arXiv subjects

Jorge Nocedal

Publications and source records attributed to Jorge Nocedal.

21 records · Page 2Linked to original sources

arXiv2015

A Second-Order Method for Convex $\ell_1$-Regularized Optimization with Active Set Prediction

We describe an active-set method for the minimization of an objective function $ϕ$ that is the sum of a smooth convex function and an $\ell_1$-regularization term. A distinctive feature of the method is the way in which active-set identification and {second-order} subspace minimization steps are integrated to combine the predictive power of the two approaches. At every iteration, the algorithm selects a candidate set of free and fixed variables, performs an (inexact) subspace phase, and then assesses the quality of the new active set. If it is not judged to be acceptable, then the set of free variables is restricted and a new active-set prediction is made. We establish global convergence for our approach, and compare the new method against the state-of-the-art code LIBLINEAR.

math.OC↗

arXiv2014

An Algorithm for Quadratic $\ell_1$-Regularized Optimization with a Flexible Active-Set Strategy

We present an active-set method for minimizing an objective that is the sum of a convex quadratic and $\ell_1$ regularization term. Unlike two-phase methods that combine a first-order active set identification step and a subspace phase consisting of a \emph{cycle} of conjugate gradient (CG) iterations, the method presented here has the flexibility of computing a first-order proximal gradient step or a subspace CG step at each iteration. The decision of which type of step to perform is based on the relative magnitudes of some scaled components of the minimum norm subgradient of the objective function. The paper establishes global rates of convergence, as well as work complexity estimates for two variants of our approach, which we call the iiCG method. Numerical results illustrating the behavior of the method on a variety of test problems are presented.

math.OC↗

arXiv2013

An Inexact Successive Quadratic Approximation Method for Convex L-1 Regularized Optimization

We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model of the objective function. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iteration. Our inexactness conditions are based on a semi-smooth function that represents a (continuous) measure of the optimality conditions of the problem, and that embodies the soft-thresholding iteration. We give careful consideration to the algorithm employed for the inner minimization, and report numerical results on two test sets originating in machine learning.

math.OC↗