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Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

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Abstract

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP).

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References

  1. Clarke F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York

    MATH  Google Scholar 

  2. Cottle R.W. (1966) Nolinear programs with positively bounded Jacobians. SIAM J. Appl. Math. 14, 147–158

    Article  MATH  MathSciNet  Google Scholar 

  3. Facchinei F., Pang J.S. (2003) Finte-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin Helidelberg New York

    Google Scholar 

  4. Fadell A.G. (1973) A generalization of Rademacher’s theorem on complete differential. Colloq. Math. 27, 126–131

    MathSciNet  Google Scholar 

  5. Fukushima M. (1992) Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Harrington J.E., Hobbs B.F., Pang J.S., Liu A., Roch G.(2005) Collusive game solutions via optimization. Math. Program. 104, 407–435

    Article  MATH  MathSciNet  Google Scholar 

  8. Huang L.R., Ng K.F. (2005) Equivalent Optimization Formulations and Error Bounds for Variational Inequality Problems. J. Optim. Theory Appl. 125, 299–314

    Article  MATH  MathSciNet  Google Scholar 

  9. Jiang H., Qi L. (1995) Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities. J. Math. Anal. Appl. 196, 314–331

    Article  MATH  MathSciNet  Google Scholar 

  10. King A.J., Rockafellar R.T. (1992) Sensitivity analysis for nonsmooth generalized equations. Math. Program. 55, 193–202

    Article  MathSciNet  Google Scholar 

  11. Kyparisis J. (1986) Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems. Math. Program. 36, 105–113

    MATH  MathSciNet  Google Scholar 

  12. Luc D.T. (2001) Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308

    Article  MATH  MathSciNet  Google Scholar 

  13. Luc D.T. (2002) The Frechet approximate jacobian and local uniqueness in variational inequalities. J. Math. Anal. Appl. 268, 629–646

    Article  MATH  MathSciNet  Google Scholar 

  14. Mordukhovich B.S. (1994) Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Am. Math. Soc. 343, 609–658

    Article  MATH  MathSciNet  Google Scholar 

  15. Ng K.F., Zheng X.Y. (2001) Error bound for lower semicontinuous functions in normed space. SIAM J. Optim. 12, 1–17

    Article  MATH  MathSciNet  Google Scholar 

  16. Pang J.S. (1994) Complementarity problems. In: Horst R., Pardalos P. (eds) Handbook on Global Optimization. Kluwer, Boston

    Google Scholar 

  17. Pang J.S., Fukushima M. (2005) Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 1, 21–56

    Article  MathSciNet  Google Scholar 

  18. Robinson S.M. (1983) Generalized equations. In: Bachem A., Grötschel M., Korte B. (eds) Mathematical Programming: The State of the Art. Springer, Berlin Helidelberg, New York, pp 346–367

    Google Scholar 

  19. Rockafellar R.T (1979) Directionally Lipschitzian functions and subdifferential calculus. Proc. Lond Math. Soc. 39, 331–355

    Article  MATH  MathSciNet  Google Scholar 

  20. Rockafellar R.T (1980) Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280

    MATH  MathSciNet  Google Scholar 

  21. Rockafellar R.T., Wets R.J-B. (1998) Variational Analysis. Springer, Berlin Helidelberg, New York

    MATH  Google Scholar 

  22. Solodov M.V., Tseng P. (1996) Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34, 1814–1830

    Article  MATH  MathSciNet  Google Scholar 

  23. Solodov M.V., Svaiter B.F.(1999) A new projection method for variational inequlity problems. SIAM J. Control Optim. 37, 765–776

    Article  MATH  MathSciNet  Google Scholar 

  24. Stampacchia G. (1964) Formes bilinaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416

    MATH  MathSciNet  Google Scholar 

  25. Ward D.E., Borwein J.M. (1987) Nonsmooth calculus in finite dimensions. SIAM J. Control Optim. 25, 1312–1340

    Article  MATH  MathSciNet  Google Scholar 

  26. Wu J.H., Florian M., Marcotte P. (1993) A general descent framework for the monotone variational inequality problem. Math. Program. 61, 281–300

    Article  MathSciNet  Google Scholar 

  27. Yamashita N., Taji K., Fukushima M. (1997) Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 92, 439–456

    Article  MATH  MathSciNet  Google Scholar 

  28. Zagrodny D. (1988) Approximate mean value theorem for upper subderivatives. Nonlinear Anal. 12, 1413–1428

    Article  MATH  MathSciNet  Google Scholar 

  29. Zalinescu C. (2002) Convex Analysis in General Vector Spaces. World Scientific, Singapore

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, People’s Republic of China

    Kung Fu Ng & Lu Lin Tan

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  1. Kung Fu Ng
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  2. Lu Lin Tan
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Correspondence to Lu Lin Tan.

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The research of this author was supported by an Earmarked Grant from the Research Council of Hong Kong.

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Ng, K.F., Tan, L.L. Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems. Math. Program. 110, 405–429 (2007). https://doi.org/10.1007/s10107-006-0007-2

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  • DOI: https://doi.org/10.1007/s10107-006-0007-2

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