## Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

HTML articles powered by AMS MathViewer

- by X. Chen, L. Qi and D. Sun PDF
- Math. Comp.
**67**(1998), 519-540 Request permission

## Abstract:

The smoothing Newton method for solving a system of nonsmooth equations $F(x)=0$, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the $k$th step, the nonsmooth function $F$ is approximated by a smooth function $f(\cdot , \varepsilon _k)$, and the derivative of $f(\cdot , \varepsilon _k)$ at $x^k$ is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if $F$ is semismooth at the solution and $f$ satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is $P$–uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).## References

- O. Axelsson and I.E. Kaporin,
*On the solution of nonlinear equations for nondifferentiable mappings*, Technical Report, Department of Mathematics, University of Nijmegen, The Netherlands, 1994. - S.C. Billups, S.P. Dirkse and M.C. Ferris,
*A comparison of algorithms for large-scale mixed complementarity problems*, Comp. Optim. Appl., 7 (1997), pp. 3-25. - B. Chen and P. Harker,
*Smooth approximations to nonlinear complementarity problems*, SIAM J. Optim., 7 (1997), pp. 403-420. - Chunhui Chen and O. L. Mangasarian,
*A class of smoothing functions for nonlinear and mixed complementarity problems*, Comput. Optim. Appl.**5**(1996), no. 2, 97–138. MR**1373293**, DOI 10.1007/BF00249052 - Xiao Jun Chen and Li Qun Qi,
*A parameterized Newton method and a quasi-Newton method for nonsmooth equations*, Comput. Optim. Appl.**3**(1994), no. 2, 157–179. MR**1273659**, DOI 10.1007/BF01300972 - Xiao Jun Chen and Tetsuro Yamamoto,
*Convergence domains of certain iterative methods for solving nonlinear equations*, Numer. Funct. Anal. Optim.**10**(1989), no. 1-2, 37–48. MR**978801**, DOI 10.1080/01630568908816289 - Frank H. Clarke,
*Optimization and nonsmooth analysis*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR**709590** - T. De Luca, F. Facchinei and C. Kanzow,
*A semismooth equation approach to the solution of nonlinear complementarity problems,*Math. Programming, 75 (1996), pp. 407-439. - S.P. Dirkse and M.C. Ferris,
*The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems*, Optim. Method. Softw., 5 (1995), pp. 123-156. - B. C. Eaves,
*On the basic theorem of complementarity*, Math. Programming**1**(1971), no. 1, 68–75. MR**287901**, DOI 10.1007/BF01584073 - F. Facchinei, A. Fischer and C. Kanzow,
*Inexact Newton methods for semismooth equations with applications to variational inequality problems,*in: G. Di Pillo and F. Giannessi, eds., “Nonlinear Optimization and Applications”, Plenum Press, New York, 1996, pp. 155-170. - F. Facchinei, A. Fischer and C. Kanzow,
*A semismooth Newton method for variational inequalities: Theoretical results and preliminary numerical experience,*Preprint 102, Institute of Applied Mathematics, University of Hamburg, Hamburg, December 1995. - F. Facchinei, A. Fischer and C. Kanzow,
*A semismooth Newton method for variational inequalities: The case of box constraints*, in: M.C. Ferris and J.S. Pang, eds., “Complementarity and Variational Problems: State of the Art,” SIAM, Philadelphia, Pennsylvania, 1997, pp. 76-90. - F. Facchinei and C. Kanzow,
*A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems,*Math. Programming, 76 (1997), pp. 493-512. - A. Fischer,
*A special Newton-type optimization method*, Optimization**24**(1992), no. 3-4, 269–284. MR**1247636**, DOI 10.1080/02331939208843795 - A. Fischer,
*An NCP-function and its use for the solution of complementarity problems,*in: D. Du, L. Qi and R. Womersley, eds., “Recent Advances in Nonsmooth Optimization”, World Scientific Publishers, New Jersey, 1995, pp. 88-105. - M. Fukushima,
