Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
Authors:
X. Chen, L. Qi and D. Sun
Journal:
Math. Comp. 67 (1998), 519540
MSC (1991):
Primary 90C33, 90C30, 65H10
MathSciNet review:
1458218
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The smoothing Newton method for solving a system of nonsmooth equations , 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 th step, the nonsmooth function is approximated by a smooth function , and the derivative of at 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 is semismooth at the solution and satisfies a Jacobian consistency property. We show that most common smooth functions, such as the GabrielMoré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).
 1.
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.
 2.
S.C. Billups, S.P. Dirkse and M.C. Ferris, A comparison of algorithms for largescale mixed complementarity problems, Comp. Optim. Appl., 7 (1997), pp. 325. CMP 97:09
 3.
B. Chen and P. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), pp. 403420. CMP 97:11
 4.
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
(96m:90102), http://dx.doi.org/10.1007/BF00249052
 5.
Xiao
Jun Chen and Li
Qun Qi, A parameterized Newton method and a quasiNewton method for
nonsmooth equations, Comput. Optim. Appl. 3 (1994),
no. 2, 157–179. MR 1273659
(95e:90090), http://dx.doi.org/10.1007/BF01300972
 6.
Xiao
Jun Chen and Tetsuro
Yamamoto, Convergence domains of certain iterative methods for
solving nonlinear equations, Numer. Funct. Anal. Optim.
10 (1989), no. 12, 37–48. MR 978801
(90a:65137), http://dx.doi.org/10.1080/01630568908816289
 7.
Frank
H. Clarke, Optimization and nonsmooth analysis, Canadian
Mathematical Society Series of Monographs and Advanced Texts, John Wiley
& Sons, Inc., New York, 1983. A WileyInterscience Publication. MR 709590
(85m:49002)
 8.
T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming, 75 (1996), pp. 407439. CMP 97:05
 9.
S.P. Dirkse and M.C. Ferris, The PATH solver: A nonmonotone stabilization scheme for mixed complementarity problems, Optim. Method. Softw., 5 (1995), pp. 123156.
 10.
B.
C. Eaves, On the basic theorem of complementarity, Math.
Programming 1 (1971), no. 1, 68–75. MR 0287901
(44 #5103)
 11.
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. 155170.
 12.
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.
 13.
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. 7690. CMP 97:11
 14.
F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of largescale nonlinear complementarity problems, Math. Programming, 76 (1997), pp. 493512. CMP 97:08
 15.
A.
Fischer, A special Newtontype optimization method,
Optimization 24 (1992), no. 34, 269–284. MR 1247636
(94g:90109), http://dx.doi.org/10.1080/02331939208843795
 16.
A. Fischer, An NCPfunction 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. 88105.
 17.
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. 155170.
 18.
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. 105116. CMP 97:11
 19.
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
(95e:90108), http://dx.doi.org/10.1137/S0895479892237859
 20.
Patrick
T. Harker and JongShi
Pang, Finitedimensional 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
(91g:90166), http://dx.doi.org/10.1007/BF01582255
 21.
Patrick
T. Harker and Baichun
Xiao, Newton’s method for the nonlinear complementarity
problem: a 𝐵differentiable equation approach, Math.
Programming 48 (1990), no. 3, (Ser. B),
339–357. MR 1078191
(91m:90176), http://dx.doi.org/10.1007/BF01582262
 22.
M.
Heinkenschloss, C.
T. Kelley, and H.
T. Tran, Fast algorithms for nonsmooth compact fixedpoint
problems, SIAM J. Numer. Anal. 29 (1992), no. 6,
1769–1792. MR 1191145
(94c:65073), http://dx.doi.org/10.1137/0729099
 23.
Chi
Ming Ip and Jerzy
Kyparisis, Local convergence of quasiNewton methods for
𝐵differentiable equations, Math. Programming
56 (1992), no. 1, Ser. A, 71–89. MR 1175560
(93c:65072), http://dx.doi.org/10.1007/BF01580895
 24.
G. Isac and M. M. Kostreva, The generalized order complementarity problem, models and iterative methods, Ann. Oper. Res., 44 (1993), pp. 6392.
