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Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

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Abstract

We consider a path following algorithm for solving linear complementarity problems with positive semi-definite matrices. This algorithm can start from any interior solution and attain a linear rate of convergence. Moreover, if the starting solution is appropriately chosen, this algorithm achieves a complexity of O(\(\sqrt m\) L}) iterations, wherem is the number of variables andL is the size of the problem encoding in binary. We present a simple complexity analysis for this algorithm, which is based on a new Lyapunov function for measuring the nearness to optimality. This Lyapunov function has itself interesting properties that can be used in a line search to accelerate convergence. We also develop an inexact line search procedure in which the line search stepsize is obtainable in a closed form. Finally, we extended this algorithm to handle directly variables which are unconstrained in sign and whose corresponding matrix is positive definite. The rate of convergence of this extended algorithm is shown to be independent of the number of such variables.

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Authors and Affiliations

  1. Center for Intelligent Control Systems, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Paul Tseng

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  1. Paul Tseng
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Additional information

This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation, grant NSF-ECS-8519058.

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Tseng, P. Complexity analysis of a linear complementarity algorithm based on a Lyapunov function. Mathematical Programming 53, 297–306 (1992). https://doi.org/10.1007/BF01585708

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  • DOI: https://doi.org/10.1007/BF01585708

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