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Augmented Lagrangian and exact penalty methods for quasi-variational inequalities

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Abstract

A variant of the classical augmented Lagrangian method was recently proposed in Kanzow (Math Program 160(1–2, Ser. A):33–63, 2016), Pang and Fukushima (Comput Manag Sci 2(1):21–56, 2005) for the solution of quasi-variational inequalities (QVIs). In this paper, we describe an improved convergence analysis to the method. In particular, we introduce a secondary QVI as a new optimality concept for quasi-variational inequalities and use this tool to prove convergence theorems for certain popular classes of QVIs under very mild assumptions. Finally, we present a modification of the augmented Lagrangian method which turns out to be an exact penalty method, and also give detailed numerical results illustrating the performance of both methods.

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Correspondence to Daniel Steck.

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This research was supported by the German Research Foundation (DFG) within the priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) under Grant Number KA 1296/24-1.

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Kanzow, C., Steck, D. Augmented Lagrangian and exact penalty methods for quasi-variational inequalities. Comput Optim Appl 69, 801–824 (2018). https://doi.org/10.1007/s10589-017-9963-0

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