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Tian, Ming; Gong, Jun-Ying

Strong convergence of modified algorithms based on the regularization for the constrained convex minimization problem. (English) Zbl 1473.47043

Abstr. Appl. Anal. 2014, Article ID 870102, 9 p. (2014).
Summary: As is known, the regularization method plays an important role in solving constrained convex minimization problems. Based on the idea of regularization, implicit and explicit iterative algorithms are proposed in this paper and the sequences generated by the algorithms can converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. As an application, we also apply the algorithm to solve the split feasibility problem.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

strong convergence; implicit iterative algorithms; split feasibility problem; regularisation method
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\textit{M. Tian} and \textit{J.-Y. Gong}, Abstr. Appl. Anal. 2014, Article ID 870102, 9 p. (2014; Zbl 1473.47043)
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References:

[1] Xu, H.-K., Averaged mappings and the gradient-projection algorithm, Journal of Optimization Theory and Applications, 150, 2, 360-378 (2011) · Zbl 1233.90280
[2] Levitin, E.-S.; Polyak, B.-T., Constrained minimization methods, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 6, 787-823 (1996) · Zbl 0184.38902
[3] Calamai, P. H.; Moré, J. J., Projected gradient methods for linearly constrained problems, Mathematical Programming, 39, 1, 93-116 (1987) · Zbl 0634.90064
[4] Polyak, B. T., Introduction to Optimization (1987), New York, NY, USA: Optimization Software, New York, NY, USA
[5] Su, M.; Xu, H.-K., Remarks on the gradient-projection algorithm, Journal of Nonlinear Analysis and Optimization, 1, 1, 35-43 (2010) · Zbl 1413.90272
[6] Moudafi, A., Viscosity approximation methods for fixed-points problems, Journal of Mathematical Analysis and Applications, 241, 1, 46-55 (2000) · Zbl 0957.47039
[7] Xu, H.-K., Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 1, 279-291 (2004) · Zbl 1061.47060
[8] Ceng, L.-C.; Ansari, Q. H.; Yao, J.-C., Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Analysis: Theory, Methods & Applications, 74, 16, 5286-5302 (2011) · Zbl 1368.47046
[9] Marino, G.; Xu, H.-K., Convergence of generalized proximal point algorithms, Communications on Pure and Applied Analysis, 3, 4, 791-808 (2004) · Zbl 1095.90115
[10] Marino, G.; Xu, H.-K., A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 1, 43-52 (2006) · Zbl 1095.47038
[11] Al-Mazrooei, A. E.; Latif, A.; Yao, J. C., Solving generalized mixed equilibria, variational inequalities, and constrained convex minimization, Abstract and Applied Analysis, 2014 (2014) · Zbl 1473.47027
[12] Censor, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8, 2-4, 221-239 (1994) · Zbl 0828.65065
[13] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 1, 103-120 (2004) · Zbl 1051.65067
[14] Censor, Y.; Elfving, T.; Kopf, N.; Bortfeld, T., The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21, 6, 2071-2084 (2005) · Zbl 1089.65046
[15] Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A., A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine and Biology, 51, 10, 2353-2365 (2006)
[16] Xu, H.-K., A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22, 6, 2021-2034 (2006) · Zbl 1126.47057
[17] Xu, H.-K., Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26, 10 (2010) · Zbl 1213.65085
[18] Tian, M., A general iterative algorithm for nonexpansive mappings in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 73, 3, 689-694 (2010) · Zbl 1192.47064
[19] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory. Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics (1990), Cambridge University Press · Zbl 0708.47031
[20] Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mapping. Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mapping, Monographs and Textbooks in Pure and Applied Mathematics, 83 (1984), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0537.46001
[21] Ceng, L.-C.; Wong, M.-M.; Petruşel, A.; Yao, J.-C., Relaxed implicit extragradient-like methods for finding minimum-norm solutions of the split feasibility problem, Fixed Point Theory, 14, 2, 327-344 (2013) · Zbl 1292.47043
[22] Ceng, L. C.; Petruşel, A.; Yao, J. C., Relaxed extragradient methods with regularization for general system of variational inequalities with constraints of split feasibility and fixed point problems, Abstract and Applied Analysis, 2013 (2013) · Zbl 1272.49061
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