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Approximation Properties of Sum-Up Rounding in the Presence of Vanishing Constraints

Paul Manns (paul.manns***at***tu-bs.de)
Christian Kirches (c.kirches***at***tu-bs.de)
Felix Lenders (felix.lenders***at***iwr.uni-heidelberg.de)

Abstract: 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 $L^\infty$-sense with respect to the considered relaxation. Moreover, we improve the bound on the control approximation error to a value that is asymptotically tight.

Keywords: Discrete approximations, error estimates, relaxations of mixed integer optimal control

Category 1: Combinatorial Optimization (Approximation Algorithms )

Category 2: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 3: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Citation:

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Entry Submitted: 04/19/2018
Entry Accepted: 04/19/2018
Entry Last Modified: 03/15/2020

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