ON THE RELAXED HYBRID-EXTRAGRADIENT METHOD FOR SOLVING CONSTRAINED CONVEX MINIMIZATION PROBLEMS IN HILBERT SPACES

• Published in: Taiwanese journal of mathematics Vol. 17; no. 3; pp. 911 - 936
• Main Authors: Ceng, L. C., Chou, C. Y.
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.06.2013
• Subjects: Academic education; Algorithms; Hilbert spaces; Lipschitz condition; Mathematical monotonicity; Mathematical sets; Perceptron convergence procedure; Variational inequalities
• ISSN: 1027-5487; 2224-6851; 2224-6851
• DOI: 10.11650/tjm.17.2013.2567

In 2006, Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,SIAM J. Optim., 16(4) (2006), 1230-1241.] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality in a real Hilbert space via combining two well-known methods: hybrid and extragradient. In this paper, motivated by Nadezhkina and Takahashi’s hybrid-extragradient method we propose and analyze a relaxed hybridextragradient method for finding a solution of a constrained convex minimization problem, which is also a common element of the solution set of a variational inclusion and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We obtain a strong convergence theorem for three sequences generated by this algorithm. Based on this result, we also construct an iterative algorithm for finding a solution of the constrained convex minimization problem, which is also a common fixed point of two mappings taken from the more general class of strictly pseudocontractive mappings. 2010Mathematics Subject Classification: 49J40, 65K05, 47H09. Key words and phrases: Constrained convex minimization, Variational inclusion, Variational inequality, Nonexpansive mapping, Inverse strongly monotone mapping, Maximal monotone mapping, Strong convergence.