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Mathematics of Computation

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

Author: Don R. Wilhelmsen
Journal: Math. Comp. 30 (1976), 48-57
MSC: Primary 52A25; Secondary 65D99
MathSciNet review: 0394439
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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

$\displaystyle Lf = \sum\limits_{i = 1}^{{N^\ast}} {{\lambda _i}f({x_i}),} \quad {\lambda _i} > 0,f \in \Phi ,$

for a positive linear functional L acting on a suitable finite-dimensional function space $ \Phi $.

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Keywords: Convex set, nearest point, projection, positive linear approximation, linear algorithm, cubature
Article copyright: © Copyright 1976 American Mathematical Society

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