The Maximax Minimax Quotient Theorem

TL;DR

This paper introduces an analytical solution to a complex optimization problem from control theory, combining fractional and max-min programming, and proves that the minimax solution occurs at a vertex of the feasible region.

Contribution

It provides a novel analytical approach to a challenging optimization problem at the intersection of fractional and max-min programming, avoiding nested nonlinear solutions.

Findings

Derived an explicit analytical solution for the optimization problem.

Proved the minimax solution is attained at a vertex of the constraint set.

Connected geometric arguments to optimization solutions.

Abstract

We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min programming on polytopes. A na\"ive solution would require solving four nested, possibly nonlinear, optimization problems. Instead, relying on numerous geometric arguments we determine an analytical solution to this problem. In the course of proving our main theorem we also establish another optimization result stating that the minimum of a specific minimax optimization is located at a vertex of the constraint set.