 25.
H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim., 35 (1997), pp. 178193. CMP 97:07
 26.
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. 197212.
 27.
C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the Dgap function for box constrained variational inequalities, Math. Programming, to appear.
 28.
C. Kanzow and H. Jiang, A continuation method for (strongly) monotone variational inequalities, Math. Programming, to appear.
 29.
Bernd
Kummer, Newton’s method for nondifferentiable functions,
Advances in mathematical optimization, Math. Res., vol. 45,
AkademieVerlag, Berlin, 1988, pp. 114–125. MR 953328
(89h:90201)
 30.
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. 204225. CMP 97:11
 31.
José
Mario Martínez and Li
Qun Qi, Inexact Newton methods for solving nonsmooth
equations, J. Comput. Appl. Math. 60 (1995),
no. 12, 127–145. Linear/nonlinear iterative methods and
verification of solution (Matsuyama, 1993). MR 1354652
(96h:65076), http://dx.doi.org/10.1016/03770427(94)00088I
 32.
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New YorkLondon, 1970. MR 0273810
(42 #8686)
 33.
Jiří
V. Outrata, On optimization problems with variational inequality
constraints, SIAM J. Optim. 4 (1994), no. 2,
340–357. MR 1273763
(95g:90075), http://dx.doi.org/10.1137/0804019
 34.
Jiří
V. Outrata and Jochem
Zowe, A Newton method for a class of quasivariational
inequalities, Comput. Optim. Appl. 4 (1995),
no. 1, 5–21. MR 1314522
(95k:49023), http://dx.doi.org/10.1007/BF01299156
 35.
JongShi
Pang, Newton’s method for 𝐵differentiable
equations, Math. Oper. Res. 15 (1990), no. 2,
311–341. MR 1051575
(91m:49029), http://dx.doi.org/10.1287/moor.15.2.311
 36.
JongShi
Pang, A Bdifferentiable equationbased, 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
(92f:90062), http://dx.doi.org/10.1007/BF01586928
 37.
JongShi
Pang, Complementarity problems, Handbook of global
optimization, Nonconvex Optim. Appl., vol. 2, Kluwer Acad. Publ.,
Dordrecht, 1995, pp. 271–338. MR 1377087
(97a:90095), http://dx.doi.org/10.1007/9781461520252_6
 38.
JongShi
Pang and Li
Qun Qi, Nonsmooth equations: motivation and algorithms, SIAM
J. Optim. 3 (1993), no. 3, 443–465. MR 1230150
(94g:90145), http://dx.doi.org/10.1137/0803021
 39.
JongShi
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
(95m:90131), http://dx.doi.org/10.1137/S0363012992241673
 40.
Li
Qun Qi, Convergence analysis of some algorithms for solving
nonsmooth equations, Math. Oper. Res. 18 (1993),
no. 1, 227–244. MR 1250115
(95f:65109), http://dx.doi.org/10.1287/moor.18.1.227
 41.
L. Qi, Cdifferentiability, CDifferential operators and generalized Newton methods, AMR 96/5, Applied Mathematics Report, University of New South Wales, Sydney, 1996.
 42.
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
(96a:90035), http://dx.doi.org/10.1137/S036301299223619X
 43.
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 (94b:90077), http://dx.doi.org/10.1007/BF01581275
 44.
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
(95e:49012), http://dx.doi.org/10.1287/moor.19.2.352
 45.
D. Ralph and S. Wright, Superlinear convergence of an interiorpoint method for monotone variational inequalities, Technical Report MCSP5560196, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, 1996.
 46.
Werner
C. Rheinboldt, A unified convergence theory for a class of
iterative processes, SIAM J. Numer. Anal. 5 (1968),
42–63. MR
0225468 (37 #1061)
 47.
Stephen
M. Robinson, Newton’s method for a class of nonsmooth
functions, SetValued Anal. 2 (1994), no. 12,
291–305. Set convergence in nonlinear analysis and optimization. MR 1285835
(95c:65093), http://dx.doi.org/10.1007/BF01027107
 48.
