Smooth Primal-Dual Coordinate Descent Algorithms for Nonsmooth Convex Optimization

We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method features the first convergence rate guarantees among the coordinate descent methods, that are the best-known under a variety of common structure assumptions on the template. We provide numerical evidence to support the theoretical results with a comparison to state-of-the-art algorithms.Comment: NIPS 201

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

A generic coordinate descent solver for nonsmooth convex optimization

International audienceWe present a generic coordinate descent solver for the minimization of a nonsmooth convex objective with structure. The method can deal in particular with problems with linear constraints. The implementation makes use of efficient residual updates and automatically determines which dual variables should be duplicated. A list of basic functional atoms is pre-compiled for efficiency and a modelling language in Python allows the user to combine them at run time. So, the algorithm can be used to solve a large variety of problems including Lasso, sparse multinomial logistic regression, linear and quadratic programs

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Hybrid Methods in Polynomial Optimisation

The Moment/Sum-of-squares hierarchy provides a way to compute the global minimizers of polynomial optimization problems (POP), at the cost of solving a sequence of increasingly large semidefinite programs (SDPs). We consider large-scale POPs, for which interior-point methods are no longer able to solve the resulting SDPs. We propose an algorithm that combines a first-order Burer-Monteiro-type method for solving the SDP relaxation, and a second-order method on a non-convex problem obtained from the POP. The switch from the first to the second-order method is based on a quantitative criterion, whose satisfaction ensures that Newton's method converges quadratically from its first iteration. This criterion leverages the point-estimation theory of Smale and the active-set identification. We illustrate the methodology to obtain global minimizers of large-scale optimal power flow problems