zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An approximate proximal-extragradient type method for monotone variational inequalities. (English) Zbl 1068.65087
One of the most known approaches to constructing solution methods for monotone variational inequalities consists in incorporating a predictor step for computing parameters of a separating hyperplane and for providing the Fejér-monotone convergence. This approach is also known as combined relaxation; see {\it I. V. Konnov} [Russ. Mathem. (Iz. VUZ), 37, No. 2, 44--51 (1993; Zbl 0835.90123)] and can be extended in several directions. {\it M. V. Solodov} and {\it B. F. Svaiter} [Math. Progr. 88, 371--389 (2000; Zbl 0963.90064)] proposed an inexact proximal point iteration as the predictor step. The authors suggest a modification of this method which involves an additional projection iteration for completing the predictor step. The method possesses the same convergence properties. Some results of numerical experiments on a network equilibrium problem are reported.

65K10Optimization techniques (numerical methods)
49J40Variational methods including variational inequalities
49M20Methods of relaxation type in calculus of variations
Full Text: DOI
[1] Auslender, A.; Teboulle, M.: Entropic proximal decomposition methods for convex programs and variational inequalities. Math. programming 91, 33-47 (2001) · Zbl 1051.90017
[2] Bertsekas, D. P.; Tsitsiklis, J. N.: Parallel and distributed computation. Numerical methods (1989) · Zbl 0743.65107
[3] Chen, X. J.; Fukushima, M.: Proximal quasi-Newton methods for nondifferentiable convex optimization. Math. programming 85, 313-334 (1998) · Zbl 0946.90111
[4] Eckstein, J.: Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. oper. Res. 18, 202-226 (1993) · Zbl 0807.47036
[5] Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. programming 83, 113-123 (1998) · Zbl 0920.90117
[6] Ferris, M. C.; Pang, J. S.: Engineering and economic applications of complementarity problems. SIAM rev. 39, 669-713 (1997) · Zbl 0891.90158
[7] He, B. S.; Liao, L. -Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. optim. Theory appl. 112, 111-128 (2002) · Zbl 1025.65036
[8] Noor, M. A.: Some recent advances in variational inequalities, part I. Basic concepts. New Zealand J. Math. 26, 53-80 (1997) · Zbl 0886.49004
[9] Noor, M. A.: Some recent advances in variational inequalities, part II. Other concepts. New Zealand J. Math. 26, 229-255 (1997) · Zbl 0889.49006
[10] Noor, M. A.: Extragradient method for pseudomonotone variational inequalities. J. optim. Theory appl. 117, 475-488 (2003) · Zbl 1049.49009
[11] Rockafellar, R. T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. oper. Res. 1, 97-116 (1976) · Zbl 0402.90076
[12] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053
[13] Solodov, M. V.; Svaiter, B. F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Math. programming, ser. B 88, 371-389 (2000) · Zbl 0963.90064
[14] Taji, K.; Fukushima, M.; Ibaraki, T.: A globally convergent Newton method for solving strongly monotone variational inequalities. Math. programming 58, 369-383 (1993) · Zbl 0792.49007
[15] Yang, H.; Bell, M. G. H.: Traffic restraint, road pricing and network equilibrium. Transportation res. B 31, 303-314 (1997)