Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints

TL;DR

This paper introduces a penalty alternating direction method for mixed-integer optimal control problems with combinatorial constraints, enabling efficient solution and convergence to partial minima through a novel lifting and decomposition approach.

Contribution

It proposes a new penalty-based decomposition method that handles combinatorial constraints efficiently and guarantees convergence to partial minima in mixed-integer optimal control.

Findings

Method converges to partial minima.

Outperforms heuristics in example tests.

Effective in optimizing electric transmission networks.

Abstract

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.