Optimal switching sequence for switched linear systems

TL;DR

This paper introduces an exact polynomial-time algorithm for optimizing switching sequences in linear systems with the oligo-vertex property, outperforming existing solvers on large instances and opening new research directions.

Contribution

The paper defines the oligo-vertex property, provides conditions for it, and develops a polynomial-time algorithm for a class of switching sequence optimization problems.

Findings

The algorithm outperforms state-of-the-art solvers on large instances.

The oligo-vertex property enables polynomial-time solutions.

Numerical results confirm the efficiency of the proposed method.

Abstract

We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of $n\times n$ matrices and an $n$-dimensional vector, find a sequence of $K$ matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the $K$ matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce in this paper for a finite set of matrices. We derive several sufficient conditions for a set of matrices to have the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global optimization solver. We also propose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest.