Isolated Calmness of Perturbation Mappings and Superlinear Convergence of Newton-Type Methods

Matúš Benko () and Patrick Mehlitz ()
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Matúš Benko: Johann Radon Institute for Computational and Applied Mathematics
Patrick Mehlitz: Philipps-Universität Marburg

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 2, No 20, 1587-1621

Abstract: Abstract In this paper, we characterize Lipschitzian properties of different multiplier-free and multiplier-dependent perturbation mappings associated with the stationarity system of a so-called generalized nonlinear program popularized by Rockafellar. Special emphasis is put on the investigation of the isolated calmness property at and around a point. The latter is decisive for the locally fast convergence of the so-called semismooth* Newton-type method by Gfrerer and Outrata. Our central result is the characterization of the isolated calmness at a point of a multiplier-free perturbation mapping via a combination of an explicit condition and a rather mild assumption, automatically satisfied e.g. for standard nonlinear programs. Isolated calmness around a point is characterized analogously by a combination of two stronger conditions. These findings are then related to so-called criticality of Lagrange multipliers, as introduced by Izmailov and extended to generalized nonlinear programming by Mordukhovich and Sarabi. We derive a new sufficient condition (a characterization for some problem classes) of nonexistence of critical multipliers, which has been also used in the literature as an assumption to guarantee local fast convergence of Newton-, SQP-, or multiplier-penalty-type methods. The obtained insights about critical multipliers seem to complement the vast literature on the topic.

Keywords: Critical multipliers; Generalized nonlinear programming; Isolated calmness; Newton-like methods; Variational analysis; 49J52; 49J53; 90C30 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02522-2

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