Complexity of a noninterior path-following method for the linear complementarity problem

  • Authors:

  • J. Burke;S. Xu

  • Affiliations:

  • Professor, Department of Mathematics, University of Washington, Seattle, Washington;Postdoctoral Assistant, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

  • Venue:
  • Journal of Optimization Theory and Applications
  • Year:
  • 2002

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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 ||min{x, Y}||∞≤ ε in at most O((2 +β)(1 + (1/l(M)))2 log((1 + (1/2)β)µ0)/ε)) Newton iterations; here, β and µ0 depend on the initial point, l(M) depends on M, and ∈ ≥ 0. When M is symmetric and positive definite, the complexity bound is O((2 +β)C2 log((1 + (1/2)β)µ0)/ε), where C= 1+ (√n/(min{λmin(M), 1/λmax(M)}), and λmin(M),λmax(M) are the smallest and largest eigenvalues of M.