MultiDimensional Sum-Up Rounding for Elliptic Control Systems

Abstract

Partial outer convexification has been used to derive relaxations of mixed-integer optimal control problems (MIOCPs) that are constrained by time-dependent differential equations. The family of sum-up rounding (SUR) algorithms provides a means to approximate feasible points of these relaxations, i.e., $[0,1]$-valued control trajectories, with $\{0,1\}$-valued points. The approximants computed by an SUR algorithm converge in a weak sense when the coarseness of the rounding grid of the SUR algorithm is driven to zero, which in turn induces norm convergence of the corresponding sequence of state vectors. We show that this approximation property can be transferred to MIOCPs with integer control variables distributed in more than one dimension when carrying out an appropriate grid refinement strategy. We deduce a norm convergence result for the state vector of elliptic PDE systems and provide computational results illustrating the applicability of the theoretical framework.