The popular BFGS quasi-Newton minimization algorithm under reasonable conditions converges globally on smooth convex functions. This result was proved by Powell in 1976: we consider its implications for functions that are not smooth. In particular, an analogous convergence result holds for functions, like the Euclidean norm, that are nonsmooth at the minimizer. Citation Manuscript: School … Read more
This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more
Trust-region algorithms have been proved to globally converge with probability one when the accuracy of the trust-region models is imposed with a certain probability conditioning on the iteration history. In this paper, we study their complexity, providing global rates and worst case complexity bounds on the number of iterations (with overwhelmingly high probability), for both … Read more
The seminal paper by Barzilai and Borwein [IMA J. Numer. Anal. 8 (1988)] has given rise to an extensive investigation aimed at developing effective gradient methods, able to deal with large-scale optimization problems. Several steplength rules have been first designed for unconstrained quadratic problems and then extended to general nonlinear problems; these rules share the … Read more
We describe the fundamentals of algorithms for minimizing a smooth nonlinear function, and extensions of these methods to the sum of a smooth function and a convex nonsmooth function. Such objective functions are ubiquitous in data analysis applications, as we illustrate using several examples. We discuss methods that make use of gradient (first-order) information about … Read more
We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{\rm Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m – 1$ containing the set of Pareto values of … Read more
Building upon multigrid methods, the framework of multilevel optimization methods was developed to solve structured optimization problems, including problems in optimal control, image processing, etc. In this paper, we give a broader view of the multilevel framework and establish some connections between multilevel algorithms and the other approaches. An interesting case of the so called … Read more
Cubic-regularization and trust-region methods with worst case first-order complexity $O(\varepsilon^{-3/2})$ and worst-case second-order complexity $O(\varepsilon^{-3})$ have been developed in the last few years. In this paper it is proved that the same complexities are achieved by means of a quadratic regularization method with a cubic sufficient-descent condition instead of the more usual predicted-reduction based descent. … Read more
The limited memory steepest descent method (LMSD) proposed by Fletcher is an extension of the Barzilai-Borwein “two-point step size” strategy for steepest descent methods for solving unconstrained optimization problems. It is known that the Barzilai-Borwein strategy yields a method with an R-linear rate of convergence when it is employed to minimize a strongly convex quadratic. … Read more
We introduce a quasi-Newton method with block updates called Block BFGS. We show that this method, performed with inexact Armijo-Wolfe line searches, converges globally and superlinearly under the same convexity assumptions as BFGS. We also show that Block BFGS is globally convergent to a stationary point when applied to non-convex functions with bounded Hessian, and … Read more