Inertial forward-backward splitting method in Banach spaces with application to compressed sensing

Prasit Cholamjiak; Yekini Shehu

Applications of Mathematics (2019)

  • Volume: 64, Issue: 4, page 409-435
  • ISSN: 0862-7940

Abstract

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We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the few available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.

How to cite

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Cholamjiak, Prasit, and Shehu, Yekini. "Inertial forward-backward splitting method in Banach spaces with application to compressed sensing." Applications of Mathematics 64.4 (2019): 409-435. <http://eudml.org/doc/294746>.

@article{Cholamjiak2019,
abstract = {We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the few available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.},
author = {Cholamjiak, Prasit, Shehu, Yekini},
journal = {Applications of Mathematics},
keywords = {inertial term; forward-backward splitting; inclusion problem; strong convergence; Banach space},
language = {eng},
number = {4},
pages = {409-435},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inertial forward-backward splitting method in Banach spaces with application to compressed sensing},
url = {http://eudml.org/doc/294746},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Cholamjiak, Prasit
AU - Shehu, Yekini
TI - Inertial forward-backward splitting method in Banach spaces with application to compressed sensing
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 409
EP - 435
AB - We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the few available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.
LA - eng
KW - inertial term; forward-backward splitting; inclusion problem; strong convergence; Banach space
UR - http://eudml.org/doc/294746
ER -