D. Sun, M. Fukushima and L. Qi, A computable generalized Hessian of the Dgap function and Newtontype 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. 452473. CMP 97:11
 49.
D. Sun and J. Han, Newton and quasiNewton methods for a class of nonsmooth equations and related problems, SIAM J. Optim., 7 (1997), pp. 463480. CMP 97:11
 50.
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. 199202. CMP 97:11
 51.
N. Yamashita and M. Fukushima, Modified Newton methods for solving semismooth reformulations of monotone complementarity problems, Math. Programming, 76 (1997), pp. 469491. CMP 97:08
 52.
Israel
Zang, A smoothingout technique for minmax optimization,
Math. Programming 19 (1980), no. 1, 61–77. MR 579403
(82c:90085), http://dx.doi.org/10.1007/BF01581628
 1.
 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.
 2.
 S.C. Billups, S.P. Dirkse and M.C. Ferris, A comparison of algorithms for largescale mixed complementarity problems, Comp. Optim. Appl., 7 (1997), pp. 325. CMP 97:09
 3.
 B. Chen and P. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), pp. 403420. CMP 97:11
 4.
 C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comp. Optim. Appl., 5 (1996), pp. 97138. MR 96m:90102
 5.
 X. Chen and L. Qi, A parameterized Newton method and a quasiNewton method for solving nonsmooth equations, Comp. Optim. Appl., 3 (1994), pp. 157179. MR 95e:90090
 6.
 X. Chen and T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optim., 10 (1989), pp. 3748. MR 90a:65137
 7.
 F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. MR 85m:49002
 8.
 T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming, 75 (1996), pp. 407439. CMP 97:05
 9.
 S.P. Dirkse and M.C. Ferris, The PATH solver: A nonmonotone stabilization scheme for mixed complementarity problems, Optim. Method. Softw., 5 (1995), pp. 123156.
 10.
 B.C. Eaves, On the basic theorem of complementarity, Math. Programming, 1 (1971), pp. 6875. MR 44:5103
 11.
 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. 155170.
 12.
 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.
 13.
 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. 7690. CMP 97:11
 14.
 F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of largescale nonlinear complementarity problems, Math. Programming, 76 (1997), pp. 493512. CMP 97:08
 15.
 A. Fischer, A special Newtontype optimization method, Optim., 24 (1992), pp. 269284. MR 94g:90109
 16.
 A. Fischer, An NCPfunction 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. 88105.
 17.
 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. 155170.
 18.
 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. 105116. CMP 97:11
 19.
 M.S. Gowda and R. Sznajder, The generalized order linear complementarity problem, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 779795. MR 95e:90108
 20.
 P.T. Harker and J.S. Pang, Finitedimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), pp. 161220. MR 91g:90166
 21.
 P.T. Harker and B. Xiao, Newton's method for the nonlinear complementarity problem: A Bdifferentiable equation approach, Math. Programming, 48 (1990), pp. 339357. MR 91m:90176
 22.
 M. Heinkenschloß, C.T. Kelley and H.T. Tran, Fast algorithms for nonsmooth compact fixed point problems, SIAM J. Numer. Anal., 29 (1992), pp. 17691792. MR 94c:65073
 23.
 C.M. Ip and J. Kyparisis, Local convergence of quasiNewton methods for Bdifferentiable equations, Math. Programming, 56 (1992), pp. 7189. MR 93c:65072
 24.
 G. Isac and M. M. Kostreva, The generalized order complementarity problem, models and iterative methods, Ann. Oper. Res., 44 (1993), pp. 6392.
 25.
 H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim., 35 (1997), pp. 178193. CMP 97:07
 26.
 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. 197212.
 27.
 C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the Dgap function for box constrained variational inequalities, Math. Programming, to appear.
 28.
 C. Kanzow and H. Jiang, A continuation method for (strongly) monotone variational inequalities, Math. Programming, to appear.
 29.
