An adaptive splitting algorithm for the sum of three operators

TL;DR

This paper introduces an adaptive splitting algorithm for finding zeros of the sum of three operators, combining forward and backward steps with adaptive parameters to handle different operator properties.

Contribution

It proposes a novel adaptive splitting method that accommodates operators with different properties, especially when some resolvents are costly or unavailable.

Findings

Algorithm guarantees convergence under specified conditions.

Handles operators with generalized monotonicity and cocoercivity.

Adapts parameters based on operator properties for improved performance.

Abstract

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.