A nearest point algorithm for convex polyhedral cones and applications to positive linear approximation

Abstract:

Suppose K is a convex polyhedral cone in ${E_n}$ and is defined in terms of some generating set $\{ {e_1},{e_2}, \ldots ,{e_N}\}$. A procedure is devised so that, given any point $q \in {E_n}$, the nearest point p in K to q can be found as a positive linear sum of ${N^\ast } \leqslant n$ points from the generating set. The procedure requires at most finitely many linear steps. The algorithm is then applied to find a positive representation \[ Lf = \sum \limits _{i = 1}^{{N^\ast }} {{\lambda _i}f({x_i}),} \quad {\lambda _i} > 0,f \in \Phi ,\] for a positive linear functional L acting on a suitable finite-dimensional function space $\Phi$.