Multivariate convex approximation and least-norm convex data-smoothing

Abstract

The main contents of this paper is two-fold. First, we present a method to approximate multivariate convex functions by piecewise linear upper and lower bounds. We consider a method that is based on function evaluations only. However, to use this method, the data have to be convex. Unfortunately, even if the underlying function is convex, this is not always the case due to (numerical) errors. Therefore, secondly, we present a multivariate data-smoothing method that smooths nonconvex data. We consider both the case that we have only function evaluations and the case that we also have derivative information. Furthermore, we show that our methods are polynomial time methods. We illustrate this methodology by applying it to some examples. © Springer-Verlag Berlin Heidelberg 2006.