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Regularized Gap Functions and D-Gap Functions for Nonsmooth Variational Inequalities

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Optimization and Related Topics

Part of the book series: Applied Optimization ((APOP,volume 47))

Abstract

In this paper, we investigate the regularized gap function due to Auchmuty (1989) and Fukushima (1992), and the D-gap function due to Peng (1997) for a nonsmooth variational inequality problem where the mapping involved is Lipschitz continuous but not necessarily continuously differentiable. Under the semismoothness and strong monotonicity, we prove that the solutions of the nonsmooth variational inequality problem coincide with the stationary points of the gap functions. A descent algorithm is proposed for finding a stationary point of the D-gap function. The algorithm differs from the usual nonsmooth descent algorithms in that at each step the computation of a descent direction involves merely the function value. In order to improve the convergence locally, a generalized Newton’s algorithm is proposed. Finally, initial numerical results are reported.

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References

  • Auchmuty, G. (1989), Variational principals for variational inequalities, Numerical Functional Analysis and Optimization, Vol. 10, 863–874.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Demyanov, V.F. (1968), Algorithms for some minimax problems, Journal of Computer and System Sciences, Vol. 2, 342–380.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferris, M.C. and Pang, J.-S. (1997), Engineering and economic applications of complementarity problems, SIAM Review, Vol. 39, 669–713.

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer, A. (1997), Solution of monotone complementarity problems with lo- cally Lipschitzian functions, Mathematical Programming, Vol. 76, 513–532.

    MathSciNet  MATH  Google Scholar 

  • Fukushima, M. (1992), Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, Vol. 53, 101–110.

    Article  MathSciNet  Google Scholar 

  • Fukushima, M. (1996), Merit functions for variational inequality and complementarity problems, In: Di Pillo,. G. and Giannessi, F. (eds.), Nonlinear Optimization and Applications, Plenum Publishing Corporation, New York, 155–170.

    Google Scholar 

  • 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, Vol. 48, 161–220.

    Article  MathSciNet  MATH  Google Scholar 

  • Jiang, H. (1996), Unconstrained minimization approaches to nonlinear complementarity problems, Journal of Global Optimization, Vol. 9, 169–181.

    Article  MATH  Google Scholar 

  • Jiang, H. and Qi, L. (1995), Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities, Journal of Mathematical Analysis and Applications, Vol. 196, 314–331.

    Article  MathSciNet  MATH  Google Scholar 

  • Kanzow, C. and Fukushima, M., Theoretical and numerical Investigation of the D-gap function for box constrained variational inequalities, to appear in Mathematical Programming.

    Google Scholar 

  • Kanzow, C. and Fukushima, M. (1997), Solving box constrained variational inequalities by using the natural residual with D-gap function globalization, preprint.

    Google Scholar 

  • Kiwiel, K.C. (1985), Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1133.

    MATH  Google Scholar 

  • Lemarechal, C. (1980), Extensions Diverses des Methodes de Gradient et Application, These d’etat, Paris.

    Google Scholar 

  • Luo, Z.-Q., Pang, J.-S. and Ralph, D. (1996), Mathematical Programs with Equilibrium Constraints,Cambridge University Press.

    Google Scholar 

  • Mangsarian, O.L. and Solodov, M.V. (1993), Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming, Vol. 62, 277–297.

    Article  MathSciNet  Google Scholar 

  • Mayne, D.Q. and Polak, E. (1984), Non-differentiable optimization adaptive smoothing, Journal of Optimization Theory and Applications, Vol. 43, 601–613.

    Article  MathSciNet  MATH  Google Scholar 

  • Pang, J.-S. (1994), Complementarity problems, In: Handbook of Global Optimization, Horst, R. and Pardalos, P.M. (eds.), Kluwer Academic Publishers, Norwell, Massachusetts, 271–338.

    Google Scholar 

  • Peng, J.M. (1997), Equivalence of variational inequality problems to unconstrained optimization, Mathematical Programming, Vol. 78, 347–355.

    MathSciNet  MATH  Google Scholar 

  • Peng, J.M. and Yuan, Y.X. (1995), Unconstrained Methods for Generalized Complementarity Problems, Technical Report, State Key Laboratory of Scientific and Engineering Computing, Academia Sinica, Beijing, China.

    Google Scholar 

  • Polak, E., Mayne, D.Q. and Wardi, Y. (1983), On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems, SIAM Journal of Control and Optimization, Vol. 21, 179–203.

    Article  MathSciNet  MATH  Google Scholar 

  • Qi, L. (1993), Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, Vol. 18, 227–244.

    Article  MathSciNet  MATH  Google Scholar 

  • Qi, L. and Sun, J. (1993), A nonsmooth version of Newton’s method, Mathematical Programming, Vol. 58, 353–367.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, D., Fukushitna, M. and Qi, L. (1997), A computable generalized Hessian of D-gap function and Newton-type methods for variational inequality problems, In: Ferris, M.C. and Pang, J.S. (eds.), Complementarity and Variational Problems: State of Art, SIAM Publications, Philadelphia, 452–473.

    Google Scholar 

  • Taji, K., Fukushima, M. and Ibaraki, T. (1993), A globally convergent Newton’s method for solving strongly monotone variational inequalities, Mathematical Programming, Vol. 58, 369–383.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, H. and Chang, X. (1997), Approximate Newton methods for nonsmooth equations, Journal of Optimization Theory and Applications, Vol. 93, 373–394.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, H. (1999), Set-valued approximation and Newton’s methods, Mathematical Programming, Vol. 84, 401–420.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, H., Rubinov, A.M. and Glover, B.M. (1999), Continuous approximations to generalized Jacobians, Optimization, Vol. 46, 221–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, H. (1998), Adaptive smoothing methods, deterministically differentiable Jacobians and Newton’s methods, preprint, Department of Mathematics and Statistics, The University of Melbourne, Parkville, Australia.

    Google Scholar 

  • Yamashita, N., Taji, K. and Fukushima, M. (1997), Unconstrained optimization reformulation of variational inequality problems, Journal of Optimization Theory and Optimization, Vol. 92, 439–456.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Australian Graduate School of Management, The University of New South Wales, Sydney, NSW, 2052, Australia

    Huifu Xu

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  1. Huifu Xu
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Editor information

Editors and Affiliations

  1. School of Information Technology & Mathematical Sciences, University of Ballarat, Victoria, Australia

    Alexander Rubinov

  2. School of Mathematics and Statistics, Curtin University of Technology, WA, Australia

    Barney Glover

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© 2001 Springer Science+Business Media Dordrecht

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Xu, H. (2001). Regularized Gap Functions and D-Gap Functions for Nonsmooth Variational Inequalities. In: Rubinov, A., Glover, B. (eds) Optimization and Related Topics. Applied Optimization, vol 47. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6099-6_11

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  • DOI: https://doi.org/10.1007/978-1-4757-6099-6_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4844-1

  • Online ISBN: 978-1-4757-6099-6

  • eBook Packages: Springer Book Archive

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