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.
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)
Dedicated to our friends Beresford and Velvel on the occasion of their sixtieth birthdays. ABSTRACT We show that a certain matrix norm ratio studied by Parlett has a supremum that is O(p n) when the chosen norm is the Frobenius norm, while it is O(log n) for the 2-norm. This ratio arises in Parlett's analysis of the Cholesky decomposition of an n by n… (More)
We present fast and accurate ways to normalize two and three dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the computational geometry and computer aided design communities are not aware of the possibility of speeding up these fundamental… (More)
We present examples of divergence for the BFGS and Gauss Newton methods. These examples have objective functions with bounded level sets and other properties concerning the examples published recently in this journal, like unit steps and convexity along the search lines. As these other examples, the iterates, function values and gradients in the new… (More)