Frugal forward-backward splitting methods with deviations

The deviation vectors provide additional degrees of freedom and effectively enhance the flexibility of algorithms. In the literature, the iterative schemes with deviations are constructed and their convergence analyses are performed on an inefficient, algorithm-by-algorithm basis. In this paper, we address these by providing a general framework of frugal forward-backward splitting methods with deviations for finding zeros in the sum of a finite number of maximally monotone operators and cocoercive operators. Our framework encompasses the Douglas--Rachford splitting method with deviations. A unified weak convergence analysis is made under mild conditions. Numerical experiments on Markowitz portfolio optimization problem are given to demonstrate the effectiveness of deviations.