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Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach

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Abstract

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality function is first approximated by a piecewise linear DC function (difference of two convex functions) and a sequence of convex subproblems is then constructed by successively linearizing the concave terms of the DC function. Under some mild assumptions, we establish that any accumulation point of the sequence generated by the method is a KKT point of the DC approximation problem. We show that the basic algorithm can be refined by adding strengthening cuts in the subproblems. Finally, we report some preliminary computational results on cardinality-constrained portfolio selection problems.

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Authors and Affiliations

  1. School of Economics and Management, Tongji University, Shanghai, 200092, P.R. China

    Xiaojin Zheng

  2. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, P.R. China

    Xiaoling Sun

  3. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Duan Li

  4. Department of Decision Sciences and Risk Management Institute, National University of Singapore, Singapore, 117245, Singapore

    Jie Sun

Authors
  1. Xiaojin Zheng
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  2. Xiaoling Sun
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  3. Duan Li
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  4. Jie Sun
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Corresponding author

Correspondence to Xiaoling Sun.

Additional information

Dedicated to Professor Masao Fukushima in celebration of his 65th birthday.

This research was supported by the NSFC grants 10971034, 11101092 and the Joint NSFC/RGC grant 71061160506, and by Hong Kong RGC Grants CUHK414610 and 2050524.

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Zheng, X., Sun, X., Li, D. et al. Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach. Comput Optim Appl 59, 379–397 (2014). https://doi.org/10.1007/s10589-013-9582-3

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  • DOI: https://doi.org/10.1007/s10589-013-9582-3

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