Abstract
In this paper, we introduce a new class of smoothing functions, which include some popular smoothing complementarity functions. We show that the new smoothing functions possess a system of favorite properties. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P 0 function are discussed. The Jacobian consistency of this class of smoothing functions is analyzed. Based on the new smoothing functions, we investigate a smoothing Newton algorithm for the NCP and discuss its global and local superlinear convergence. Some preliminary numerical results are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Harker P.T., Pang J.S.: Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)
Ferris M.C., Pang J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
Chinchuluun A., Pardalos P.M., Migdalas A., Pitsoulis L.: Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)
Isac G.: Nonlinear analysis and complementarity theory. J. Glob. Optim. 40, 129–146 (2008)
Pardalos P.M., Ye Y.Y., Han C.G., Kaliski J.A.: Solution of P 0 matrix linear complementarity problems using reduction algorithm. SIAM J. Matrix Anal. Appl. 14, 1048–1060 (1993)
Ma C.F., Chen X.H.: The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function. J. Comput. Appl. Math. 216, 1–13 (2008)
Chen J.S.: The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem. J. Glob. Optim. 36, 565–580 (2006)
Huang Z.H., Han J., Xu D., Zhang L.: The non-linear continuation methods for solving the P 0-functions non-linear complementarity problem. Sci. China 44(2), 1107–1114 (2001)
Huang Z.H., Han J., Chen Z.: Predictor-corrector smoothing newton method, based on a new smoothing function, for solving the nonlinear complementarity problem with a P0 function. J. Optim. Theory Appl. 117(1), 39–68 (2003)
Qi L., Sun D., Zhou G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87(1), 1–35 (2000)
Yu H.D., Pu D.G.: Smoothing Newton method for NCP with the identification of degenerate indices. J. Comput. Appl. Math. 234, 3424–3435 (2010)
Chen J.S., Huang Z.H., She C.Y.: A new class of penalized NCP-functions and its properties. Comput. Optim. Appl. 50, 49–73 (2011)
Hu S.L., Huang Z.H., Chen J.S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69–82 (2009)
Kojima M., Megiddo N., Noma T.: Homotopy continuation methods for nonlinear complementarity problems. Math. Oper. Res. 16, 754–774 (1991)
Kanzow C., Pieper H.: Jacobian smoothing methods for nonlinear complementarity problems. SIAM J. Optim. 9, 342–373 (1999)
Zhang L.P., Wu S.Y., Gao T.R.: Improved smoothing Newton methods for P0 nonlinear complementarity problems. Appl. Math. Comput. 215, 324–332 (2009)
Kanzow C.: Some equation-based methods for the nonlinear complementarity problem. Optim. Methods Softw. 3, 327–340 (1994)
Jiang H., Qi L.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 35, 178–193 (1997)
Liu X., Wu W.: Coerciveness of some merit functions over symmetric cones. J. Ind. Manag. Optim. 5, 603–613 (2009)
Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)
Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Gowa M.S., Tawhid M.A.: Existence and limiting behavior of trajectories associated with P 0-equations. Comput. Optim. Appl. 12, 229–251 (1999)
Kojima M., Megiddo N., Noma T., Yoshise A.: A Unified Approach to Interior-Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science. Springer, Berlin (1991)
Chen X., Qi L.Q., Sun D.F.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67, 519–540 (1998)
Facchinei F., Kanzow C.: Beyond monotonicity in regularization methods for non-linear complementarity problems. SIAM J. Control Optim. 37, 1150–1161 (1999)
Fang L.: A new one-step smoothing Newton method for nonlinear complementarity problem with P0-function. Appl. Math. Comput. 216, 1087–1095 (2010)
Pang J.S., Gabriel S.A.: NE/SQP: a robust algorithm for the nonlinear complementarity problem. Math. Program. 60, 295–337 (1993)
Mangasarian O.L., Solodov M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Program. 62, 277–297 (1993)
Dirkse S.P., Ferris M.C.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319–345 (1995)
Oberlin C., Wright S.J.: An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems. Math. Program. 117, 355–386 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, J., Hao, B. A new class of smoothing functions and a smoothing Newton method for complementarity problems. Optim Lett 7, 481–497 (2013). https://doi.org/10.1007/s11590-011-0432-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0432-x