Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces

Abstract

In this paper, we study strong convergence of some proximal-type algorithms to a solution of split minimization problem in complete p-uniformly convex metric spaces. We also analyse asymptotic behaviour of the sequence generated by Halpern-type proximal point algorithm and extend it to approximate a common solution of a finite family of minimization problems in the setting of complete p-uniformly convex metric spaces. Furthermore, numerical experiments of our algorithms in comparison with other algorithms are given to show the applicability of our results.

On Mixed Equilibrium Problems in Hadamard Spaces

“…In this case, x * is called a minimizer of Ψ and argmin y∈C Ψ(y) denotes the set of minimizers of Ψ. MPs are very useful in optimization theory and convex and nonlinear analysis. One of the most popular and effective methods for solving MPs is the proximal point algorithm (PPA) which was introduced in Hilbert space by Martinet [1] in 1970 and was further extensively studied in the same space by Rockafellar [2] in 1976. e PPA and its generalizations have also been studied extensively for solving MP (1) and related optimization problems in Banach spaces and Hadamard manifolds (see [3][4][5][6][7] and the references therein), as well as in Hadamard and p-uniformly convex metric spaces (see [8][9][10][11][12][13] and the references therein).…”

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

“…for all x, y, v ∈ X, t ∈ [0, 1]. The notion of p-uniformly convex metric spaces is a natural generalization of the notion of p-uniformly convex Banach spaces (see [9,14,41,42]). A typical example of p-uniformly convex metric spaces are L p spaces with p ≥ 2, CAT(0) spaces (with p = 2, and parameter c = 2) and CAT(k) spaces (k > 0) with diam(X) < π [14,35,42]).…”

“…al. [14] adopted (6) to study the Split Minimization Problem (SMP) in p-uniformly convex metric spaces. First, they studied some fundamental properties of the p-resolvent operator, which includes the unique existence, firmly nonexpansivity, nonexpansivity and the monotonicity properties of the p-resolvent of a proper convex and lower semicontinuous functional.…”

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

“…Variational inequality theory is an important tool in economics, engineering, mathematical programming, transportation, and in other fields (see, for example, [1][2][3][4][5][6][7][8]). Many numerical methods have been constructed for solving variational inequalities and related optimization problems, see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and references therein.…”