Abstract
Let f and g be continuously differentiable functions on R n. The nonlinear complementarity problem NCP(f,g), 0≤f(x)⊥g(x)≥0, arises in many applications including discrete Hamilton-Jacobi-Bellman equations and nonsmooth Dirichlet problems. A popular method to find a solution of the NCP(f,g) is the generalized Newton method which solves an equivalent system of nonsmooth equations F(x)=0 derived by an NCP function. In this paper, we present a sufficient and necessary condition for F to be Fréchet differentiable, when F is defined by the “min” NCP function, the Fischer-Burmeister NCP function or the penalized Fischer-Burmeister NCP function. Moreover, we give an explicit formula of an element in the Clarke generalized Jacobian of F defined by the “min” NCP function, and the B-differential of F defined by other two NCP functions. The explicit formulas for generalized differentials of F lead to sharper global error bounds for the NCP(f,g).
Similar content being viewed by others
References
Alefeld, G., Chen, X.: A regularized projection method for complementarity problems with non-Lipschitzian functions. Math. Comput. 77, 379–395 (2008)
Alefeld, G., Wang, Z.: Error bounds for complementarity problems with tridiagonal nonlinear functions. Computing 83, 175–192 (2008)
Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam (1982)
Berman, A., Plemmons, R.J.: Nonnegative Matrix in the Mathematical Sciences. SIAM, Philadelphia (1994)
Chen, B.: Error bounds for R 0-type and monotone nonlinear complementarity problems. J. Optim. Theory Appl. 108, 297–316 (2001)
Chen, X.: First order conditions for nonsmooth discretized constrained optimal control problems. SIAM J. Control Optim. 42, 2004–2015 (2004)
Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)
Chen, X., Xiang, S.: Computation of error bounds for P-matrix linear complementarity problems. Math. Program. 106, 513–525 (2006)
Chen, X., Xiang, S.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18, 1250–1265 (2007)
Chen, X., Ye, Y.: On homotopy-smoothing methods for box-constrained variational inequalities. SIAM J. Control Optim. 37, 589–616 (1999)
Chen, X., Nashed, Z., Qi, L.: Convergence of Newton’s method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim. 7, 445–462 (1997)
Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67, 519–540 (1998)
Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function. Math. Program. 88, 211–216 (2000)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)
De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)
De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16, 173–205 (2000)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Ferris, M.C., Mangasarian, O.L.: Error bounds and strong upper semicontinuity for monotone affine variational inequalities. Ann. Oper. Res. 47, 293–305 (1993)
Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)
Fischer, A.: An NCP-function and its use for the solution of complementarity problems. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 11–105. World Scientific, Singapore (1995)
Gao, Y.: Newton methods for solving nonsmooth equations via a new subdifferential. Math. Methods Oper. Res. 54, 239–257 (2001)
García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 22, 1071–1075 (2009)
Gowda, M.S., Tawhid, N.A.: Existence and limiting behavior of trajectories associated with P 0-equations. Comput. Optim. Appl. 12, 229–251 (1999)
Hanker, P.T., Xiao, B.: Newton’s method for the nonlinear complementarity problem: A B-differentiable approach. Math. Program. 48, 339–357 (1990)
Hoppe, R.H.: Multigrid methods for Hamilton-Jacobi-Bellman equations. Numer. Math. 49, 239–254 (1986)
Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer Academic, Amsterdam (2002)
Mangasarian, O.L., Shiau, T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J. Control Optim. 25, 583–595 (1987)
Mathias, R., Pang, J.-S.: Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra Appl. 132, 123–136 (1990)
Pang, J.-S.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–341 (1990)
Pang, J.-S.: A B-differentiable equation-based, globally, and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems. Math. Program. 51, 101–131 (1991)
Pang, J.-S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)
Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 295–337 (1993)
Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)
Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Robinson, S.M.: Local structure of feasible sets in nonlinear programming. Part III. Stability and sensitivity. Math. Program. Stud. 30, 45–66 (1987)
Robinson, S.M.: An implicit function theorem for B-differentiable functions. Industrial Engineering Working Paper, University of Wisconsin (Madison, WI, 1998)
Robinson, S.M.: Newton’s method for a class of nonsmooth functions. Industrial Engineering Working Paper, University of Wisconsin (Madison, WI, 1998)
Rudin, W.: Functional Analysis. McGraw-Hill Science, New York (1991)
Śmietanski, M.J.: A generalized Jocobian based Newton method for semismooth block-triangular system of equations. J. Comput. Appl. Math. 205, 305–313 (2007)
Sun, J., Han, J.: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7, 463–480 (1997)
Teng, P.: Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89, 17–37 (1996)
Wang, Z., Yuan, Y.: A new residual function and componentwise error estimation for linear complementarity problem. IMA J. Numer. Anal. To appear
Xiu, N., Zhang, J.: A characteristic quantity of P-matrices. Appl. Math. Lett. 15, 41–46 (2002)
Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627–649 (2009)
Zhang, C., Chen, X., Xiu, N.: Global error bounds for the extended vertical LCP. Comput. Optim. Appl. 42, 335–352 (2009)
Zhou, S.Z., Zou, Z.: A new iterative method for discrete HJB equations. Numer. Math. 111, 159–167 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicate to Professor Liqun Qi on his 65 birthday.
This paper is supported partly by the Hong Kong Research Grant Council, NSF of China (Nos. 10771218, 11071260) and the Program for New Century Excellent Talents in University, State Education Ministry, China.
Rights and permissions
About this article
Cite this article
Xiang, S., Chen, X. Computation of generalized differentials in nonlinear complementarity problems. Comput Optim Appl 50, 403–423 (2011). https://doi.org/10.1007/s10589-010-9349-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-010-9349-z