In this work, we show that for linearly constrained optimization problems the primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can be written as an entirely primal algorithm. This allows us to prove convergence of the iterates even in the degenerate cases when the linear system is inconsistent or when the strong duality … Read more
We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov’s accelerated convex gradient method. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satis es the … Read more
Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming one of the most flexible operator splitting frameworks available. While weak convergence of the iterates to a solution has been established, … Read more
Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimization problems with exploitable special structures. … Read more
We investigate a second order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov’s accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the … Read more
We propose two numerical algorithms for minimizing the sum of a smooth function and the composition of a nonsmooth function with a linear operator in the fully nonconvex setting. The iterative schemes are formulated in the spirit of the proximal and, respectively, proximal linearized alternating direction method of multipliers. The proximal terms are introduced through … Read more
We generalize the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions via a new measure of steepness. For the deterministic projected subgradient method, we derive a global $O(1/\sqrt{T})$ convergence rate for any function with at most exponential growth. Our approach implies generalizations of the standard convergence rates for gradient descent on … Read more
We consider an accelerated proximal gradient algorithm for the composite optimization with “independent errors” (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing … Read more
We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be inde nite which plays … Read more
The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more