Abstract
The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.
Highlights
Let H be a real Hilbert space, the associated product is denoted by h·, ·i and the corresponding norm is k · k
Let T : H → 2 H be a maximally monotone operator with its domain and range denoted by Dom( T ) and R( T )
And in what follows, let I be the identity operator, the resolvent of T with parameter λ > 0 is defined by JλT = ( I + λT )−1, and the Yoshida approximation of T with index λ is denoted by λ T = λ1 ( I − JλT ), respectively
Summary
Let H be a real Hilbert space, the associated product is denoted by h·, ·i and the corresponding norm is k · k. It is worth mentioning that the scaled proximity operator (10) was extensively used in [32,33] for deriving effective iterative algorithms to solve structural convex optimization problems These works didn’t consider the general scaled proximity operator of φ ◦ A and the related resolvent operator problem. We employ the obtained results to solve the problem of computing scaled proximity operators of a convex function composed by a linear operator and a finite sum of proper, lower semi-continuous convex functions, respectively. We employ the proposed fixed point algorithm to compute the resolvent of the sum of a finite number of maximally monotone operators with U. We give some conclusions and future work in the last section
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