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This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor… (More)
We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in different contexts. The method converges after O(n) iterations with overall arithmetic complexity O(n 2). Numerical experiments show that in practice the method converges in a small number of iterations with… (More)
This work shows that the BFGS method and other methods in the Broyden class, with exact line searches, may fail for non-convex objective functions.
We discuss the convergence of line search methods for minimization. We explain how Newton's method and the BFGS method can fail even if the restrictions of the objective function to the search lines are strictly convex functions, the level sets of the objective functions are compact, the line searches are exact and the Wolfe conditions are satisfied. This… (More)
In this note we discuss the convergence of Newton's method for minimization. We present examples in which the Newton iterates satisfy the Wolfe conditions and the Hessian is positive definite at each step and yet the iterates converge to a non-stationary point. These examples answer a question posed by Fletcher in his 1987 book Practical methods of… (More)
A new method is introduced for packing objects in convex regions of the Euclidian n-dimensional space. By means of this approach the packing problem becomes a global finite-dimensional continuous optimization problem. The strategy is based on the new concept of sentinels sets. Sentinels sets are finite subsets of the objects to be packed such that when two… (More)