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Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem

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  • J. Burke

    (University of Washington)

  • S. Xu

    (University of Waterloo)

Abstract

We study the complexity of a noninterior path-following method for the linear complementarity problem. The method is based on the Chen–Harker–Kanzow–Smale smoothing function. It is assumed that the matrix M is either a P-matrix or symmetric and positive definite. When M is a P-matrix, it is shown that the algorithm finds a solution satisfying the conditions Mx-y+q=0 and $$\left\| {{\text{min\{ }}x,y{\text{\} }}} \right\|_\infty \leqslant \varepsilon $$ in at most $$\mathcal{O}((2 + \beta )(1 + (1/l(M)))^2 \log ((1 + (1/2)\beta )\mu _0 )/\varepsilon ))$$ Newton iterations; here, β and µ0 depend on the initial point, l(M) depends on M, and ɛ> 0. When Mis symmetric and positive definite, the complexity bound is $$\mathcal{O}((2 + \beta )C^2 \log ((1 + (1/2)\beta )\mu _0 )/\varepsilon ),$$ where $$C = 1 + (\sqrt n /(\min \{ \lambda _{\min } (M),1/\lambda _{\max } (M)\} ),$$ and $$\lambda _{\min } (M),\lambda _{\max } (M)$$ are the smallest and largest eigenvalues of M.

Suggested Citation

  • J. Burke & S. Xu, 2002. "Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 53-76, January.
  • Handle: RePEc:spr:joptap:v:112:y:2002:i:1:d:10.1023_a:1013040428127
    DOI: 10.1023/A:1013040428127
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    References listed on IDEAS

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    1. James V. Burke & Song Xu, 1998. "The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 719-734, August.
    2. Z.-Q. Luo & O. L. Mangasarian & J. Ren & M. V. Solodov, 1994. "New Error Bounds for the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 880-892, November.
    3. Houyuan Jiang, 1999. "Global Convergence Analysis of the Generalized Newton and Gauss-Newton Methods of the Fischer-Burmeister Equation for the Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 529-543, August.
    4. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
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    Cited by:

    1. Da Tian, 2014. "An entire space polynomial-time algorithm for linear programming," Journal of Global Optimization, Springer, vol. 58(1), pages 109-135, January.
    2. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.
    3. Jean-Pierre Dussault & Mathieu Frappier & Jean Charles Gilbert, 2019. "A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(4), pages 359-380, December.

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