 B. Kummer, Newton's method for nondifferentiable functions, in: J. Guddat, B. Bank, H. Hollatz, P. Kall, D. Klatte, B. Kummer, K. Lommatzsch, L. Tammer, M. Vlach and K. Zimmerman, eds., ``Advances in Mathematical Optimization'', AkademiVerlag, Berlin, 1988, pp. 114125. MR 89h:90201
 30.
 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. 204225. CMP 97:11
 31.
 J.M. Martínez, and L. Qi, Inexact Newton methods for solving nonsmooth equations, J. Comp. Appl. Math., 60 (1995), pp. 127145. MR 96h:65076
 32.
 J.M. Ortega and W.C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 42:8686
 33.
 J.V. Outrata, On optimization problems with variational inequality constraints, SIAM J. Optim., 4 (1994), pp. 340357. MR 95g:90075
 34.
 J.V. Outrata and J. Zowe, A Newton method for a class of quasivariational inequalities, Comp. Optim. Appl., 4 (1995), pp. 521. MR 95k:49023
 35.
 J.S. Pang, Newton's method for Bdifferentiable equations, Math. Oper. Res., 15 (1990), pp. 311341. MR 91m:49029
 36.
 J.S. Pang, A Bdifferentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Programming, 51 (1991), pp. 101131. MR 92f:90062
 37.
 J.S. Pang, Complementarity problems, in: R. Horst and P. Pardalos, eds., ``Handbook of Global Optimization'', Kluwer Academic Publishers, Boston, pp. 271338, 1995. MR 97a:90095
 38.
 J.S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM J. Optim., 3 (1993), pp. 443465. MR 94g:90145
 39.
 J.S. Pang and J.C. Yao, On a generalization of a normal map and equation, SIAM J. Control Optim., 33 (1995), pp. 168184. MR 95m:90131
 40.
 L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227244. MR 95f:65109
 41.
 L. Qi, Cdifferentiability, CDifferential operators and generalized Newton methods, AMR 96/5, Applied Mathematics Report, University of New South Wales, Sydney, 1996.
 42.
 L. Qi and X. Chen, A globally convergent successive approximation method for severely nonsmooth equations, SIAM J. Control Optim., 33 (1995), pp. 402418. MR 96a:90035
 43.
 L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353367. MR 94b:90077
 44.
 D. Ralph, Global convergence of damped Newton's method for nonsmooth equations via the path search, Math. Oper. Res., 19 (1994), pp. 352389. MR 95e:49012
 45.
 D. Ralph and S. Wright, Superlinear convergence of an interiorpoint method for monotone variational inequalities, Technical Report MCSP5560196, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, 1996.
 46.
 W.C. Rheinboldt, A unified convergence theory for a class of iterative processes, SIAM J. Numer. Anal., 5 (1968), pp. 4263. MR 37:1061
 47.
 S. M. Robinson, Newton's method for a class of nonsmooth functions, SetValued Anal., 2 (1994), pp. 291305. MR 95c:65093
 48.
 D. Sun, M. Fukushima and L. Qi, A computable generalized Hessian of the Dgap function and Newtontype 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. 452473. CMP 97:11
 49.
 D. Sun and J. Han, Newton and quasiNewton methods for a class of nonsmooth equations and related problems, SIAM J. Optim., 7 (1997), pp. 463480. CMP 97:11
 50.
 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. 199202. CMP 97:11
 51.
 N. Yamashita and M. Fukushima, Modified Newton methods for solving semismooth reformulations of monotone complementarity problems, Math. Programming, 76 (1997), pp. 469491. CMP 97:08
 52.
 I. Zang, A smoothingout technique for minmax optimization, Math. Programming, 19 (1980), pp. 6177. MR 82c:90085
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
90C33,
90C30,
65H10
Retrieve articles in all journals
with MSC (1991):
90C33,
90C30,
65H10
Additional Information
X. Chen
Affiliation:
School of Mathematics The University of New South Wales\ Sydney 2052, Australia
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
DOI:
http://dx.doi.org/10.1090/S0025571898009326
PII:
S 00255718(98)009326
Keywords:
Variational inequalities,
nonsmooth equations,
smoothing approximation,
smoothing Newton method,
convergence
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.
Article copyright:
© Copyright 1998
American Mathematical Society
