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An Analytic Center Cutting Plane Method for Solving Semi-Infinite Variational Inequality Problems

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Abstract

We study a variational inequality problem VI(X,F) with X being defined by infinitely many inequality constraints and F being a pseudomonotone function. It is shown that such problem can be reduced to a problem of finding a feasible point in a convex set defined by infinitely many constraints. An analytic center based cutting plane algorithm is proposed for solving the reduced problem. Under proper assumptions, the proposed algorithm finds an ∈-optimal solution in O*(n 2/ρ2) iterations, where O*(·) represents the leading order, n is the dimension of X, ε is a user-specified tolerance, and ρ is the radius of a ball contained in the ε-solution set of VI(X,F).

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References

  1. Fang, S.-C. and Tsao, H.-S. J. (1997), Entropy Optimization and Mathematical Programming, Kluwer Academic Publishers, Boston, MA.

    Google Scholar 

  2. Goffin, J.-L., Luo, Z.-Q. and Ye, Y. (1993), On the complexity of a column generation algorithm for convex or quasiconvex feasibility problems, in: Large Scale Optimization: State of The Art, Hager, W.W., Hearn, D.W. and Pardalos, P.M. (eds.), Kluwer Academic Publishers, New York.

    Google Scholar 

  3. Goffin, J-L., Luo, Z. and Ye, Y. (1996), Complexity analysis of an interior cutting plane method for convex feasibility problems, SIAM Journal on Optimization 6, 638–652.

    Google Scholar 

  4. Goffin, J.-L., Marcotte, P. and Zhu, D. (1997), An analytic center cutting plane method for pseudomonotone variational inequalities, Operations Research Letters 20, 1–6.

    Google Scholar 

  5. Goffin, J.-L. and Vial, J.-P. (2002), Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method, Optimization Methods and Software 179, 805–867.

    Google Scholar 

  6. Goffin, J.-L. and Vial, J.-P. (2001), Multiple cuts in analytic cutting plane method, SIAM Journal on Optimization 11, 266–288.

    Google Scholar 

  7. Harker, P. T. and Pang, J.-S. (1990), Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming 48, 161–220.

    Google Scholar 

  8. Holmes, R.B. (1975), Geometric Functional Analysis and Its Applications, Springer, New York.

    Google Scholar 

  9. Luo, Z.-Q. Analysis of a cutting plane method that uses weighted analytic center and multiple cuts, SIAM J. Optimization 7, 697–716.

  10. Luo, Z.-Q. and Sun, J. (1999), An analytic center based column generation algorithm for convex quadratic feasibility problems, SIAM J. Optimization 9, 217–235.

    Google Scholar 

  11. Luo, Z.-Q. and Sun, J. (2000), A polynomial cutting surfaces algorithm for the convex feasibility problem defined by self-concordant inequalities, Computational Optimization and Applications 15, 167–191.

    Google Scholar 

  12. Lüthi, H.-J. and Büeler, B. (2000), An analytical center quadratic cut method for strongly monotone variational inequality problems, SIAM J. Optimization 10, 415–426.

    Google Scholar 

  13. Magnanti, T. L. and Perakis, G. (1995), A unifying geometric solution framework and complexity analysis for variational inequalities, Mathematical Programming 71, 327–351.

    Google Scholar 

  14. Marcotte, P. and Zhu, D. (2001), A cutting plane method for solving quasimonotone variational inequalities, Computational Optimization and Applications 20, 317–324.

    Google Scholar 

  15. Mokhtarian, F. S. and Goffin, J-L. An analytic center quadratic cut method for the convex quadratic feasibility problem, to appear in Mathematical Programming.

  16. Nesterov, Y. and Vial, J.-P. (1999), Homogeneous analytic center cutting plane methods for convex problems and variational inequalities, SIAM J. Optimization 9, 707–728.

    Google Scholar 

  17. Rockafellar, R.T., (1970), Convex Analysis, Princeton University Press, N.J.

    Google Scholar 

  18. Ye, Y. (1997), Complexity analysis of the analytic center cutting plane method that uses multiple cuts, Mathematical Programming 78, 85–104.

    Google Scholar 

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Authors and Affiliations

  1. Industrial Engineering and Operations Research, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Shu-Cherng Fang

  2. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan ROC;700 (E-mail

    Soon-Yi Wu

  3. Department of Decision Sciences and Singapore-MIT Alliance, National University of Singapore, Republic of Singapore

    Jie Sun

Authors
  1. Shu-Cherng Fang
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  2. Soon-Yi Wu
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  3. Jie Sun
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Fang, SC., Wu, SY. & Sun, J. An Analytic Center Cutting Plane Method for Solving Semi-Infinite Variational Inequality Problems. Journal of Global Optimization 28, 141–152 (2004). https://doi.org/10.1023/B:JOGO.0000015308.49676.ea

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  • DOI: https://doi.org/10.1023/B:JOGO.0000015308.49676.ea

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