*Merit functions for variational inequality and complementarity problems,*in: G. Di Pillo and F. Giannessi eds., “Nonlinear Optimization and Applications”, Plenum Publishing Corporation, New York, 1996, pp. 155-170. - S.A. Gabriel and J.J. Moré,
*Smoothing of mixed complementarity problems*, in: M.C. Ferris and J.S. Pang, eds., “Complementarity and Variational Problems: State of the Art,” SIAM, Philadelphia, Pennsylvania, 1997, pp. 105-116. - M. Seetharama Gowda and Roman Sznajder,
*The generalized order linear complementarity problem*, SIAM J. Matrix Anal. Appl.**15**(1994), no. 3, 779–795. MR**1282694**, DOI 10.1137/S0895479892237859 - Patrick T. Harker and Jong-Shi Pang,
*Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications*, Math. Programming**48**(1990), no. 2, (Ser. B), 161–220. MR**1073707**, DOI 10.1007/BF01582255 - Patrick T. Harker and Baichun Xiao,
*Newton’s method for the nonlinear complementarity problem: a $\textrm {B}$-differentiable equation approach*, Math. Programming**48**(1990), no. 3, (Ser. B), 339–357. MR**1078191**, DOI 10.1007/BF01582262 - M. Heinkenschloss, C. T. Kelley, and H. T. Tran,
*Fast algorithms for nonsmooth compact fixed-point problems*, SIAM J. Numer. Anal.**29**(1992), no. 6, 1769–1792. MR**1191145**, DOI 10.1137/0729099 - Chi Ming Ip and Jerzy Kyparisis,
*Local convergence of quasi-Newton methods for $B$-differentiable equations*, Math. Programming**56**(1992), no. 1, Ser. A, 71–89. MR**1175560**, DOI 10.1007/BF01580895 - G. Isac and M. M. Kostreva,
*The generalized order complementarity problem, models and iterative methods*, Ann. Oper. Res., 44 (1993), pp. 63-92. - H. Jiang and L. Qi,
*A new nonsmooth equations approach to nonlinear complementarity problems,*SIAM J. Control Optim., 35 (1997), pp. 178-193. - H. Jiang, L. Qi, X. Chen and D. Sun,
*Semismoothness and superlinear convergence in nonsmooth optimization and nonsmooth equations,*in: G. Di Pillo and F. Giannessi eds., “Nonlinear Optimization and Applications”, Plenum Press, New York, 1996, pp. 197-212. - C. Kanzow and M. Fukushima,
*Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,*Math. Programming, to appear. - C. Kanzow and H. Jiang,
*A continuation method for (strongly) monotone variational inequalities,*Math. Programming, to appear. - Bernd Kummer,
*Newton’s method for nondifferentiable functions*, Advances in mathematical optimization, Math. Res., vol. 45, Akademie-Verlag, Berlin, 1988, pp. 114–125. MR**953328** - Z.Q. Luo and P. Tseng,
*A new class of merit functions for the nonlinear complementarity problem*, in: M.C. Ferris and J.S. Pang, eds., “Complementarity and Variational Problems: State of the Art,” SIAM, Philadelphia, Pennsylvania, 1997, pp. 204-225. - José Mario Martínez and Li Qun Qi,
*Inexact Newton methods for solving nonsmooth equations*, J. Comput. Appl. Math.**60**(1995), no. 1-2, 127–145. Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993). MR**1354652**, DOI 10.1016/0377-0427(94)00088-I - J. M. Ortega and W. C. Rheinboldt,
*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810** - Jiří V. Outrata,
*On optimization problems with variational inequality constraints*, SIAM J. Optim.**4**(1994), no. 2, 340–357. MR**1273763**, DOI 10.1137/0804019 - Jiří V. Outrata and Jochem Zowe,
*A Newton method for a class of quasi-variational inequalities*, Comput. Optim. Appl.**4**(1995), no. 1, 5–21. MR**1314522**, DOI 10.1007/BF01299156 - Jong-Shi Pang,
*Newton’s method for $B$-differentiable equations*, Math. Oper. Res.**15**(1990), no. 2, 311–341. MR**1051575**, DOI 10.1287/moor.15.2.311 - Jong-Shi Pang,
*A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems*, Math. Programming**51**(1991), no. 1, (Ser. A), 101–131. MR**1119247**, DOI 10.1007/BF01586928 - Jong-Shi Pang,
