Dearing and Zeck presented a dual algorithm for the problem of the minimum covering ball in $\mathbb{R}^n$. Each iteration of their algorithm has a computational complexity of at least $\mathcal O(n^3)$. In this paper we propose a modification to their algorithm that, together with an implementation that uses updates to the QR factorization of a … Read more
In the first part of the paper we focus on two problems: (a) regularized least squares and (b) nonsmooth minimization over an affine subspace. For these problems we establish the connection between the primal-dual method of Chambolle-Pock and Tseng’s proximal gradient method. For problem (a) it allows us to derive a nonergodic $O(1/k^2)$ convergence rate … Read more
The alternating direction method of multipliers (ADMM) was proposed by Glowinski and Marrocco in 1975; and it has been widely used in a broad spectrum of areas, especially in some sparsity-driven application domains. In 1982, Fortin and Glowinski suggested to enlarge the range of the step size for updating the dual variable in ADMM from … Read more
Designing distributed algorithms for empirical risk minimization (ERM) has become an active research topic in recent years because of the practical need to deal with the huge volume of data. In this paper, we propose a general framework for training an ERM model via solving its dual problem in parallel over multiple machines. Our method … Read more
The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well-known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive … Read more
We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the … Read more
The Lojasievicz inequality shows that sharpness bounds on the minimum of convex optimization problems hold almost generically. Here, we show that sharpness directly controls the performance of restart schemes. The constants quantifying sharpness are of course unobservable, but we show that optimal restart strategies are fairly robust, and searching for the best scheme only increases … Read more
We show that accelerated optimization methods can be seen as particular instances of multi-step integration schemes from numerical analysis, applied to the gradient flow equation. In comparison with recent advances in this vein, the differential equation considered here is the basic gradient flow and we show that multi-step schemes allow integration of this differential equation … Read more
In this paper, we analyze in depth a simplicial decomposition like algorithmic framework for large scale convex quadratic programming. In particular, we first propose two tailored strategies for handling the master problem. Then, we describe a few techniques for speeding up the solution of the pricing problem. We report extensive numerical experiments on both real … Read more
Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we study the oracle complexity of such methods, or equivalently, the number of iterations required to optimize a function to a given accuracy. Focusing on smooth and convex functions, we derive … Read more