We study the class of nonsmooth nonconvex problems in which the objective is to minimize the difference between a continuously differentiable function (possibly nonconvex) and a convex (possibly nonsmooth) function over a convex polytope. This general class contains many types of problems, including difference of convex functions (DC) problems, and as such, can be used … Read more
Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call \emph{delay}, is the number of times it has been … Read more
We present a filter line-search algorithm for nonconvex continuous optimization that combines an augmented Lagrangian function and a constraint violation metric to accept and reject steps. The approach is motivated by real-time optimization applications that need to be executed on embedded computing platforms with limited memory and processor speeds. In particular, the proposed method enables … Read more
We introduce the Stochastic Asynchronous Proximal Alternating Linearized Minimization (SAPALM) method, a block coordinate stochastic proximal-gradient method for solving nonconvex, nonsmooth optimization problems. SAPALM is the first asynchronous parallel optimization method that provably converges on a large class of nonconvex, nonsmooth problems. We prove that SAPALM matches the best known rates of convergence — among … Read more
We introduce the Asynchronous PALM algorithm, a new extension of the Proximal Alternating Linearized Minimization (PALM) algorithm for solving nonconvex nonsmooth optimization problems. Like the PALM algorithm, each step of the Asynchronous PALM algorithm updates a single block of coordinates; but unlike the PALM algorithm, the Asynchronous PALM algorithm eliminates the need for sequential updates … Read more
We present a filter line-search algorithm that does not require inertia information about the linear system to ensure global convergence. The proposed approach performs curvature tests along the search step to ensure descent. This feature permits more modularity in the linear algebra, enabling the use of a wider range of iterative and decomposition strategies. We … Read more
The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … Read more
We investigate the BFGS algorithm with an inexact line search when applied to nonsmooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, assuming only absolute continuity. We also prove that the line search terminates … Read more
The idea of a finite collection of closed sets having “strongly regular intersection” at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably “regular” (special cases being convex … Read more
We prove that if two smooth manifolds intersect transversally, then the method of alternating projections converges locally at a linear rate. We bound the speed of convergence in terms of the angle between the manifolds, which in turn we relate to the modulus of metric regularity for the intersection problem, a natural measure of conditioning. … Read more