31 research outputs found
Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers
In this work, using Moreau envelopes, we define a complete metric for the set
of proper lower semicontinuous convex functions. Under this metric, the
convergence of each sequence of convex functions is epi-convergence. We show
that the set of strongly convex functions is dense but it is only of the first
category. On the other hand, it is shown that the set of convex functions with
strong minima is of the second category
Special Libraries, May-June 1941
Volume 32, Issue 5https://scholarworks.sjsu.edu/sla_sl_1941/1004/thumbnail.jp
Using generalized simplex methods to approximate derivatives
This paper presents two methods for approximating a proper subset of the
entries of a Hessian using only function evaluations. These approximations are
obtained using the techniques called \emph{generalized simplex Hessian} and
\emph{generalized centered simplex Hessian}. We show how to choose the matrices
of directions involved in the computation of these two techniques depending on
the entries of the Hessian of interest. We discuss the number of function
evaluations required in each case and develop a general formula to approximate
all order- partial derivatives. Since only function evaluations are required
to compute the methods discussed in this paper, they are suitable for use in
derivative-free optimization methods.Comment: arXiv admin note: text overlap with arXiv:2304.0322