zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
A relaxed projection method for variational inequalities. (English) Zbl 0598.49024
Let K be a closed convex set in R n , f a mapping from R n into itself, then consider the problem of finding uK such that (*) (f(u),v-u)0, for all vK. It is well known that the projection methods can be used to find the approximate solution of (*) numerically. In this paper, the author presents a modification. In the proposed modified algorithm, each iteration consists of a projection onto a half space containing the given closed set rather the latter set itself, which makes the implementation of the algorithm easy as compared to the standard projection methods. Convergence criteria are also discussed along with some examples.
Reviewer: M.A.Noor

MSC:
49M20Methods of relaxation type in calculus of variations
49J40Variational methods including variational inequalities
65K10Optimization techniques (numerical methods)
47H05Monotone operators (with respect to duality) and generalizations
References:
[1]B.-H. Ahn and W.W. Hogan, ”On convergence of the PIES algorithm for computing equilibria”,Operations Research 30 (1982) 281–300. · Zbl 0481.90011 · doi:10.1287/opre.30.2.281
[2]A. Auslender,Optimisation: Méthodes numériques (Masson, Paris, 1976).
[3]A.B. Bakusinskii and B.T. Polyak, ”On the solution of variational inequalities”,Soviet Mathematics Doklady 15 (1974) 1705–1710.
[4]D.P. Bertsekas and E.M. Gafni, ”Projection methods for variational inequalities with application to the traffic assignment problem”,Mathematical Programming Study 17 (1982) 139–159.
[5]S. Dafermos, ”Traffic equilibrium and variational inequalities”,Transportation Science 14 (1980) 42–54. · doi:10.1287/trsc.14.1.42
[6]S. Dafermos, ”An iterative scheme for variational inequalities”,Mathematical Programming 26 (1983) 40–47. · Zbl 0506.65026 · doi:10.1007/BF02591891
[7]S.C. Fang, ”An iterative method for generalized complementarity problems”,IEEE Transactions on Automatic Control AC-25 (1980) 1225–1227.
[8]S.C. Fang, ”A linearization method for generalized complementarity problems”,IEEE Transactions on Automatic Control AC-29 (1984) 930–933.
[9]S. Fisk and S. Nguyen, ”Solution algorithms for network equilibrium models with asymmetric user costs”,Transportation Science 16 (1982) 361–381. · doi:10.1287/trsc.16.3.361
[10]M. Florian and M. Los, ”A new look at static spatial price equilibrium models”,Regional Science and Urban Economics 12 (1982) 579–597. · doi:10.1016/0166-0462(82)90008-4
[11]M. Florian and H. Spiess, ”The convergence of diagonalization algorithms for asymmetric network equilibrium problems”,Transportation Research 16B (1982) 477–483.
[12]M. Fukushima, ”An outer approximation algorithm for solving general convex programs”,Operations Research 31 (1983) 101–113. · Zbl 0495.90066 · doi:10.1287/opre.31.1.101
[13]M. Fukushima, ”On the convergence of a class of outer approximation algorithms for convex programs”,Journal of Computational and Applied Mathematics 10 (1984) 147–156. · Zbl 0532.65047 · doi:10.1016/0377-0427(84)90051-7
[14]D. Kinderlehrer and G. Stampacchia,An introduction to variational inequalities and their applications (Academic Press, New York, 1980).
[15]J.S. Pang and D. Chan, ”Iterative methods for variational and complementarity problems”,Mathematical Programming 24 (1982) 284–313. · Zbl 0499.90074 · doi:10.1007/BF01585112
[16]S.M. Robinson, ”A subgradient algorithm for solvingK-convex inequalities”, in: W. Oettli and K. Ritter, eds.,Optimization and Operations Research (Springer-Verlag, Berlin, 1976) pp. 237–245.
[17]S.M. Robinson, ”Regularity and stability for convex multivalued functions”,Mathematics of Operations Research 1 (1976) 130–143. · Zbl 0418.52005 · doi:10.1287/moor.1.2.130
[18]S.M. Robinson, ”Some continuity properties of polyhedral multifunctions”,Mathematical Programming Study 14 (1981) 206–214.
[19]R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, 1970).