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

Author

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  • S. H. Pan

    (South China University of Technology)

  • J. S. Chen

    (National Taiwan Normal University)

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.

Suggested Citation

  • S. H. Pan & J. S. Chen, 2008. "Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 95-113, July.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:1:d:10.1007_s10957-008-9380-8
    DOI: 10.1007/s10957-008-9380-8
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    References listed on IDEAS

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    1. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    2. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    3. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
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