Search

In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more. [ABSTRACT FROM AUTHOR]

In this paper, we present two nonmonotone versions of adaptive cubic regularized (ARC) method for unconstrained optimization problems. The proposed methods are a combination of the ARC algorithm with the nonmonotone line search methods introduced by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056] and Ahookhosh et al. [A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl. Math. Model. 36 (2012), pp. 478–487]. The global convergence analysis for these iterative algorithms is established under suitable conditions. Several numerical examples are given to illustrate the efficiency and robustness of the newly suggested methods. The obtained results show the satisfactory performance of the proposed algorithms when compared to the basic ARC algorithm. [ABSTRACT FROM AUTHOR]