A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial … Read more
In this paper we extend naturally the scalarization proximal point method to solve multiobjective unconstrained minimization problems, proposed by Apolinario et al.(2016), from Euclidean spaces to Hadamard manifolds for locally Lipschitz and quasiconvex vector objective functions. Moreover, we present a convergence analysis, under some mild assumptions on the multiobjective function, for two inexact variants of … 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
We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to select the most relevant Benders cuts. We propose a limited memory variant of Level 1 and show the convergence of DDP … Read more
In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge.Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly … Read more