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A unifying geometric solution framework and complexity analysis for variational inequalities

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Abstract

In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result, we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs.

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Authors and Affiliations

  1. Sloan School of Management and School of Engineering, MIT, 02139, Cambridge, MA, USA

    Thomas L. Magnanti

  2. Operations Research Center, MIT, 02139, Cambridge, MA, USA

    Thomas L. Magnanti & Georgia Perakis

Authors
  1. Thomas L. Magnanti
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  2. Georgia Perakis
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Additional information

Preparation of this paper was supported, in part, by NSF Grant 9312971-DDM from the National Science Foundation.

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Magnanti, T.L., Perakis, G. A unifying geometric solution framework and complexity analysis for variational inequalities. Mathematical Programming 71, 327–351 (1995). https://doi.org/10.1007/BF01590959

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  • DOI: https://doi.org/10.1007/BF01590959

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