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Stability of the Duality Gap in Linear Optimization

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Abstract

In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.

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Acknowledgments

This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-01 and SECTyP-UNCuyo Res. 4540/13-R.

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Correspondence to M. A. Goberna.

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Goberna, M.A., Ridolfi, A.B. & Vera de Serio, V.N. Stability of the Duality Gap in Linear Optimization. Set-Valued Var. Anal 25, 617–636 (2017). https://doi.org/10.1007/s11228-017-0405-z

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