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Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming

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Abstract

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ++. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538–543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China

    S. H. Pan

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    J. S. Chen

Authors
  1. S. H. Pan
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  2. J. S. Chen
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Corresponding author

Correspondence to J. S. Chen.

Additional information

Communicated by M. Fukushima.

Research of Shaohua Pan was partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

Jein-Shan Chen is a member of the Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work was partially supported by National Science Council of Taiwan.

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Pan, S.H., Chen, J.S. Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming. J Optim Theory Appl 138, 95–113 (2008). https://doi.org/10.1007/s10957-008-9380-8

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