  Vertex direction algorithms have been around for a few decades in the
experimental design and mixture models literature. We briefly review this type
of algorithm and describe a new member of the family: the support reduction
algorithm. The support reduction algorithm is applied to the problem of
computing nonparametric estimates in two inverse problems: convex density
estimation and the Gaussian deconvolution problem. Usually, VD algorithms solve
a finite dimensional (version of the) optimization problem of interest. We
introduce a method to solve the true infinite dimensional optimization problem.
