Date of Award

June 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Lixin Shen

Keywords

Convex Analysis, Moreau Envelope, Optimization, Sparse Optimization, Thresholding Operator

Subject Categories

Physical Sciences and Mathematics

Abstract

Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing.

Access

Open Access

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