×

Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. (English) Zbl 0883.47063

Summary: A generalized “measure of distance” defined by \(D_f(x,y):= f(x)- f(y)-\langle\nabla f(y),x- y\rangle\), is generated from any member \(f\) of the class of Bregman functions. Although it is not, technically speaking, a distance function, it has been used in the past to define and study projection operators. In this paper, we give new definitions of paracontractions, convex combinations, and firmly nonexpansive operators, based on \(D_f(x,y)\), and study sequential and simultaneous iterative algorithms employing them for the solution of the problem of finding a common asymptotic fixed point of a family of operators. Applications to the convex feasibility problem, to optimization and to monotone operator theory are also included.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H05 Monotone operators and generalizations
90C25 Convex programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0196-8858(83)90019-2 · Zbl 0489.52005 · doi:10.1016/0196-8858(83)90019-2
[2] Bauschke H.H. Borwein J.M. On projection algorithms for solving convex feasibility problems Dept. of Mathematics and Statistics. Simon Fraser University Burnaby, B.C. 1993 Research Report 93-12 Canada, June. SIAM Reciew, to appear
[3] DOI: 10.1016/0022-247X(89)90128-5 · Zbl 0675.26007 · doi:10.1016/0022-247X(89)90128-5
[4] DOI: 10.1016/0041-5553(67)90040-7 · doi:10.1016/0041-5553(67)90040-7
[5] DOI: 10.1080/00207169008803865 · Zbl 0708.90064 · doi:10.1080/00207169008803865
[6] DOI: 10.1109/83.210869 · doi:10.1109/83.210869
[7] DOI: 10.1137/1023097 · Zbl 0469.65037 · doi:10.1137/1023097
[8] DOI: 10.1007/BF01589408 · Zbl 0658.90099 · doi:10.1007/BF01589408
[9] DOI: 10.1007/BF00934676 · Zbl 0431.49042 · doi:10.1007/BF00934676
[10] Censor Y., In Numerical Anslysis and Mathematical Modelling pp 145– (1990)
[11] DOI: 10.1016/0168-9274(91)90059-9 · Zbl 0722.65032 · doi:10.1016/0168-9274(91)90059-9
[12] DOI: 10.1007/BF00940051 · Zbl 0794.90058 · doi:10.1007/BF00940051
[13] Censor Y., Numerical 8 pp 221– (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692
[14] Censor Y. Iusem A. N. Zenios S. A. An interior point method with Bregman functions for the variationul inequality problem with paramo not one operators 1994 Technical Report
[15] DOI: 10.1137/0803026 · Zbl 0808.90103 · doi:10.1137/0803026
[16] DOI: 10.1109/5.214546 · doi:10.1109/5.214546
[17] DOI: 10.1214/aos/1176348385 · Zbl 0753.62003 · doi:10.1214/aos/1176348385
[18] DOI: 10.1007/BF00940283 · Zbl 0581.90069 · doi:10.1007/BF00940283
[19] Dye J.M., Optimization and Nonlinear Analysis pp 106– (1992)
[20] DOI: 10.1287/moor.18.1.202 · Zbl 0807.47036 · doi:10.1287/moor.18.1.202
[21] DOI: 10.1016/0024-3795(90)90204-P · Zbl 0715.65037 · doi:10.1016/0024-3795(90)90204-P
[22] DOI: 10.1016/0895-7177(89)90358-0 · Zbl 0691.68102 · doi:10.1016/0895-7177(89)90358-0
[23] DOI: 10.1007/BF01396232 · Zbl 0763.65035 · doi:10.1007/BF01396232
[24] Goeble K., Uniform Convexity. Hyperbolic Geometry and Noncxpansire Mappings (1984)
[25] DOI: 10.1016/0041-5553(67)90113-9 · Zbl 0199.51002 · doi:10.1016/0041-5553(67)90113-9
[26] Hurt N.E., Phase Retrieval and Zero Crossings Mathematical Methods in Image Reconstruetion (1989) · Zbl 0687.94001 · doi:10.1007/978-94-010-9608-9
[27] DOI: 10.1137/0801025 · Zbl 0753.90051 · doi:10.1137/0801025
[28] DOI: 10.1016/0024-3795(93)00089-I · Zbl 0821.65037 · doi:10.1016/0024-3795(93)00089-I
[29] DOI: 10.1007/BF02612715 · Zbl 0523.49022 · doi:10.1007/BF02612715
[30] Stark H., Image Recovery theory and Applications (1987) · Zbl 0627.94001
[31] DOI: 10.1287/moor.17.3.670 · Zbl 0766.90071 · doi:10.1287/moor.17.3.670
[32] DOI: 10.1137/0802021 · Zbl 0763.49011 · doi:10.1137/0802021
[33] Censor Y., Parallel Optimizution: Theory Algorithms and Applications (1996)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.