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Smoothing Newton Method for Minimizing the Sum of p-Norms

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  • S. H. Pan

    (South China University of Technology)

  • Y. X. Jiang

    (Dalian Jiaotong University)

Abstract

Consider the problem of minimizing the sum of p-norms, where p is a fixed real number in the interval [1,2]. This nondifferentiable problem arises in many applications, including the VLSI (very-large-scale-integration) layout problem, the facilities location problem and the Steiner minimum tree problem under a given topology. In this paper, we establish the optimality conditions, duality and uniqueness results for the problem. We then present a smoothing Newton method by the semismooth equations which are derived from the optimality conditions. The method is globally and superlinearly convergent, and moreover, it is quadratically convergent when p∈[1,3/2]∪{2}. Particularly, the quadratic convergence is proved for the case wherep∈(1,3/2]∪{2} without requiring strict complementarity. Preliminary numerical results are reported, which indicate that the method proposed is extremely promising.

Suggested Citation

  • S. H. Pan & Y. X. Jiang, 2008. "Smoothing Newton Method for Minimizing the Sum of p-Norms," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 255-275, May.
    • Handle: RePEc:spr:joptap:v:137:y:2008:i:2:d:10.1007_s10957-008-9364-8
    DOI: 10.1007/s10957-008-9364-8
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    References listed on IDEAS

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    1. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    2. Stephen M. Robinson, 1992. "Normal Maps Induced by Linear Transformations," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 691-714, August.
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