Related papers: Linear convergence of the Douglas-Rachford algorit…
We study the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex, and inconsistent feasibility problems. This algorithm can be viewed as a convex relaxation between the cyclic Douglas-Rachford algorithm first introduced by…
We study the optimal lower and upper complexity bounds for finding approximate solutions to the composite problem $\min_x\ f(x)+h(Ax-b)$, where $f$ is smooth and $h$ is convex. Given access to the proximal operator of $h$, for strongly…
More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas-Rachford algorithm. Today, this method is well recognized as a classical and highly successful splitting method to find minimizers of the sum of two (not…
We show that the weak convergence of the Douglas--Rachford algorithm for finding a zero of the sum of two maximally monotone operators cannot be improved to strong convergence. Likewise, we show that strong convergence can fail for the…
This paper proposes a universal algorithm for convex minimization problems of the composite form $g_0(x)+h(g_1(x),\dots, g_m(x)) + u(x)$. We allow each $g_j$ to independently range from being nonsmooth Lipschitz to smooth, from convex to…
In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a…
This paper is devoted to the design of efficient primal-dual algorithm (PDA) for solving convex optimization problems with known saddle-point structure. We present a new PDA with larger acceptable range of parameters and correction, which…
In recent years, the Douglas-Rachford splitting method has been shown to be effective at solving many non-convex optimization problems. In this paper we present a local convergence analysis for non-convex feasibility problems and show that…
We consider the problem of finding a saddle point for the convex-concave objective $\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly…
We study the solution of minimax problems $\min_x \max_y G(x) + \langle K(x),y\rangle - F^*(y)$ in finite-dimensional Hilbert spaces. The functionals $G$ and $F^*$ we assume to be convex, but the operator $K$ we allow to be non-linear. We…
Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization…
We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection to the minimization problem of the sum of a convex lower semi-continuous function $f$ and a…
In this paper, we propose a unified primal-dual algorithm framework based on the augmented Lagrangian function for composite convex problems with conic inequality constraints. The new framework is highly versatile. First, it not only covers…
We study the composite convex optimization problems with a Quasi-Self-Concordant smooth component. This problem class naturally interpolates between classic Self-Concordant functions and functions with Lipschitz continuous Hessian.…
In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our…
We propose a new primal-dual algorithmic framework for a prototypical constrained convex optimization template. The algorithmic instances of our framework are universal since they can automatically adapt to the unknown Holder continuity…
We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization $\min_{x \in S} f(x)$ with a proximally smooth set $S =…
Error bound analysis, which estimates the distance of a point to the solution set of an optimization problem using the optimality residual, is a powerful tool for the analysis of first-order optimization algorithms. In this paper, we use…
Recently, heuristics based on the Douglas-Rachford splitting algorithm and the alternating direction method of multipliers (ADMM) have found empirical success in minimizing convex functions over nonconvex sets, but not much has been done to…
Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…