Approximation Algorithms for Optimization of Combinatorial Dynamical Systems

TL;DR

This paper introduces scalable approximation algorithms with provable bounds for optimizing large combinatorial dynamical systems, enabling efficient solutions without solving complex dynamical equations.

Contribution

It develops a novel linear approximation approach for large-scale binary decision optimization in dynamical systems, with provable suboptimality guarantees.

Findings

Algorithms perform well on refrigeration system control problems.

Proposed methods are computationally efficient for high-dimensional systems.

Suboptimality bounds are linked to the concavity of the objective function.

Abstract

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems.