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A smoothing homotopy method for variational inequality problems on polyhedral convex sets

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  • Zhengyong Zhou
  • Bo Yu

Abstract

In this paper, based on the Robinson’s normal equation and the smoothing projection operator, a smoothing homotopy method is presented for solving variational inequality problems on polyhedral convex sets. We construct a new smoothing projection operator onto the polyhedral convex set, which is feasible, twice continuously differentiable, uniformly approximate to the projection operator, and satisfies a special approximation property. It is computed by solving nonlinear equations in a neighborhood of the nonsmooth points of the projection operator, and solving linear equations with only finite coefficient matrices for other points, which makes it very efficient. Under the assumption that the variational inequality problem has no solution at infinity, which is a weaker condition than several well-known ones, the existence and global convergence of a smooth homotopy path from almost any starting point in $$R^n$$ are proven. The global convergence condition of the proposed homotopy method is same with that of the homotopy method based on the equivalent KKT system, but the starting point of the proposed homotopy method is not necessarily an interior point, and the efficiency is more higher. Preliminary test results show that the proposed method is practicable, effective and robust. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Zhengyong Zhou & Bo Yu, 2014. "A smoothing homotopy method for variational inequality problems on polyhedral convex sets," Journal of Global Optimization, Springer, vol. 58(1), pages 151-168, January.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:1:p:151-168
    DOI: 10.1007/s10898-013-0033-6
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    References listed on IDEAS

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    1. Qing Xu & Bo Yu & Guo-Chen Feng, 2005. "Homotopy Methods for Solving Variational Inequalities in Unbounded Sets," Journal of Global Optimization, Springer, vol. 31(1), pages 121-131, January.
    2. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    3. L. Qi & D. Sun, 2002. "Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 121-147, April.
    4. Y. B. Zhao & J. Y. Han & H. D. Qi, 1999. "Exceptional Families and Existence Theorems for Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 475-495, May.
    5. Stephen M. Robinson, 1992. "Normal Maps Induced by Linear Transformations," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 691-714, August.
    6. Z. Lin & Y. Li, 1999. "Homotopy Method for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 100(1), pages 207-218, January.
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    2. Yang Zhan & Peixuan Li & Chuangyin Dang, 2020. "A differentiable path-following algorithm for computing perfect stationary points," Computational Optimization and Applications, Springer, vol. 76(2), pages 571-588, June.
    3. Cao, Yiyin & Dang, Chuangyin & Xiao, Zhongdong, 2022. "A differentiable path-following method to compute subgame perfect equilibria in stationary strategies in robust stochastic games and its applications," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1032-1050.
    4. Biao Qu & Changyu Wang & Naihua Xiu, 2017. "Analysis on Newton projection method for the split feasibility problem," Computational Optimization and Applications, Springer, vol. 67(1), pages 175-199, May.

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