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Critical multipliers in variational systems via second-order generalized differentiation

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Abstract

In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal–dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.

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Acknowledgements

The first author gratefully acknowledges numerous discussions with Alexey Izmailov and Mikhail Solodov on critical multipliers and related topics. We particularly appreciate sharing with us Izmailov’s instructive notes [17]. We are also indebted to two anonymous referees and the handling editor for their very careful reading of the paper and making helpful remarks that allowed us to improve the original presentation.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

  2. RUDN University, Moscow, 117198, Russia

    Boris S. Mordukhovich

  3. Department of Mathematics, Miami University, Oxford, OH, 45065, USA

    M. Ebrahim Sarabi

Authors
  1. Boris S. Mordukhovich
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  2. M. Ebrahim Sarabi
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Correspondence to Boris S. Mordukhovich.

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Research of this author was partly supported by the National Science Foundation under Grants DMS-1007132 and DMS-1512846, by the Air Force Office of Scientific Research under Grant #15RT0462, and by the Ministry of Education and Science of the Russian Federation (Agreement Number 02.a03.21.0008 of 24 June 2016).

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Mordukhovich, B.S., Sarabi, M.E. Critical multipliers in variational systems via second-order generalized differentiation. Math. Program. 169, 605–648 (2018). https://doi.org/10.1007/s10107-017-1155-2

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