Octahedral Projections of a Point onto a Polyhedron
In computational methods and mathematical modeling, it is often required to find vectors of a linear manifold or a polyhedron that are closest to a given point. The “closeness” can be understood in different ways. In particular, the distances generated by octahedral, Euclidean, and Hölder norms can be used. In these norms, weight coefficients can also be introduced and varied. This paper presents the results on the properties of a set of octahedral projections of the origin of coordinates onto a polyhedron. In particular, it is established that any Euclidean and Hölder projection can be obtained as an octahedral projection due to the choice of weights in the octahedral norm. It is proven that the set of octahedral projections of the origin of coordinates onto a polyhedron coincides with the set of Pareto-optimal solutions of the multicriterion problem of minimizing the absolute values of all components.
Keywordslinear inequalities polyhedron octahedral projections Euclidean projections Paretooptimal solutions
Unable to display preview. Download preview PDF.
- 1.J. C. F. Gauss, Selected Geophysical Works (Geodezizdat, Moscow, 1967) [in Russian].Google Scholar
- 5.V. I. Zorkal’tsev and M. A. Kiseleva, Systems of Linear Inequalities (Irkutsk. Gos. Univ., Irkutsk, 2007) [in Russian].Google Scholar
- 6.V. I. Mudrov and V. L. Kushko, Methods for Measurement Processing (Sovetskoe Radio, Moscow, 1976) [in Russian].Google Scholar
- 7.V. I. Mudrov and V. L. Kushko, Methods for Measurement Processing: Quasi-Likelihood Estimates (Radio i Svyaz’, Moscow, 1983) [in Russian].Google Scholar