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http://hdl.handle.net/10397/74150
| Title: | Optimality and complexity for constrained optimization problems with nonconvex regularization | Authors: | Bian, W Chen, X |
Issue Date: | Nov-2017 | Source: | Mathematics of operations research, Nov. 2017, v. 42, no. 4, p. 1063-1084 | 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 Lp 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. | Keywords: | Constrained nonsmooth nonconvex optimization Directional derivative consistency Generalized directional derivative Numerical property Optimality condition |
Publisher: | Institute for Operations Research and the Management Sciences | Journal: | Mathematics of operations research | ISSN: | 0364-765X | EISSN: | 1526-5471 | DOI: | 10.1287/moor.2016.0837 | Rights: | Copyright © 2017, INFORMS This is the accepted manuscript of the following article: Bian, W., & Chen, X. (2017). Optimality and complexity for constrained optimization problems with nonconvex regularization. Mathematics of Operations Research, 42(4), 1063-1084, which has been published in final form at https://doi.org/10.1287/moor.2016.0837 |
| Appears in Collections: | Journal/Magazine Article |
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