Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings

Author:

Levy A. B.,Poliquin R.,Thibault L.

Abstract

Attouch’s Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch’s Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch’s Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference40 articles.

1. Applicable Mathematics Series;Attouch, H.,1984

2. Epigraphical analysis;Attouch, H.;Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire,1989

3. Epigraphical convergence of functions and convergence of their derivatives in Banach spaces;Attouch, H.;S\'{e}m. Anal. Convexe,1990

4. The topology of the 𝜌-Hausdorff distance;Attouch, Hédy;Ann. Mat. Pura Appl. (4),1991

5. On the convergence of subdifferentials of convex functions;Attouch, Hédy;Arch. Math. (Basel),1993

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. V -Prox-Regular Functions in Smooth Banach Spaces;Journal of Function Spaces;2020-11-24

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3