A simple algorithm for computing projection onto intersection of finite level sets

Abstract

We consider the problem of computing the projection P C u, where u is chosen in a real Hilbert space H arbitrarily and the closed convex subset C of H is the intersection of finite level sets of convex functions given as follows:, where m is a positive integer and c i : H → R is a convex function for i = 1, . . . , m. A relaxed Halpern-type algorithm is proposed for computing the projection P C u in this paper, which is defined by xn , n ≥ 0, where the initial guess x 0 ∈ H is chosen arbitrarily, the sequence (λ n ) is in (0, 1) and (C i n ) is a sequence of half-spaces containing C i for i = 1, . . . , m. Since calculations of the projections onto half-spaces C i n (i = 1, . . . , m; n = 1, 2, . . .) are easy in practice, this algorithm is quite implementable. Strong convergence of our algorithm is proved under some ordinary conditions. Some numerical experiments are provided which show advantages of our algorithm. MSC: 58E35; 47H09; 65J15

The Combination Projection Method for Solving Convex Feasibility Problems

“…Further results on the SOP were obtained by Gubin et al [26] and Bruck et al [27]. The SOP is the most fundamental method to solve CFP, and many existing algorithms [24,28] can be regarded as generalizations or variants of the SOP. Let {C i } m i=1 be a finite family of level sets of convex functions {c i } m i=1 (i.e., [9], He et al [28] introduced a contraction type sequential projection algorithm which generates the iterating process:…”