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)
Iterative algorithm for solving triple-hierarchical constrained optimization problem. (English) Zbl 1216.90092
Summary: Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.
MSC:
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
References:
[1]Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, pp. 473–504. Elsevier, Amsterdam (2001)
[2]Combettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process. 51(7), 1771–1782 (2003) · doi:10.1109/TSP.2003.812846
[3]Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009) · Zbl 1176.47064 · doi:10.1137/070702497
[4]Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descent method. IEEE Trans. Signal Process. 55, 4511–4522 (2007) · doi:10.1109/TSP.2007.896252
[5]Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. (2010, to appear). doi: 10.1007/s10107-010-0427-x
[6]Maingé, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)
[7]Moudafi, A.: Krasnoselski–Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007) · Zbl 1128.47060 · doi:10.1088/0266-5611/23/4/015
[8]Zeidler, E.: Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators. Springer, New York (1985)
[9]Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[10]Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003)
[11]Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
[12]Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)
[13]Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, New York (1993)
[14]Baillon, J.B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977) · Zbl 0352.47023 · doi:10.1007/BF03007664
[15]Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics Appl. Math., vol. 28. SIAM, Philadelphia (1999)
[16]Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Classics Appl. Math., vol. 31. SIAM, Philadelphia (2000)
[17]Vasin, V.V., Ageev, A.L.: Ill-Posed Problems with A Priori Information. V.S.P. Intl Science, Utrecht (1995)
[18]Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)
[19]Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[20]Stark, H., Yang, Y.: Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley, New York (1998)
[21]Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005) · Zbl 1079.90098 · doi:10.1137/S105262340343467X
[22]Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996)
[23]Izmaelov, A.F., Solodov, M.V.: An active set Newton method for mathematical program with complementary constraints. SIAM J. Optim. 19, 1003–1027 (2008) · Zbl 1201.90193 · doi:10.1137/070690882
[24]Hirstoaga, S.A.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020–1035 (2006) · Zbl 1106.47057 · doi:10.1016/j.jmaa.2005.12.064
[25]Iiduka, H.: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 71, 1292–1297 (2009) · Zbl 1238.65061 · doi:10.1016/j.na.2009.01.133
[26]Iiduka, H.: A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59, 873–885 (2010) · Zbl 1236.47064 · doi:10.1080/02331930902884158