Multi‐dimensional Sum‐Up Rounding using Hilbert curve iterates

Mixed‐integer optimal control problems can be reformulated by means of partial outer convexification, which introduces binary‐valued switching functions for the different realizations of a discrete‐valued control variable. They can be relaxed naturally by allowing them to take values in [0, 1]. Sum‐Up Rounding (SUR) algorithms approximate feasible switching functions of the relaxation with binary ones. If the controls are distributed in one dimension, the approximants are known to converge in the weak∗ topology of L∞. We show that this still holds true for controls that are distributed in more than one dimension if an appropriate grid refinement strategy that is coupled with a deliberate ordering of the grid cells is chosen. This condition is satisfied by the iterates of space‐filling curves, e.g. the Hilbert curve.


Introduction
We rephrase the SUR algorithm from [7] and the known convergence properties [3][4][5]7] for multi-dimensional domains. We state an abstract recursive property for grid refinements and the order of the grid cells as they are approached by SUR in subsequent iterations, which is sufficient to prove the desired weak * convergence from the rephrased convergence properties. We note that the discretization into square cells induced by subsequent Hilbert curve iterates indeed has the desired properties.

The multi-dimensional SUR algorithm
We consider a finite partition of a bounded domain Ω ⊂ R d and compute a binary control ω ∈ L ∞ (Ω, M i=1 α i = 1 and 0 ≤ α i a.e., using the SUR algorithm, which follows a given order of the cells that partition Ω. If a tie arises with respect to the maximizing index k, the smallest applicable index is chosen. 3,5,7]) Let α be a relaxed control and ω(α) be computed by the SUR algorithm of Def. 2.1. Then, ω(α) is a binary control and there exists C > 0 such that for φ := α − ω(α) the following estimate holds:

Weak * approximation for suitable grid refinements
We consider three Hilbert curve iterates on the unit square and its induced partitions in Fig. 1, a facsimile of the figure in [2]. The ordering of the squares along the curve is preserved from an iterate to the next, which allows to conserve a weighted mean of the quantity φ because the successive averaging always happens on sub-cells of cells we have already under control. We formalize this property below.  N (n) of Ω be given. Let (ω (n) ) n denote the binary controls, which are computed by the SUR algorithm on the partitions along the subscript orderings. We define φ (n) := α − ω (n) for all n.
P r o o f. We abbreviate φ := φ i and have to show Ω φ (n) f → 0, where the codomains of f and φ (n) are subsets of R. As φ (n) ∈ L ∞ (Ω), the products φ (n) f are integrable. Let ε > 0. Then, f can be approximated with a simple function f (1) (1) can be approximated with a simple function f (2) that is defined on the generator from Ass. 3.2, i.e. there exist n 0 ∈ N such that f (2) We study the product f (3) φ (n) . Let n ≥ n 0 . Then, property 3. of Ass. 3.2 gives Here, j(i, n) is the starting and k(i, n) the end index the sets S  k(i,j) exactly once during all iterations n ≥ n 0 . We plug the considerations above together and use an ε 3 -argument to obtain for n ≥ n 0 large enough.
We summarize the weak * convergence, which follows easily from the previous lemma, into the following theorem. Theorem 3.4 Let Ass. 3.2 hold. Then, φ (n) 0 in L p (Ω, R M ) for p ∈ [1, ∞) and φ (n) * 0 for p ∈ (1, ∞]. Using e.g. the monograph by Sagan [6], one can easily verify the following proposition, which we skip for sake of brevity. Proposition 3.5 The sequence of sets of squares induced by the Hilbert curve iterates satisfies Def. 3.1.