joint work with Andreas Themelis, Panagiotis Patrinos
We establish a link between the block-coordinate proximal gradient for minimizing the sum of two nonconvex functions (none of which is necessarily separable) and incremental aggregated proximal gradient methods for finite sum problems. The main tool for establishing this connection is the forward-backward envelope (FBE), which serves as a Lyapunov function. This result greatly simplifies the rate-of-convergence analysis and opens up the possibility to develop accelerated variants of incremental aggregated proximal gradient methods.
joint work with Yura Malitsky
The forward-backward splitting algorithm is a method for finding a zero in the sum of two monotone operators when one assumed single-valued and cocoercive. Unfortunately, the latter property, which is equivalent to strong monotonicity of the inverse, is too strong to hold in many monotone inclusions of interest. In this talk, I will report on recently discovered modifications of the forward-backward splitting algorithm which converge without requiring the cocoercivity assumption. Based on joint work with Yura Malitsky (University of Göttingen)
joint work with Robert Hannah, Wotao Yin
Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a rigorous correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of contractiveness of the corresponding operator.