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Approximation properties of sum-up rounding in the presence of vanishing constraints

Authors: Paul Manns, Christian Kirches and Felix Lenders
Journal: Math. Comp. 90 (2021), 1263-1296
MSC (2020): Primary 90C59; Secondary 49M20, 49M25
Published electronically: February 18, 2021
MathSciNet review: 4232224
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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.

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Additional Information

Paul Manns
Affiliation: Institute for Mathematical Optimization, Technische Universität Braunschweig, 38106 Braunschweig, Germany
MR Author ID: 1201468
ORCID: 0000-0003-0654-6613

Christian Kirches
Affiliation: Institute for Mathematical Optimization, Technische Universität Braunschweig, 38106 Braunschweig, Germany
MR Author ID: 899522
ORCID: 0000-0002-3441-8822

Felix Lenders
Affiliation: ABB Corporate Research, ABB AG, 68526 Ladenburg, Germany.
ORCID: 0000-0003-3152-4221

Keywords: Discrete approximations, error estimates, relaxations of mixed integer optimal control
Received by editor(s): December 14, 2017
Received by editor(s) in revised form: March 15, 2020, and September 25, 2020
Published electronically: February 18, 2021
Additional Notes: The first and second authors acknowledge funding by Deutsche Forschungsgemeinschaft through Priority Programme 1962, grants no KI1839/1-1 and KI1839/1-2. The second author acknowledges financial support by the German Federal Ministry of Education and Research, program “Mathematics for Innovations in Industry and Service”, grants no 05M2017-MoPhaPro, 05M2018-MOReNet, 05M2020-LEOPLAN, and program “IKT 2020: Software Engineering”, grant 01/S17089C-ODINE. The third author acknowledges funding by the German National Academic Foundation.
Article copyright: © Copyright 2021 American Mathematical Society