References

top
  1. Alvarez, F., Attouch, H., 10.1023/A:1011253113155, Set-Valued Anal. 9 (2001), 3-11. (2001) Zbl0991.65056MR1845931DOI10.1023/A:1011253113155
  2. Attouch, H., Cabot, A., Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions, Available at https://hal.archives-ouvertes.fr/hal-01782016 (2018), Hal ID: 01782016, 35 pages. (2018) MR4026592
  3. Baillon, J.-B., Haddad, G., 10.1007/BF03007664, Isr. J. Math. 26 French (1977), 137-150. (1977) Zbl0352.47023MR0500279DOI10.1007/BF03007664
  4. Beck, A., Teboulle, M., 10.1137/080716542, SIAM J. Imaging Sci. 2 (2009), 183-202. (2009) Zbl1175.94009MR2486527DOI10.1137/080716542
  5. Bertsekas, D. P., Tsitsiklis, J. N., Parallel and Distributed Computation: Numerical Methods, Athena Scientific, Belmont (2014). (2014) Zbl1325.65001MR3587745
  6. Boţ, R. I., Csetnek, E. R., An inertial alternating direction method of multipliers, Minimax Theory Appl. 1 (2016), 29-49. (2016) Zbl1337.90082MR3477895
  7. Boţ, R. I., Csetnek, E. R., Hendrich, C., 10.1016/j.amc.2015.01.017, Appl. Math. Comput. 256 (2015), 472-487. (2015) Zbl1338.65145MR3316085DOI10.1016/j.amc.2015.01.017
  8. Brézis, H., Lions, P.-L., 10.1007/BF02761171, Isr. J. Math. 29 (1978), 329-345 French. (1978) Zbl0387.47038MR0491922DOI10.1007/BF02761171
  9. Byrne, C., 10.1088/0266-5611/20/1/006, Inverse Probl. 20 (2004), 103-120. (2004) Zbl1051.65067MR2044608DOI10.1088/0266-5611/20/1/006
  10. Chen, C., Chan, R. H., Ma, S., Yang, J., 10.1137/15100463X, SIAM J. Imaging Sci. 8 (2015), 2239-2267. (2015) Zbl1328.65134MR3404682DOI10.1137/15100463X
  11. Chen, G. H.-G., Rockafellar, R. T., 10.1137/S1052623495290179, SIAM J. Optim. 7 (1997), 421-444. (1997) Zbl0876.49009MR1443627DOI10.1137/S1052623495290179
  12. Chidume, C., 10.1007/978-1-84882-190-3, Lecture Notes in Mathematics 1965, Springer, Berlin (2009). (2009) Zbl1167.47002MR2504478DOI10.1007/978-1-84882-190-3
  13. Cholamjiak, P., 10.1007/s11075-015-0030-6, Numer. Algorithms 71 (2016), 915-932. (2016) Zbl1342.47079MR3479747DOI10.1007/s11075-015-0030-6
  14. Cholamjiak, P., Cholamjiak, W., Suantai, S., 10.1186/s13660-015-0739-8, J. Inequal. Appl. 2015 (2015), Article ID 220, 10 pages. (2015) Zbl1338.47077MR3367213DOI10.1186/s13660-015-0739-8
  15. Cholamjiak, W., Cholamjiak, P., Suantai, S., 10.1007/s11784-018-0526-5, J. Fixed Point Theory Appl. 20 (2018), Article ID 42, 17 pages. (2018) Zbl06858734MR3764675DOI10.1007/s11784-018-0526-5
  16. Cioranescu, I., 10.1007/978-94-009-2121-4, Mathematics and Its Applications 62, Kluwer Academic Publishers, Dordrecht (1990). (1990) Zbl0712.47043MR1079061DOI10.1007/978-94-009-2121-4
  17. Combettes, P. L., Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal. 16 (2009), 727-748. (2009) Zbl1193.47067MR2583892
  18. Combettes, P. L., Wajs, V. R., 10.1137/050626090, Multiscale Model. Simul. 4 (2005), 1168-1200. (2005) Zbl1179.94031MR2203849DOI10.1137/050626090
  19. Dong, Q., Jiang, D., Cholamjiak, P., Shehu, Y., 10.1007/s11784-017-0472-7, J. Fixed Point Theory Appl. 19 (2017), 3097-3118. (2017) Zbl06817792MR3720497DOI10.1007/s11784-017-0472-7
  20. Dunn, J. C., 10.1016/0022-247X(76)90152-9, J. Math. Anal. Appl. 53 (1976), 145-158. (1976) Zbl0321.49025MR0388176DOI10.1016/0022-247X(76)90152-9
  21. Güler, O., 10.1137/0329022, SIAM J. Control Optim. 29 (1991), 403-419. (1991) Zbl0737.90047MR1092735DOI10.1137/0329022
  22. Halpern, B., 10.1090/S0002-9904-1967-11864-0, Bull. Am. Math. Soc. 73 (1967), 957-961. (1967) Zbl0177.19101MR0218938DOI10.1090/S0002-9904-1967-11864-0
  23. Haugazeau, Y., Sur la minimisation des formes quadratiques avec contraintes, C. R. Acad. Sci., Paris, Sér. A 264 (1967), 322-324 French. (1967) Zbl0149.36201MR0215113
  24. Kazarinoff, N. D., Analytic Inequalities, Holt, Rinehart and Winston, New York (1961). (1961) Zbl0097.03801MR0260957
  25. Lions, P.-L., Mercier, B., 10.1137/0716071, SIAM J. Numer. Anal. 16 (1979), 964-979. (1979) Zbl0426.65050MR0551319DOI10.1137/0716071
  26. López, G., Martín-Márquez, V., Wang, F., Xu, H.-K., 10.1155/2012/109236, Abstr. Appl. Anal. 2012 (2012), Article ID 109236, 25 pages. (2012) Zbl1252.47043MR2955015DOI10.1155/2012/109236
  27. Lorenz, D. A., Pock, T., 10.1007/s10851-014-0523-2, J. Math. Imaging Vis. 51 (2015), 311-325. (2015) Zbl1327.47063MR3314536DOI10.1007/s10851-014-0523-2
  28. Maingé, P.-E., 10.1016/j.jmaa.2005.12.066, J. Math. Anal. Appl. 325 (2007), 469-479. (2007) Zbl1111.47058MR2273538DOI10.1016/j.jmaa.2005.12.066
  29. Martinet, B., Régularisation d'inéquations variationnelles par approximations successives, Rev. Franç. Inform. Rech. Opér. 4 (1970), 154-158 French. (1970) Zbl0215.21103MR0298899
  30. Moudafi, A., Oliny, M., 10.1016/S0377-0427(02)00906-8, J. Comput. Appl. Math. 155 (2003), 447-454. (2003) Zbl1027.65077MR1984300DOI10.1016/S0377-0427(02)00906-8
  31. Passty, G. B., 10.1016/0022-247X(79)90234-8, J. Math. Anal. Appl. 72 (1979), 383-390. (1979) Zbl0428.47039MR0559375DOI10.1016/0022-247X(79)90234-8
  32. Pesquet, J.-C., Pustelnik, N., A parallel inertial proximal optimization method, Pac. J. Optim. 8 (2012), 273-306. (2012) Zbl1259.47080MR2954380
  33. Polyak, B. T., 10.1016/0041-5553(64)90137-5, U.S.S.R. Comput. Math. Math. Phys. 4 (1967), 1-17 translation from Zh. Vychisl. Mat. Mat. Fiz. 4 1964 791-803. (1967) Zbl0147.35301MR0169403DOI10.1016/0041-5553(64)90137-5
  34. Reich, S., 10.1016/0022-247X(80)90323-6, J. Math. Anal. Appl. 75 (1980), 287-292. (1980) Zbl0437.47047MR0576291DOI10.1016/0022-247X(80)90323-6
  35. Rockafellar, R. T., 10.1137/0314056, SIAM J. Control Optim. 14 (1976), 877-898. (1976) Zbl0358.90053MR0410483DOI10.1137/0314056
  36. Shehu, Y., 10.1155/2016/5973468, J. Funct. Spaces 2016 (2016), Article ID 5973468, 9 pages. (2016) Zbl1337.47096MR3459658DOI10.1155/2016/5973468
  37. Shehu, Y., Cai, G., 10.1007/s13398-016-0366-3, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 112 (2018), 71-87. (2018) Zbl06836237MR3742991DOI10.1007/s13398-016-0366-3
  38. Suantai, S., Pholasa, N., Cholamjiak, P., 10.3934/jimo.2018023, J. Ind. Manag. Optim. 14 (2018), 1595-1615. (2018) MR3917875DOI10.3934/jimo.2018023
  39. Sunthrayuth, P., Cholamjiak, P., 10.1007/s11075-017-0411-0, Numer. Algorithms 78 (2018), 1019-1044. (2018) Zbl06916361MR3827320DOI10.1007/s11075-017-0411-0
  40. Tseng, P., 10.1137/S0363012998338806, SIAM J. Control Optim. 38 (2000), 431-446. (2000) Zbl0997.90062MR1741147DOI10.1137/S0363012998338806
  41. Wei, L., Agarwal, R. P., 10.1186/s13663-015-0495-y, Fixed Point Theory Appl. 2016 (2016), Article ID 7, 22 pages. (2016) Zbl1346.47071MR3441542DOI10.1186/s13663-015-0495-y
  42. Xu, H.-K., 10.1016/0362-546X(91)90200-K, Nonlinear Anal., Theory Methods Appl. 16 (1991), 1127-1138. (1991) Zbl0757.46033MR1111623DOI10.1016/0362-546X(91)90200-K

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