Paper Info
Title
Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization.
Abstract
In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set, and the objective function has a nonsmooth, nonconvex regularizer. Such a regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding, and non-Lipschitz L-p penalties as special cases. Using the theory of the generalized directional derivative and the tangent cone, we derive a first order necessary optimality condition for local minimizers of the problem, and define the generalized stationary point of it. We show that the generalized stationary point is the Clarke stationary point when the objective function is Lipschitz continuous at this point, and satisfies the existing necessary optimality conditions when the objective function is not Lipschitz continuous at this point. Moreover, we prove the consistency between the generalized directional derivative and the limit of the classic directional derivatives associated with the smoothing function. Finally, we establish a lower bound property for every local minimizer and show that finding a global minimizer is strongly NP-hard when the objective function has a concave regularizer.
Year
DOI
Venue
2017
10.1287/moor.2016.0837
MATHEMATICS OF OPERATIONS RESEARCH
Keywords
Field
DocType
constrained nonsmooth nonconvex optimization,optimality condition,generalized directional derivative,directional derivative consistency,numerical property
Mathematical optimization,Convex set,Regularization (mathematics),Smoothing,Feasible region,Stationary point,Lipschitz continuity,Tangent cone,Directional derivative,Mathematics
Journal
Volume
Issue
ISSN
42
4
0364-765X
Citations 
PageRank 
References 
6
0.44
18
Authors
2
Name
Order
Citations
PageRank
Wei Bian128614.65
Xiaojun Chen21298107.51