Note on a Differential-Geometrical Construction of Optimal Directions in Linearly-Constrained Systems

This note presents an analytic construction of the optimal unit-norm direction hat(x) = x/|x| that maximizes or minimizes the objective linear expression, B . hat{x}, subject to a system of linear constraints of the form [A] . x = 0, where x is an unknown n-dimensional real vector to be determined up to an overall normalization constant, 0 is an m-dimensional null vector, and the n-dimensional real vector B and the m\times n-dimensional real matrix [A] (with m < n and n >= 2) are given. The analytic solution to this problem can be expressed in terms of a combination of double wedge and Hodge-star products of differential forms.