ON THE SET OF HYPERPLANES MINIMIZING THE SUM OF ABSOLUTE DEVIATIONS

抄録

We shall consider the optimal hyperplanes which minimize the sum of absolute deviations with respect to the m-dimensional data. Though similar problem was treated in the linear programming, we shall discuss the problem from algebraic viewpoint, and we shall show that the set U of the coefficient vectors of all optimal hyperplanes of degree m-1 becomes a bounded and closed convex cell in E^m. Further, we shall obtain a necessary and sufficient condition for the extreme point of U, and then we establish a method for finding the set U. Thus we find that the optimal set U is completely determined as a hull of extreme points of U. Finally, we shall apply these results to the numerical examples and determine their optimal sets.