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A linear bound on the integrality gap for 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: In this article we consider the integrality gap between a relaxed continuous control trajectory and an integer feasible one that is constructively obtained in linear runtime by a family of sum-up-rounding algorithms. Such algorithms are typically invoked when solving Mixed-Integer Optimal Control Problems (MIOCPs) and serve to construct integer feasible approximations from optimal solutions of a particular relaxation. We give a constructive proof of a bound on the integrality gap in the presence of additional constraints on the discrete control. Our bound is linear in both the time discretization granularity and the number of discrete choices of the control. This result completes recent work on the approximation of feasible points of the relaxed problem by points of the combinatorial problem.

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 )


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Entry Submitted: 04/19/2018
Entry Accepted: 04/19/2018
Entry Last Modified: 09/24/2018

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