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A first order method for finding minimal norm-like solutions of convex optimization problems

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Abstract

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values. The paper ends with several illustrating numerical examples.

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, 32000 , Haifa, Israel

    Amir Beck

  2. School of Mathematical Sciences, Tel-Aviv University, 69978 , Ramat-Aviv, Israel

    Shoham Sabach

Authors
  1. Amir Beck
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  2. Shoham Sabach
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Corresponding author

Correspondence to Amir Beck.

Additional information

The research of Amir Beck was partially supported by the Israel Science Foundation under Grant ISF No.253/12.

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Beck, A., Sabach, S. A first order method for finding minimal norm-like solutions of convex optimization problems. Math. Program. 147, 25–46 (2014). https://doi.org/10.1007/s10107-013-0708-2

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  • DOI: https://doi.org/10.1007/s10107-013-0708-2

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