STRONGLY CONVEX FUNCTIONS, MOREAU ENVELOPES, AND THE GENERIC NATURE OF CONVEX FUNCTIONS WITH STRONG MINIMIZERS

被引：20
|
作者：
Planiden, C. [1 ]
Wang, X. [1 ]
机构：
[1] Univ British Columbia Okanagan, Math, Kelowna, BC V1V 1V7, Canada
来源：
SIAM JOURNAL ON OPTIMIZATION | 2016年 / 26卷 / 02期
基金：
加拿大自然科学与工程研究理事会;
关键词：
Attouch-Wets metric; Baire category; complete metric space; convex function; epi-convergence; epi-topology; generic set; meager set; Moreau envelope; proximal mapping; strong minimizer; strongly convex; EPI-DISTANCE TOPOLOGY; OPTIMIZATION PROBLEMS; MONOTONE-OPERATORS; SPACES;
D O I：
10.1137/15M1035550
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions in a finite-dimensional space. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category.
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页码：1341 / 1364
页数：24
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