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)
On variable-step relaxed projection algorithm for variational inequalities. (English) Zbl 1056.49018
Summary: Projection algorithms are practically useful for solving variational inequalities (VI). However some among them require the knowledge related to VI in advance, such as Lipschitz constant. Usually it is impossible in practice. This paper studies the variable-step basic projection algorithm and its relaxed version under a weak co-coercivity condition. The algorithms discussed need not know constant/function associated with the co-coercivity or weak co-coercivity and the step-size is varied from one iteration to the next. Under certain conditions the convergence of the variable-step basic projection algorithm is established. For the practical consideration, we also give the relaxed version of this algorithm, in which the projection onto a closed convex set is replaced by another projection at each iteration and the latter is easy to calculate. The convergence of the relaxed scheme is also obtained under certain assumptions. Finally, we apply these two algorithms to the Split Feasibility Problem (SFP).

49J40Variational methods including variational inequalities
65K10Optimization techniques (numerical methods)
Full Text: DOI
[1] Facchinei, F.; Pang, J. S.: Finite-dimensional variational inequality and complementarity problems, vols. 1 and 2. (2003) · Zbl 1062.90002
[2] Harker, P.; Pang, J. S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. program. 48, 161-220 (1990) · Zbl 0734.90098
[3] Pang, J. S.: Inexact Newton methods for the nonlinear complementarity problems. Math. program. 36, 54-71 (1986) · Zbl 0613.90097
[4] Auslender, A.: Optimisation: méthodes numérique. (1976)
[5] Fukushima, M.: An outer approximation algorithm for solving general convex programs. Oper. res. 31, 101-113 (1983) · Zbl 0495.90066
[6] Fukushima, M.: A relaxed projection method for variational inequalities. Math. program. 35, 58-70 (1986) · Zbl 0598.49024
[7] Kinderlehrer, D.; Stampacchia, G.: An introduction to variational inequalities and their applications. (1980) · Zbl 0457.35001
[8] Byrne, C.: Iterative oblique projection onto convex sets and the split feasibilities problem. Inverse problems 18, 441-453 (2002) · Zbl 0996.65048
[9] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse problems 20, 103-120 (2004) · Zbl 1051.65067
[10] Byrne, C.: Bregman -- Legendre multidistances projection algorithm for convex feasibility and optimization. Inherently parallel algorithms in feasility and optimization and their applications, 87-100 (2001) · Zbl 0990.90094
[11] Censor, Y.; Elfving, T.: A multiprojection algorithm using Bregman projections in a production space. Numer. algorithms 8, 221-239 (1994) · Zbl 0828.65065
[12] Censor, Y.; Iusem, A.; Zenios, S.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. program. 81, 370-400 (1998) · Zbl 0919.90123
[13] Dolidze, Z.: Solution of variational inequalities associated with a class of monotone maps. Ekonomik i matem. Metody 18, 925-927 (1982)
[14] He, B. S.: Inexact implicit methods for monotone general variational inequalities. Math. progam. Ser. A 86, 199-217 (1999) · Zbl 0979.49006
[15] Solodov, M.; Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control optim. 34, 1814-1830 (1996) · Zbl 0866.49018
[16] Sun, D.: A class of iterative methods for solving nonlinear projection equations. J. optim. Theory appl., 123-140 (1996) · Zbl 0871.90091
[17] Tseng, P.: Alternating projection-proximal methods for variational inequalities. SIAM J. Optim. 7, 951-965 (1997) · Zbl 0914.90218
[18] Xiu, N.; Zhang, J.: Some recent advances in projection-type methods for variational inequalities. J. comput. Appl. math. 152, 559-587 (2003) · Zbl 1018.65083
[19] Noor, M.; Wang, Y.; Xiu, N.: Some projection methods for variational inequalities. Appl. math. Comput. 137, 423-435 (2003) · Zbl 1031.65078
[20] Marcotte, P.; Wu, J.: On the convergence of projection methods. Applications to the decomposition of affine variational inequalities. J. optim. Theory appl. 85, 347-362 (1995) · Zbl 0831.90104
[21] Golshtein, E.; Tretyakov, N.: Modified Lagrangians and monotone maps in optimization. (1996) · Zbl 0848.49001
[22] Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse problems 20, 1261-1266 (2004) · Zbl 1066.65047