$$\alpha $$-Firmly nonexpansive operators on metric spaces

We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.

Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond

In the setting of $\operatorname{CAT}(\kappa)$ CAT ( κ ) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.