The goal of this paper is to study the effect of inexact first-order information on the first-order methods designed for smooth strongly convex optimization problems. We introduce the notion of (delta,L,mu)-oracle, that can be seen as an extension of the inexact (delta,L)-oracle previously introduced, taking into account strong convexity. We consider different examples of (delta,L,mu)-oracle: … Read more
This paper introduces a parallel and distributed extension to the alternating direction method of multipliers (ADMM). The algorithm decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This Jacobian-type algorithm is well suited for distributed computing and is particularly attractive for solving certain large-scale problems. This paper introduces … Read more
We propose a novel method, namely the accelerated mirror-prox (AMP) method, for computing the weak solutions of a class of deterministic and stochastic monotone variational inequalities (VI). The main idea of this algorithm is to incorporate a multi-step acceleration scheme into the mirror-prox method. For both deterministic and stochastic VIs, the developed AMP method computes … Read more
We describe an asynchronous parallel stochastic proximal coordinate descent algorithm for minimizing a composite objective function, which consists of a smooth convex function plus a separable convex function. In contrast to previous analyses, our model of asynchronous computation accounts for the fact that components of the unknown vector may be written by some cores simultaneously … Read more
The problem of finding sparse solutions to underdetermined systems of linear equations arises in several real-world problems (e.g. signal and image processing, compressive sensing, statistical inference). A standard tool for dealing with sparse recovery is the l1-regularized least-squares approach that has been recently attracting the attention of many researchers. In this paper, we describe an … Read more
In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for … Read more
We propose a new family of subgradient- and gradient-based methods which converges with optimal complexity for convex optimization problems whose feasible region is simple enough. This includes cases where the objective function is non-smooth, smooth, have composite/saddle structure, or are given by an inexact oracle model. We unified the way of constructing the subproblems which … Read more
This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the … Read more
In this paper, we rigorously study tractable models for provably recovering low-rank tensors. Unlike their matrix-based predecessors, current convex approaches for recovering low-rank tensors based on incomplete (tensor completion) and/or grossly corrupted (tensor robust principal analysis) observations still suffer from the lack of theoretical guarantees, although they have been used in various recent applications and … Read more
We propose an algorithm for solving the surrogate dual of a mixed integer program. The algorithm uses a trust region method based on a piecewise affine model of the dual surrogate value function. A new and much more flexible way of updating bounds on the surrogate dual’s value is proposed, which numerical experiments prove to … Read more