DC formulations and algorithms for sparse optimization problems

We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest- k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem in...

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Bibliographic Details
Published in:Mathematical programming Vol. 169; no. 1; pp. 141 - 176
Main Authors: Gotoh, Jun-ya, Takeda, Akiko, Tono, Katsuya
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2018
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Constraints
Formulations
Full Length Paper
Iterative methods
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Penalty function
Theoretical
Sparse optimization
90C20
DCA
Ky Fan
Cardinality constraint
90C90
Rank constraint
Largest
90C26
Proximal operation
47A30
norm
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest- k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem into a penalty function form and derive exact penalty parameter values for some optimization problems, especially for quadratic minimization problems which often appear in practice. A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest- k norm. Furthermore, we can solve each DCA subproblem in linear time via a soft thresholding operation if there are no additional constraints. The framework is extended to the rank-constrained problem as well as the cardinality- and the rank-minimization problems. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1181-0