Volume 28, 2022
|Number of page(s)||24|
|Published online||25 May 2022|
Convergence of quasi-Newton methods for solving constrained generalized equations*
Department of Applied Mathematics, University of Campinas, Rua Sérgio Buarque de Holanda, 651, 13083-859 Campinas, SP, Brazil
2 Federal Institute of Piauí, Picos, PI, Brazil
3 Department of Applied Mathematics, Federal University of Espírito Santo, Rodovia BR 101, Km 60, 29932-540 São Mateus, ES, Brazil
4 Center of Exact Sciences and Technologies, Federal University of the West of Bahia, 47808-021 Barreiras, BA, Brazil
** Corresponding author: firstname.lastname@example.org
Accepted: 5 April 2022
In this paper, we focus on quasi-Newton methods to solve constrained generalized equations. As is well-known, this problem was firstly studied by Robinson and Josephy in the 70’s. Since then, it has been extensively studied by many other researchers, specially Dontchev and Rockafellar. Here, we propose two Broyden-type quasi-Newton approaches to dealing with constrained generalized equations, one that requires the exact resolution of the subproblems, and other that allows inexactness, which is closer to numerical reality. In both cases, projections onto the feasible set are also inexact. The local convergence of general quasi-Newton approaches is established under a bounded deterioration of the update matrix and Lipschitz continuity hypotheses. In particular, we prove that a general scheme converges linearly to the solution under suitable assumptions. Furthermore, when a Broyden-type update rule is used, the convergence is superlinearly. Some numerical examples illustrate the applicability of the proposed methods.
Mathematics Subject Classification: 90C53 / 65K15
Key words: Inexact quasi-Newton / Constrained generalized equations / Inexact projection / Broyden update
© The authors. Published by EDP Sciences, SMAI 2022
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