Switching Cost Aware Rounding for Relaxations of Mixed-Integer Optimal Control Problems: The 2-D Case

Abstract

This article is concerned with a recently proposed switching cost aware rounding (SCARP) strategy in the combinatorial integral approximation for mixed-integer optimal control problems (MIOCPs). We consider the case of a control variable that is discrete-valued and distributed on a two-dimensional domain. While the theoretical results from the one-dimensional case directly apply to the multidimensional setting, the structure of the cost function in the graph-based rounding computation is significantly more involved in the two-dimensional case. We describe a set up of the computational graph and the traversal algorithm underlying the SCARP strategy that enable a transfer to the two-dimensional setting. We demonstrate the SCARP strategy in this two-dimensional setting using the example of a MIOCP from topology optimization. We compare the graph-based approach to a ground truth computation using an integer linear programming (ILP) solver. The graph-based approach becomes computationally intractable for medium grid sizes. We show that the one-dimensional SCARP algorithm can be employed on a serialization of the grid cells in these cases and still provides an efficient heuristic that yields superior performance compared with that of other rounding heuristics such as sum-up rounding (SUR).