×

zbMATH — the first resource for mathematics

Averaged mappings and the gradient-projection algorithm. (English) Zbl 1233.90280
Author’s abstract: “It is well-known that the gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this article, we first provide an alternative averaged mapping approach to the GPA. This approach is operator-oriented in nature. Since, in general, in infinite-dimensional Hilbert spaces, GPA has only weak convergence, we provide two modifications of GPA so that strong convergence is guaranteed. Regularization is also applied to find the minimum-norm solution of the minimization problem under investigation.”

MSC:
90C48 Programming in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Levitin, E.S., Polyak, B.T.: Constrained minimization methods. Zh. Vychisl. Mat. Mat. Fiz. 6, 787–823 (1966) · Zbl 0184.38902
[2] Calamai, P.H., Moré, J.J.: Projected gradient methods for linearly constrained problems. Math. Program. 39, 93–116 (1987) · Zbl 0634.90064
[3] Polyak, B.T.: Introduction to optimization. In: Optimization Software, New York (1987) · Zbl 0708.90083
[4] Su, M., Xu, H.K.: Remarks on the gradient-projection algorithm. J. Nonlinear Anal. Optim. 1, 35–43 (2010) · Zbl 1413.90272
[5] Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) · Zbl 0828.65065
[6] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004) · Zbl 1051.65067
[7] Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005) · Zbl 1089.65046
[8] Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
[9] Xu, H.K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006) · Zbl 1126.47057
[10] Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010) · Zbl 1213.65085
[11] Lopez, G., Martin, V., Xu, H.K.: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal., Real World Appl. 10, 2369–2383 (2009) · Zbl 1163.47306
[12] Lopez, G., Martin, V., Xu, H.K.: Iterative algorithms for the multiple-sets split feasibility problem. In: Censor, Y., Jiang, M., Wang, G. (eds.) Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, pp. 243–279. Medical Physics Publishing, Madison (2009)
[13] Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004) · Zbl 1153.47305
[14] Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
[15] Brezis, H.: Operateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)
[16] Bertsekas, D.P., Gafni, E.M.: Projection methods for variational inequalities with applications to the traffic assignment problem. Math. Program. Stud. 17, 139–159 (1982) · Zbl 0478.90071
[17] Han, D., Lo, H.K.: Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities. Eur. J. Oper. Res. 159, 529–544 (2004) · Zbl 1065.90015
[18] Martinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed-point iteration processes. Nonlinear Anal. 64, 2400–2411 (2006) · Zbl 1105.47060
[19] Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) · Zbl 1013.47032
[20] Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967) · Zbl 0179.19902
[21] Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Féjer-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001) · Zbl 1082.65058
[22] Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967) · Zbl 0149.36301
[23] Baillon, J.B., Haddad, G.: Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977) · Zbl 0352.47023
[24] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979) · Zbl 0423.47026
[25] Hundal, H.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004) · Zbl 1070.46013
[26] Bauschke, H.H., Burke, J.V., Deutsch, F.R., Hundal, H.S., Vanderwerff, J.D.: A new proximal point iteration that converges weakly but not in norm. Proc. Am. Math. Soc. 133, 1829–1835 (2005) · Zbl 1071.65082
[27] Bauschke, H.H., Matouskova, E., Reich, S.: Projections and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004) · Zbl 1059.47060
[28] Matouskova, E., Reich, S.: The Hundal example revisited. J. Nonlinear Convex Anal. 4, 411–427 (2003) · Zbl 1065.47048
[29] Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program., Ser. A 87, 189–202 (2000) · Zbl 0971.90062
[30] Marino, G., Xu, H.K.: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791–808 (2004) · Zbl 1095.90115
[31] Xu, H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006) · Zbl 1131.90062
[32] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053
[33] Güler, O.: On the convergence of the proximal point algorithm for convex optimization. SIAM J. Control Optim. 29, 403–419 (1991) · Zbl 0737.90047
[34] Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000) · Zbl 0957.47039
[35] Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060
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.