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)
Convergence results of the ERM method for nonlinear stochastic variational inequality problems. (English) Zbl 1187.90295
The authors consider the expected residual minimization (ERM) method proposed by M. J. Luo and G. H. Lin [J. Optim. Theory Appl. 140, 103–116 (2009; Zbl 1190.90112)] and continue to study the proposed method for a stochastic variational inequality problem. The function involved is assumed to be nonlinear in this paper. The authors first consider a quasi-Monte Carlo method for the case where the underlying sample space is compact and show that the ERM method is convergent under very mild conditions. Then, a compact approximation approach is presented for the case where the sample space is noncompact.
MSC:
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
References:
[1]Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
[2]Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005) · Zbl 1162.90527 · doi:10.1287/moor.1050.0160
[3]Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009) · Zbl 1165.90012 · doi:10.1007/s10107-007-0163-z
[4]Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007) · Zbl 1151.90052 · doi:10.1137/050630805
[5]Lin, G.H., Chen, X., Fukushima, M.: New restricted NCP function and their applications to stochastic NCP and stochastic MPEC. Optimization 56, 641–753 (2007) · Zbl 1172.90455 · doi:10.1080/02331930701617320
[6]Lin, G.H., Fukushima, M.: New reformulations for stochastic complementarity problems. Optim. Methods Softw. 21, 551–564 (2006) · Zbl 1113.90110 · doi:10.1080/10556780600627610
[7]Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009) · Zbl 1190.90112 · doi:10.1007/s10957-008-9439-6
[8]Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295 (2008) · Zbl 1163.90034 · doi:10.1007/s10957-008-9358-6
[9]Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992) · Zbl 0756.90081 · doi:10.1007/BF01585696
[10]Patrick, B.: Probability and Measure. A Wiley-Interscience Publication. Wiley, New York (1995)
[11]Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)