Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T07:57:12.775Z Has data issue: false hasContentIssue false
  • English
  • Français

A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

Part of: Nonlinear operators and their properties Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  26 September 2018

C. E. Chidume ,
M. O. Uba ,
M. I. Uzochukwu ,
E. E. Otubo  and
K. O. Idu
C. E. Chidume
Affiliation:
African University of Science and Technology, Abuja, Nigeria ([email protected]; [email protected]; [email protected])
M. O. Uba
Affiliation:
Department of Mathematics, University of Nigeria, Nsukka, Nigeria ([email protected])
M. I. Uzochukwu
Affiliation:
African University of Science and Technology, Abuja, Nigeria ([email protected]; [email protected]; [email protected])
E. E. Otubo
Affiliation:
Ebonyi State University, Abakaliki, Nigeria ([email protected])
K. O. Idu
Affiliation:
African University of Science and Technology, Abuja, Nigeria ([email protected]; [email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

Keywords

monotone mappingmaximal monotone mappingsstrong convergence

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 47H05: Monotone operators and generalizations 47J05: Equations involving nonlinear operators (general)
Secondary: 47H14: Perturbations of nonlinear operators 47J20: Variational and other types of inequalities involving nonlinear operators (general)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 241 - 257
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alber, Ya., Metric and generalized projection operators in Banach spaces: properties and applications, In Theory and applications of nonlinear operators of accretive and monotone type (ed. Kartsatos, A. G.), pp. 15–50 (Marcel Dekker, New York, 1996).Google Scholar
2.Alber, Ya. and Guerre-Delabriere, S., On the projection methods for fixed point problems, Anal. (Munich) 21(1) (2001), 17–39.Google Scholar
3.Alber, Ya. and Ryazantseva, I., Nonlinear Ill posed problems of monotone type (Springer, London, 2006).Google Scholar
4.Bauschke, H. H., Matoušková, E. and Reich, S., Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal. 56 (2004), 715–738.Google Scholar
5.Berinde, V., Iterative approximation of fixed points, Lecture Notes in Mathematics (Springer, London, 2007).Google Scholar
6.Bot, R. I. and Csetnek, E. R., A hybrid proximal-extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim. 36 (2015), 951–963.Google Scholar
7.Bot, R. I. and Csetnek, E. R., An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algorithm 71 (2016), 519–540.Google Scholar
8.Browder, F. E., Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875–882.Google Scholar
9.Bruck, R. E. Jr, A strongly convergent iterative solution of 0 ∈ U(x) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114–126.Google Scholar
10.Bruck, R. E. and Reich, S., Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977), 459–470.Google Scholar
11.Burachik, R. S. and Scheimberg, S., A proximal point method for the variational inequality problem in Banach spaces, SIAM J. Control Optim. 39 (5) (2000), 1633–1649.Google Scholar
12.Chidume, C. E., The iterative solution of the equation f ∈ x + Tx for a monotone operator T in L p spaces, J. Math. Anal. Appl. Vol. 116(2) (1986), 531–537.Google Scholar
13.Chidume, C. E., Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, Volume 1965 (Springer, London, 2009).Google Scholar
14.Chidume, C. E., Strong convergence theorems for zeros of m-accretive bounded operators in uniformly smooth real Banach spaces, Contemporary Mathematics, Volume 659, pp. 31–41 (American Mathematical Society, Providence, RI, 2016).Google Scholar
15.Chidume, C. E. and Djitte, N., Iterative algorithm for zeros of multivalued accretive operators in certain Banach spaces, Afr. Mat. 26 (2015), 357–368.Google Scholar
16.Chidume, C. E. and Osilike, M. O., Equilibrium points for a system involving m-accretive operators, Proc. Edinb. Math. Soc. 44 (2001), 187–199.Google Scholar
17.Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (Kluwer Academic Publishers, 1990).Google Scholar
18.Fonseca, I. and Leoni, G. (EDS), Modern methods in the calculus of variations: L p spaces, Springer Monographs in Mathematics (Springer, New York, 2007).Google Scholar
19.Güler, O., On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29 (1991), 403–419.Google Scholar
20.Kamimura, S. and Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13(3) (2003), 938–945.Google Scholar
21.Kato, T., Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520.Google Scholar
22.Lehdili, N. and Moudafi, A., Combining the proximal algorithm and Tikhonov regularization, Optim. 37 (1996), 239–252.Google Scholar
23.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II: function spaces, Ergebnisse Mathematik Grenzgebiete, Volume 97 (Springer-Verlag, Berlin, 1979).Google Scholar
24.Martinet, B., Régularization d'inéquations variationelles par approximations successives, Revue Francaise d'informatique et de Recherche operationelle 4 (1970), 154–159.Google Scholar
25Matsushita, S-Y. and Xu, L., On convergence of the proximal point algorithm in Banach spaces, Proc. Amer. Math. Soc. 139(11) (2011), 4087–4095.Google Scholar
26.Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346.Google Scholar
27.Moreau, J. J., Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math., France 93 (1965), 273–299.Google Scholar
28.Nevanlinna, O. and Reich, S., Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), 44–58.Google Scholar
29.Pascali, D. and Sburian, S., Nonlinear mappings of monotone type (Editura Academia Bucuresti, Romania, 1978).Google Scholar
30.Reich, S., Approximating zeros of accretive operators, Proc. Amer. Math. Soc. 51 (1975), 381–384.Google Scholar
31.Reich, S., Extension problems for accretive sets in Banach spaces, J. Funct. Anal. 26 (1977), 387–395.Google Scholar
32.Reich, S., Iterative methods for accretive sets, In Nonlinear equation in abstract spaces, pp. 317–326 (Academic Press, New York, 1978).Google Scholar
33.Reich, S., Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, pp. 335–345 (Academic Press, New York, 1979).Google Scholar
34.Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287–292.Google Scholar
35.Reich, S., A weak convergence theorem for the alternating method with Bergman distance, In Theory and applications of nonlinear operators of accretive and monotone type, in Lecture Notes in Pure and Applied Mathematics, vol. 178 (ed. Kartsatos, A. G.), pp. 313–318 (Dekker, New York, 1996).Google Scholar
36.Reich, S. and Sabach, S., Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 22–44.Google Scholar
37.Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877–898.Google Scholar
38.Reich, S., Geometry of Banach spaces, duality mappings and nonlinear mappings, Bull. Amer. Math. Soc. 26 (1992), 367–370.Google Scholar
39Solodov, M. V. and Svaiter, B. F., Forcing strong convergence of proximal point iterations in a Hilber space, Math. Program., Ser. A 87 (5000), 189–202.Google Scholar
40.Xu, H. K., Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66(2) (2002), 240–256.Google Scholar
41.Xu, H. K., A regularization method for the proximal point algorithm, J. Glob. Opt. 36 (2006), 115–125.Google Scholar
42.Xu, Z. B., Jiang, Y. L. and Roach, G. F., A necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces, Proc. Edinb. Math. Soc. 38(2) (1995), 1–12.Google Scholar
43.Zeidler, E., Nonlinear functional analysis and its applications part II: monotone operators (Springer-Verlag, Berlin, 1985).Google Scholar