THE CHEBYSHEV PROJECTIONS ON POLYHEDRON

Abstract

The problem of minimizing weighted Chebyshev norm on a convex polyhedron defined as the set of solutions to a system of linear inequalities may have a non-unique solution. Moreover, among the solutions to this problem, there may be clearly not suitable points of the polyhedron for the role of the closest points to the zero vector. It complicates, in particular, the Chebyshev approximation. In order to overcome the problems arising from this, the Haar condition is used, which means the requirement for the uniqueness of the solution of the indicated problem. This requirement is not always easy to verify and it is not clear what to do if it is not true. An algorithm is presented that always generates a unique solution to the indicated problem, based on the search with respect to interior points for optimal solutions of a finite sequence of linear programming problems. The solution developed is called the Chebyshev projection of the origin onto the polyhedron. It is proved that this solution is a vector of a polyhedron with Pareto-minimal absolute values of the components. It is proved that the sets of Chebyshev (according to the introduced algorithm) and Euclidean projections of the origin of coordinates onto the polyhedron, formed by varying the positive weight coefficients in the minimized Euclidean and Chebyshev norms, coincide.