Binary optimal control by trust-region steepest descent

Abstract

We present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control function as an indicator function of a measurable set and makes set-valued adjustments derived from the sublevel sets of a topological gradient function. By combining this type of update with a trust-region framework, we are able to show by theoretical argument that our method achieves asymptotic stationarity despite possible discretization errors and truncation errors during step determination. To demonstrate the practical applicability of our method, we solve two optimal control problems constrained by ordinary and partial differential equations, respectively, and one topological optimization problem.

Sequential Linear Integer Programming for Integer Optimal Control with Total Variation Regularization

“…Existence of Optimal Solutions. In contrast to problem ( P) [21,22,26] or problems with a regularizer induced by a convex, but not strictly convex, integral superposition operator [11,20,24], we will briefly show that the problem (P) has a solution under the following mild assumptions. Assumption 2.1.…”

“…However, the approach in [5] does not scale well if very fine discretization meshes are used. A similar approach to the proposed one was analyzed and demonstrated recently in [20], where the authors take the perspective of modifying sets instead of functions and their algorithm also seeks for a point that satisfies a local stationarity condition. However, there is no guarantee that a feasible point that satisfies this condition exists, and highly oscillating control functions may occur if (a subsequence of) the iterates converge weakly- * to an infeasible (i.e., fractional-valued) limit function.…”

On Convergence of Binary Trust-Region Steepest Descent

“…Binary trust-region steepest descent [5,24,27] The BTR method applies to the binary case V = {0, 1} (generalizations to the case V = {0, 1} k for k ∈ N are conceivable but have not been considered in the literature so far). It solves trust-region subproblems in which the level sets of the control x, corresponding to the values in V , are manipulated to greedily improve the linearized objective.…”

“…A trust-region constraint limits the volume of the level sets, which is the L 1 -norm of the control function, and can change from one accepted iterate to the next. The analysis in [5] shows that BTR iterates eventually satisfy a condition called ε-stationarity under a regularity assumption used to obtain sufficient decrease of the aggregated volume of level set manipulations.…”

“…The concept ε-stationarity as introduced in [5] resembles first-order necessary optimality conditions for (R). Moreover, similar to CIA, the structural assumptions on the quantities that appear in (P) are in general not able to prevent a fine microstructure from developing over the iterations.…”