A New Randomized Block-Coordinate Primal-Dual Proximal Algorithm for Distributed Optimization

This paper proposes TriPD, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the block-coordinate scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is weaker than strong convexity). Moreover, we apply the developed algorithms to the problem of multi-agent optimization on a graph, thus obtaining novel synchronous and asynchronous distributed methods. The proposed algorithms are fully distributed in the sense that the updates and the stepsizes of each agent only depend on local information. In fact, no prior global coordination is required. Finally, we showcase an application of our algorithm in distributed formation control.status: publishe

On the Exact Solution of the Distance Geometry with Interval Distances in Dimension 1

International audienceDistance Geometry consists in embedding a simple weighted undirected graph in a given space so that the distances between embedded vertices correspond to the edge weights. Weights can be either exact real values, or real-valued intervals. In this work, the focus is on problems where the embedding space is the Euclidean 1-dimensional space, and the general situation where distances can be represented by intervals is taken into consideration. A previously proposed branch-and-prune algorithm is adapted to the 1-dimensional case, and the proposed variant turns out to be deterministic even in presence of interval distances. Backtracking pruning is introduced in the algorithm for guaranteeing that all vertex positions in a given solution are actually feasible. The proposed algorithm is tested on a set of artificially generated instances in dimension 1