## Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.(English)Zbl 0913.47048

Let $$T_i$$ $$(i= 1,2,\dots, N)$$ be nonexpansive mappings on a Hilbert space $${\mathcal H}$$, and let $$\Theta:{\mathcal H}\to \mathbb{R}\cup \{\infty\}$$ be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of $$T_i({\mathcal H})$$ for some $$i$$. The authors prove that $$\Theta$$ has a unique minimum over the intersection of the fixed point sets of all the $$T_i$$’s at some point $$u^*$$. Then a cyclic hybrid steepest descent algorithm is proposed and it is shown that it converges to $$u^*$$.

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 90C25 Convex programming 65K05 Numerical mathematical programming methods 65K10 Numerical optimization and variational techniques 90C30 Nonlinear programming
Full Text:

### References:

 [1] Barbu V., Convexity and Optimization in Banach Spaces (1986) · Zbl 0594.49001 [2] Bauschke H.H., Dykstra’s alternating projection algorithm for two sets 79 pp 418– (1994) · Zbl 0833.46011 [3] Bauschke H.H., On projection algorithms for solving convex feasibility problems 38 pp 367– (1996) · Zbl 0865.47039 [4] Bauschke H.H., The approximation of fixed points of compositions of nonexpan-sive mappings in Hilbert spac 202 pp 150– (1996) [5] Boyle J.P., A method for finding projections onto the intersection of convex sets in Hilbert spaces, in pp 28– (1985) [6] Browder, F.E. Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Nat. Acad. Sci. Vol. 53, pp.1272–1276. U.S.A · Zbl 0125.35801 [7] Censor Y., A multiprojection algorithm using Bregman projections in a product space 8 pp 221– (1994) · Zbl 0828.65065 [8] Cheney, W. and Goldstein, A.A. Proximity maps for convex sets. Proc. Amer, Math. Soc. Vol. 10, pp.448–450. · Zbl 0092.11403 [9] Combettes, P.L. The foundations of set theoretic estimation. Proc. IEEE. Vol. 81, pp.182–208. [10] Combettes P.L., Inconsistent signal feasibility problems: least squares solutions in a product space 42 pp 2955– (1994) [11] Combettes, P.L. Constrained image recovery in a product space. Proc. of 1995 Internat. Conf. Image Processing. pp.25–28. [12] Combettes, P.L. 1995.Construction d’un point fixe commun à une famille de contractions fermes, Vol. 320, 1385–1390. C.R. Acad, Sci. Paris Sér. I Math. · Zbl 0830.65047 [13] Combettes, P.L. and Bondon, P. Adaptive linear filtering with convex constraints. Proc. of 1995 Internat. Conf. Aeoust., Speech, Signal Processing. pp.1372–1375. [14] Crombez G., Finding projections onto the intersection of convex sets in Hilbert spaces 15 pp 637– (1996) · Zbl 0827.46017 [15] Deutsch F., The method of alternating orthogonal projections, in pp 105– (1992) · Zbl 0751.41031 [16] Deutsch F., The rate of convergence of Dykstra’s cyclic projections algorithms: the polyhedral case 15 pp 537– (1994) · Zbl 0807.41019 [17] Deutsch F., The rate of convergence for the method of alternating projections II 205 pp 381– (1997) · Zbl 0890.65053 [18] Dykstra R.L., An algorithm for restricted least squares regression 78 pp 837– (1983) · Zbl 0535.62063 [19] DOI: 10.1017/CBO9780511526152 [20] Halperin I., The product of projection operators 23 pp 96– (1962) · Zbl 0143.16102 [21] Halpern B., Fixed points of nonexpanding maps 73 pp 957– (1967) · Zbl 0177.19101 [22] Han S.P., A successive projection method 40 pp 1– (1988) · Zbl 0685.90074 [23] Hundal H., Two generalizations of Dykstra’s cyclic projections algorithm 77 pp 335– (1997) · Zbl 0891.90164 [24] DOI: 10.1007/BF01582891 · Zbl 0744.90066 [25] Lions, P.L. 1977.Approximation de points fixes de contradictions, A Vol. 284, 1357–1359. C. R. Acad. Sci. Paris. · Zbl 0349.47046 [26] Mangasarian O.L., Nonlinear Programming (1994) [27] von Neumann J., Functional operators, The Geometry of Orthogonal Spaces (1950) · Zbl 0039.11701 [28] Opial Z., Weak Convergence of the sequence of successive approximations for nonexpansive mappings 73 pp 591– (1967) · Zbl 0179.19902 [29] Pierra G., Eclatement de contraintes en parallèle pour la minimisaition d’une forme quadratique 40 pp 200– (1976) · Zbl 0346.49032 [30] Pierra G., Decomposition through formalization in a product space 28 pp 96– (1984) · Zbl 0523.49022 [31] Rudin W., Real and complex analysis (1987) [32] Wittmann R., Approximation of fixed points of nonexpansive mappings 58 pp 486– (1992) · Zbl 0797.47036 [33] Yamada I., Quadratic Optimization of Fixed Points of Nonexpansive Mappings in Hilbert Space 58 (1992) [34] Yosida K., Functional analysis (1974) · Zbl 0286.46002 [35] Youla D., Mathematical theory of image restoration by the method of convex pro-jections, in pp 29– (1987) [36] Zeidler E., Nonlinear Functional Analysis and Its Applications II-B: Nonlinear Monotone Operators (1990) · Zbl 0684.47029 [37] Zeidler E., Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization (1985) · Zbl 0583.47051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.