Strongly Convex Functions, Moreau Envelopes, and the Generic Nature of Convex Functions with Strong Minimizers
By Chayne Planiden, Shawn Xianfu Wang. 2015
39 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis
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In Computational Theory and Mathematics
In Geometry and Topology
In Numerical Analysis
Citations in the corpus
- Personalized Federated Learning with Moreau Envelopes (2020, methodology): “Finally, we review several useful properties of the Moreau envelope such as smoothing and preserving convexity as follows (see the review and proof for the convex case in [40, 46, 47] and for nonconvex smooth case in [48], respectively): Proposition 1.”