed form.