Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity with applications to variational inequality, conic constrained saddle point, and convex conic optimization problems

In this paper we consider a class of structured monotone inclusion (MI) problems that consist of finding a zero in the sum of two monotone operators, in which one is maximal monotone while another is locally Lipschitz continuous. In particular, we first propose a primal-dual extrapolation (PDE) method for solving a structured strongly MI problem … Read more

On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions

In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated. The FBHF splitting method adjusts and improves Tseng’s forward-backward-forward (FBF) split- ting method when the inclusion problem has a third-part operator that is cocoercive. The FBHF method recovers the FBF iteration (when this aforementioned … Read more

Exterior-point Optimization for Nonconvex Learning

In this paper we present the nonconvex exterior-point optimization solver (NExOS)—a novel first-order algorithm tailored to constrained nonconvex learning problems. We consider the problem of minimizing a convex function over nonconvex constraints, where the projection onto the constraint set is single-valued around local minima. A wide range of nonconvex learning problems have this structure including … Read more

A relative-error inertial-relaxed inexact projective splitting algorithm

For solving structured monotone inclusion problems involving the sum of finitely many maximal monotone operators, we propose and study a relative-error inertial-relaxed inexact projective splitting algorithm. The proposed algorithm benefits from a combination of inertial and relaxation effects, which are both controlled by parameters within a certain range. We propose sufficient conditions on these parameters … Read more

A parallel splitting ALM-based algorithm for separable convex programming

The augmented Lagrangian method (ALM) provides a benchmark for tackling the canonical convex minimization problem with linear constraints. We consider a special case where the objective function is the sum of $m$ individual subfunctions without coupled variables. The recent study reveals that the direct extension of ALM for separable convex programming problems is not necessarily … Read more

Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms

This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based on a new inexact version of the proximal point algorithm that includes both an inertial step and overrelaxation. We apply our new inexact ADMM method … Read more

Projective Hedging for Stochastic Programming

We propose a decomposition algorithm for multistage stochastic programming that resembles the progressive hedging method of Rockafellar and Wets, but is provably capable of several forms of asynchronous operation. We derive the method from a class of projective operator splitting methods fairly recently proposed by Combettes and Eckstein, significantly expanding the known applications of those … Read more

Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting

In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primal-dual hybrid gradient method, allows the presence of … Read more

Convergence Rates for Projective Splitting

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

Projective Splitting with Forward Steps: Asynchronous and Block-Iterative Operator Splitting

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the … Read more