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

TL;DR

This paper introduces a differential-geometrical method to analytically determine the optimal unit-norm direction in linearly-constrained systems, maximizing or minimizing a linear objective, using wedge and Hodge-star products.

Contribution

It provides a novel analytic construction for optimal directions in constrained systems utilizing differential forms and geometric algebra techniques.

Findings

Explicit analytic solution using differential forms.

Applicable to systems with linear constraints and objective functions.

Enhances understanding of geometric structure in optimization problems.

Abstract

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 0 =< m < n) 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.