Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak∗ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, a feature that is tied to the infinite-dimensional vantage point. In this work, we consider a class of MIOCPs that are constrained by pointwise mixed state-control constraints. We show that a continuous relaxation that involves so-called vanishing constraints has beneficial properties for the described approximation methodology. Moreover, we complete recent work on a variant of the sum-up rounding algorithm for this problem class. In particular, we prove that the observed infeasibility of the produced integer-valued controls vanishes in an L∞-sense with respect to the considered relaxation. Moreover, we improve the bound on the control approximation error to a value that is asymptotically tight.

中文翻译：

---

存在消失约束时求和舍入的近似性质

近似算法，如求和舍入，允许计算弱 * 意义上的连续控制的整数值近似值，最近引起了人们的兴趣。它们允许对混合整数控制问题 (MIOCP) 的连续松弛进行近似（最佳）可行解，其中整数控制任意接近。为此，他们使用了底层状态方程的紧凑性特性，这是一个与无限维有利位置相关的特征。在这项工作中，我们考虑了一类受逐点混合状态控制约束约束的 MIOCP。我们表明，涉及所谓的消失约束的连续松弛对于所描述的近似方法具有有益的特性。此外，我们完成了针对该问题类别的汇总舍入算法变体的最新工作。特别是，我们证明了所观察到的所产生的整数值控制的不可行性在 L∞ 意义上相对于所考虑的松弛消失了。此外，我们将控制近似误差的界限改进为渐近紧的值。