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Error bounds for mixed integer nonlinear optimization problems

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Abstract

We introduce a-posteriori and a-priori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the NLP relaxation of a mixed-integer nonlinear optimization problem. Our analysis mainly bases on the construction of a tractable approximation of the so-called grid relaxation retract. Under appropriate Lipschitz assumptions on the defining functions, we thereby generalize and slightly improve results for the mixed-integer linear case from Stein (Mathematical Programming, 2015, doi:10.1007/s10107-015-0872-7). In particular, we identify cases in which the optimality and feasibility errors tend to zero at an at least linear rate for increasingly refined meshes.

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Acknowledgments

The author is grateful to the anonymous referee and the associate editor for their precise and substantial remarks on an earlier version of this manuscript.

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  1. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Oliver Stein

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  1. Oliver Stein
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Correspondence to Oliver Stein.

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Stein, O. Error bounds for mixed integer nonlinear optimization problems. Optim Lett 10, 1153–1168 (2016). https://doi.org/10.1007/s11590-016-1011-y

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  • DOI: https://doi.org/10.1007/s11590-016-1011-y

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