Algebraic solution of tropical optimization problems via matrix sparsification with application to scheduling

TL;DR

This paper develops an algebraic approach using matrix sparsification to solve tropical optimization problems, providing explicit solutions and applying them to project scheduling scenarios.

Contribution

It introduces a novel matrix sparsification technique for solving tropical optimization problems and offers explicit solutions applicable to scheduling.

Findings

Explicit solutions for tropical optimization problems are derived.

Matrix sparsification effectively extends partial solutions to complete ones.

The methods are demonstrated with real-world scheduling examples.

Abstract

Optimization problems are considered in the framework of tropical algebra to minimize and maximize a nonlinear objective function defined on vectors over an idempotent semifield, and calculated using multiplicative conjugate transposition. To find the minimum of the function, we first obtain a partial solution, which explicitly represents a subset of solution vectors. We characterize all solutions by a system of simultaneous equation and inequality, and show that the solution set is closed under vector addition and scalar multiplication. A matrix sparsification technique is proposed to extend the partial solution, and then to obtain a complete solution described as a family of subsets. We offer a backtracking procedure that generates all members of the family, and derive an explicit representation for the complete solution. As another result, we deduce a complete solution of the maximization problem, given in a compact vector form by the use of sparsified matrices. The results obtained are illustrated with illuminating examples and graphical representations. We apply the results to solve real-world problems drawn from project (machine) scheduling, and give numerical examples.