*Complementarity problems*, Handbook of global optimization, Nonconvex Optim. Appl., vol. 2, Kluwer Acad. Publ., Dordrecht, 1995, pp. 271–338. MR**1377087**, DOI 10.1007/978-1-4615-2025-2_{6} - Jong-Shi Pang and Li Qun Qi,
*Nonsmooth equations: motivation and algorithms*, SIAM J. Optim.**3**(1993), no. 3, 443–465. MR**1230150**, DOI 10.1137/0803021 - Jong-Shi Pang and Jen Chih Yao,
*On a generalization of a normal map and equation*, SIAM J. Control Optim.**33**(1995), no. 1, 168–184. MR**1311665**, DOI 10.1137/S0363012992241673 - Li Qun Qi,
*Convergence analysis of some algorithms for solving nonsmooth equations*, Math. Oper. Res.**18**(1993), no. 1, 227–244. MR**1250115**, DOI 10.1287/moor.18.1.227 - L. Qi,
*C-differentiability, C-Differential operators and generalized Newton methods*, AMR 96/5, Applied Mathematics Report, University of New South Wales, Sydney, 1996. - Li Qun Qi and Xiao Jun Chen,
*A globally convergent successive approximation method for severely nonsmooth equations*, SIAM J. Control Optim.**33**(1995), no. 2, 402–418. MR**1318657**, DOI 10.1137/S036301299223619X - Li Qun Qi and Jie Sun,
*A nonsmooth version of Newton’s method*, Math. Programming**58**(1993), no. 3, Ser. A, 353–367. MR**1216791**, DOI 10.1007/BF01581275 - Daniel Ralph,
*Global convergence of damped Newton’s method for nonsmooth equations via the path search*, Math. Oper. Res.**19**(1994), no. 2, 352–389. MR**1290505**, DOI 10.1287/moor.19.2.352 - D. Ralph and S. Wright,
*Superlinear convergence of an interior-point method for monotone variational inequalities*, Technical Report MCS-P556-0196, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, 1996. - Werner C. Rheinboldt,
*A unified convergence theory for a class of iterative processes*, SIAM J. Numer. Anal.**5**(1968), 42–63. MR**225468**, DOI 10.1137/0705003 - Stephen M. Robinson,
*Newton’s method for a class of nonsmooth functions*, Set-Valued Anal.**2**(1994), no. 1-2, 291–305. Set convergence in nonlinear analysis and optimization. MR**1285835**, DOI 10.1007/BF01027107 - D. Sun, M. Fukushima and L. Qi,
*A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problem*, in: M.C. Ferris and J.S. Pang, eds., “Complementarity and Variational Problems: State of the Art,” SIAM, Philadelphia, Pennsylvania, 1997, pp. 452-473. - D. Sun and J. Han,
*Newton and quasi-Newton methods for a class of nonsmooth equations and related problems*, SIAM J. Optim., 7 (1997), pp. 463-480. - T. Yamamoto,
*Split nonsmooth equations and verification of solution*, in: “Numerical Analysis, Scientific Computing, Computer Science”, the Zeitschrift fuer Angewandte Mathematik und Mechanik (ZAMM) with the Akademie Verlag, Berlin, 1996, pp. 199-202. - N. Yamashita and M. Fukushima,
*Modified Newton methods for solving semismooth reformulations of monotone complementarity problems,*Math. Programming, 76 (1997), pp. 469-491. - Israel Zang,
*A smoothing-out technique for min-max optimization*, Math. Programming**19**(1980), no. 1, 61–77. MR**579403**, DOI 10.1007/BF01581628

## Additional Information

**X. Chen**- Affiliation: School of Mathematics The University of New South Wales Sydney 2052, Australia
- MR Author ID: 196364
- Email: X.Chen@unsw.edu.au
**L. Qi**- Affiliation: School of Mathematics The University of New South Wales Sydney 2052, Australia
- Email: L.Qi@unsw.edu.au
**D. Sun**- Affiliation: School of Mathematics The University of New South Wales Sydney 2052, Australia
- Email: sun@alpha.maths.unsw.edu.au
- Received by editor(s): June 17, 1996
- Received by editor(s) in revised form: January 9, 1997
- Additional Notes: This work is supported by the Australian Research Council.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 519-540 - MSC (1991): Primary 90C33, 90C30, 65H10
- DOI: https://doi.org/10.1090/S0025-5718-98-00932-6
- MathSciNet review: 1458218