We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or … Read more
A block version of the BFGS variable metric update formula and its modifications are investigated. In spite of the fact that this formula satisfies the quasi-Newton conditions with all used difference vectors and that the improvement of convergence is the best one in some sense for quadratic objective functions, for general functions it does not … Read more
Adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity amongst second-order methods. In this paper, we propose and analyze the use of an iteration dependent scaled norm in the adaptive regularized framework using cubics. Within such scaled norm, the obtained method … Read more
There has been much recent interest in finding unconstrained local minima of smooth functions, due in part of the prevalence of such problems in machine learning and robust statistics. A particular focus is algorithms with good complexity guarantees. Second-order Newton-type methods that make use of regularization and trust regions have been analyzed from such a … Read more
We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic … Read more
The Levenberg-Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under … Read more
We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the solution of a so-called “subproblem” in which a trial step is computed by solving an optimization problem involving an approximation of the objective function, called “the model”. The … Read more
In order to be provably convergent towards a second-order stationary point, optimization methods applied to nonconvex problems must necessarily exploit both first and second-order information. However, as revealed by recent complexity analyzes of some of these methods, the overall effort to reach second-order points is significantly larger when compared to the one of approaching first-order … Read more
Automatic Differentiation (AD) is applied to the open-source CAD system Open CASCADE Technology using the AD software tool ADOL-C (Automatic Differentiation by OverLoading in C++). The differentiated CAD system is coupled with a discrete adjoint CFD solver, thus providing the first example of a complete differentiated design chain built from generic, multi-purpose tools. The design … Read more
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