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A one-parametric class of merit functions for the second-order cone complementarity problem

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Abstract

We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular Fischer–Burmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter τ equals 2, whereas as τ tends to zero, its limit will become a multiple of the natural residual merit function. In this paper, we show that this class of merit functions enjoys several favorable properties as the FB merit function holds, for example, the smoothness. These properties play an important role in the reformulation method of an unconstrained minimization or a nonsmooth system of equations for the SOCCP. Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions, which indicate that the FB merit function is not the best. For the sparse linear SOCPs, the merit function corresponding to τ=2.5 or 3 works better than the FB merit function, whereas for the dense convex SOCPs, the merit function with τ=0.1, 0.5 or 1.0 seems to have better numerical performance.

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Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, 88 Section 4, Ting-Chou Road, 11677, Taipei, Taiwan

    Jein-Shan Chen

  2. Mathematics Division, National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwain

    Jein-Shan Chen

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

    Shaohua Pan

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  1. Jein-Shan Chen
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  2. Shaohua Pan
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Correspondence to Jein-Shan Chen.

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Chen, JS., Pan, S. A one-parametric class of merit functions for the second-order cone complementarity problem. Comput Optim Appl 45, 581–606 (2010). https://doi.org/10.1007/s10589-008-9182-9

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  • DOI: https://doi.org/10.1007/s10589-008-9182-9

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