Fast Forward-Backward splitting for monotone inclusions with a convergence rate of the tangent residual of $o(1/k)$
arxiv(2023)
摘要
We address the problem of finding the zeros of the sum of a maximally
monotone operator and a cocoercive operator. Our approach introduces a
modification to the forward-backward method by integrating an inertial/momentum
term alongside a correction term. We demonstrate that the sequence of
iterations thus generated converges weakly towards a solution for the monotone
inclusion problem. Furthermore, our analysis reveals an outstanding attribute
of our algorithm: it displays rates of convergence of the order $o(1/k)$ for
the discrete velocity and the tangent residual approaching zero. These rates
for tangent residuals can be extended to fixed-point residuals frequently
discussed in the existing literature. Specifically, when applied to minimize a
nonsmooth convex function subject to linear constraints, our method evolves
into a primal-dual full splitting algorithm. Notably, alongside the convergence
of iterates, this algorithm possesses a remarkable characteristic of
nonergodic/last iterate $o(1/k)$ convergence rates for both the function value
and the feasibility measure. Our algorithm showcases the most advanced
convergence and convergence rate outcomes among primal-dual full splitting
algorithms when minimizing nonsmooth convex functions with linear constraints.
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