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<p>A generalized version of an important theorem called Hoffman’s lemma in the book by Bonnans and Shapiro (Perturbation analysis of optimization problems, Springer, Berlin, 2000), which deals with generalized polyhedral convex multifunctions, is obtained in this paper. Under a mild assumption, the result allows us to demonstrate that the domain of a generalized polyhedral convex multifunction is closed and the multifunction is Lipschitz continuous on its effective domain. As concrete applications of the results, we prove some local error bounds for generalized affine variational inequalities and a theorem on the (strong) convergence of feasible descent methods for solving generalized quadratic programming problems.</p>

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A Generalization of Hoffman’s Lemma in Banach Spaces and Applications

  • 期刊论文

  • 发表日期：

    2025年2月24日

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  • Nguyen Quang Huy,
  • Hoang Ngoc Tuan,
  • Nguyen Dong Yen

摘要

A generalized version of an important theorem called Hoffman’s lemma in the book by Bonnans and Shapiro (Perturbation analysis of optimization problems, Springer, Berlin, 2000), which deals with generalized polyhedral convex multifunctions, is obtained in this paper. Under a mild assumption, the result allows us to demonstrate that the domain of a generalized polyhedral convex multifunction is closed and the multifunction is Lipschitz continuous on its effective domain. As concrete applications of the results, we prove some local error bounds for generalized affine variational inequalities and a theorem on the (strong) convergence of feasible descent methods for solving generalized quadratic programming problems.