Algorithm for the generalized Φ-strongly monotone mappings and application to the generalized convex optimization problem

Authors

Keywords:

Generalized Φ-strongly monotone, Surjective property, Generalized convex, Optimization

Abstract

Let E be a uniformly smooth and uniformly convex real Banach space and E∗ be its dual space. We consider a multivalued mapping A : E → 2E∗ which is bounded, generalized Φ-strongly monotone and such that for all t > 0, the range R(Jp+tA) = E∗, where Jp (p > 1) is the generalized duality mapping from E into 2E∗ . Suppose A−1(0) = ∅, we construct an algorithm which converges strongly to the solution of 0 ∈ Ax. The result is then applied to the generalized convex optimization problem.

Author Biographies

M. O. Aibinu, University of KwaZulu-Natal.

School of Mathematics, Statistics and Computer Science.

O. T. Mewomo, University of KwaZulu-Natal.

School of Mathematics, Statistics and Computer Science.

References

H. A. Abass, F.U. Ogbuisi, O.T. Mewomo,Common solution of split equilibrium problem and fixed point problem with no prior knowledge of operator norm, U.P.B. Sci. Bull., Series A, 80(1) (2018), 175-190.

M. O. Aibinu, O. T. Mewomo, Algorithm for Zeros of monotone maps in Banach spaces, Proceedings of Southern Africa Mathematical Sciences Association (SAMSA2016) Annual Conference, 21-24 November 2016, University of Pretoria, South Africa, pp. 35-44, (2017).

M. O. Aibinu, O. T. Mewomo, Strong convergence theorems for strongly monotone mappings in Banach spaces, (Submitted to Hacettepe University J. Math & Stat).

Ya. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, In: Kartsatos AG (ed) Theory and applications of nonlinear operators of accretive and monotone type, Marcel Dekker, New York, pp. 15-50, (1996).

Y. Alber and I. Ryazantseva, Nonlinear Ill Posed Problems of Monotone Type, Springer, London, (2006).

D. Aussel, J. N. Corvellec and M. Lassonde, Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Am. Math. Soc., 347, pp. 4147-4161, (1995).

D. Aussel, J. N. Corvellec and M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal., 1, pp. 195-201, (1994).

D. Aussel and M. Fabian, Single-directional properties of quasimonotone operators, Set-Valued Var. Anal., 21, pp. 617-626, (2013). DOI 10.1007/s11228-013-0253-4.

C. E. Chidume and K. O. Idu, Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems, Fixed Point Theory Appl., (97), (2016), DOI 10.1186/s13663-016-0582-8.

I. Cioranescu,Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Publishers Group, Dordrecht, (1990).

F. H. Clarke, Optimization and Nonsmooth Analysis, WileyInterscience, New-York, (1983).

C. Diop, T. M. M. Sow, N. Djitte and C. E. Chidume, Constructive techniques for zeros of monotone mappings in certain Banach spaces, SpringerPlus, 383 (4), (2015), DOI 10.1186/s40064-015-1169-2.

B. de Finetti, Sulle stratificazioni convesse, Annali di Matematica Pura ed Applicata, 30, pp. 173-183, (1949).

A.P. Farajzadeh1, A. Karamian1 and S. Plubtieng, On the translations of quasimonotone maps and monotonicity, J. of Ineq. Appl., 1 (192), (2012).

A. P. Farajzadeh, S. Plubtieng When a vector quasimonotone mapping is a vector monotone mapping, Optim Lett., 8, (2014), 2127-2134, (2014) DOI 10.1007/s11590-013-0716-4.

W. Fenchel, Convex cones, sets and functions, Lecture Notes, Princeton University, Princeton, (1953).

N. Hadjisavvas, Translations of quasimonotone maps and monotonicity, Appl. Math. Lett., 19, pp. 913-915, (2006).

A. Hassouni, Quasi-monotone multifunctions; Applications to optimality conditions in quasi-convex programming, Numer. Funct. Anal. Optim., 13 (3-4), pp. 267-275, (1992).

A. Hassouni and A. Jaddar, Quasi-convex functions and applications to optimality conditions in nonlinear programming, Appl. Math. Lett., 14, pp. 241-244, (2001).

Y. He, A relationship between pseudomonotone and monotone mappings, Appl. Math. Lett., 17, pp. 459-461, (2004).

G. Isac and D. Motreanu, Pseudomonotonicity and quasimonotonicity by translations versus monotonicity in Hilbert spaces, Aust. J. Math. Anal. Appl., 1 (1), pp. 1-8, (2004).

L. O. Jolaoso, F. U. Ogbuisi, O. T. Mewomo, An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces, Adv. Pure Appl. Math., 9 (3), pp. 167-184, (2017).

L.O. Jolaoso, K.O. Oyewole, C. C. Okeke, O.T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51, pp. 211-232, (2018).

S. Kamimura and W. Takahashi, Strong convergence of proximaltype algorithm in Banach space, SIAMJ Optim., 13 (3), pp. 938-945, (2002).

S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl., 66, pp. 37-46, (1990).

S. Karamardian, S. Schaible and J.P Crouuzeix, Characterizations of generalized monotone maps, J. Optim. Theory Appl., 76, pp. 399-413, (1993).

S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics, Pergamon Press, London, (1959).

K. Kido, Strong convergence of resolvents of monotone operators in Banach spaces, Proc. Amer. Math. Soc., 103 (3), pp. 755-7588, (1988).

V. L. Levin, Quasiconvex functions and quasimonotone operators, J. Conv. Anal., 2 (1-2), pp. 167-172, (1995).

O.T. Mewomo and F.U. Ogbuisi, Convergence analysis of iterative method for multiple set split feasibility problems in certain Banach spaces, Quaest. Math., 41 (1), pp. 129-148, (2018).

F. U. Ogbuisi, O.T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19 (1), pp. 335-358, (2018).

J. P. Penot, Are generalized derivatives useful for generalized convex functions?, University of Pau, pp. 1-44, (1997).

J. P. Penot and P. H. Quang, Generalized convexity of functions and generalized monotonicityof set-valued maps, J. Optim. Theory Appl., 92 (2), pp. 343-356, (1997).

R. R. Phelps, Convex functions, monotone operators, and differentiability, 2nd edn. In: Lecture Notes in Math, vol. 1364. Springer-Verlag, Berlin, (1993).

S. Reich, A weak convergence theorem for the alternating method with Bregman distances, In: A. G, Kartsatos (Ed.), Theory and Applications of nonlinear operators of accretive and monotone type, Lecture Notes Pure Appl. Math., Dekker, New York, 178, pp. 313-318, (1996). Anal. Appl. 75, pp. 287-292, (1980).

R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149, pp. 75-88, (1970).

R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math., 32, pp. 257-280, (1980).

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (12), pp. 1127-1138, (1991).

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (1), pp. 240-256, (2002).

Z. B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 157, pp. 189-210, (1991).

Y. Shehu and O.T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math, Sin. (Engl. Ser.), 32 (11), pp. 1357-1376, (2016).

C. Zaˇlinescu, On uniformly convex functions, J. Math. Anal. Appl., 95, pp. 344-374, (1983).

Published

2019-02-25

How to Cite

[1]
M. O. Aibinu and O. T. Mewomo, “Algorithm for the generalized Φ-strongly monotone mappings and application to the generalized convex optimization problem”, Proyecciones (Antofagasta, On line), vol. 38, no. 1, pp. 59-82, Feb. 2019.

Issue

Section

Artículos