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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose *K* is a convex polyhedral cone in and is defined in terms of some generating set . A procedure is devised so that, given any point , the nearest point *p* in *K* to *q* can be found as a positive linear sum of points from the generating set. The procedure requires at most finitely many linear steps.

The algorithm is then applied to find a positive representation

*L*acting on a suitable finite-dimensional function space .

**[1]**Philip J. Davis,*A construction of nonnegative approximate quadratures*, Math. Comp.**21**(1967), 578–582. MR**0222534**, 10.1090/S0025-5718-1967-0222534-4**[2]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[3]**A. H. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR**0327006****[4]**Vladimir Tchakaloff,*Formules de cubatures mécaniques à coefficients non négatifs*, Bull. Sci. Math. (2)**81**(1957), 123–134 (French). MR**0094632****[5]**Don R. Wilhelmsen,*Nonnegative cubature on convex sets*, SIAM J. Numer. Anal.**11**(1974), 332–346. MR**0347062****[6]**M. Wayne Wilson,*A general algorithm for nonnegative quadrature formulas*, Math. Comp.**23**(1969), 253–258. MR**0242374**, 10.1090/S0025-5718-1969-0242374-1**[7]**M. Wayne Wilson,*Necessary and sufficient conditions for equidistant quadrature formula.*, SIAM J. Numer. Anal.**7**(1970), 134–141. MR**0282530****[8]**Philip Wolfe,*Finding the nearest point in a polytope*, Math. Programming**11**(1976), no. 2, 128–149. MR**0452683**

Retrieve articles in *Mathematics of Computation*
with MSC:
52A25,
65D99

Retrieve articles in all journals with MSC: 52A25, 65D99

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1976-0394439-5

Keywords:
Convex set,
nearest point,
projection,
positive linear approximation,
linear algorithm,
cubature

Article copyright:
© Copyright 1976
American Mathematical Society