This note contains some ideas about linear programming and most-orthogonal faces. They’re mostly on an intuitive level and not very formal.
Postscriptum: The ideas here don’t work.
Linear programming
Maximize \(\t\x\) subject to \(A\x \leq \b\).
- \(\x\) is a vector of \(n\) variables \(x_i\).
- \(A\) is a \(m\times n\) matrix: there are \(m\) constraints \(A_j \x \leq b_j\).
Assumptions
We make the following assumptions:
- The system \(A\x\leq \b\) has a solution.
- None of the \(m\) constraints is redundant: for each constraint there is a solution such that equality holds in \(A_j\x \leq b_j\).
- All of the constraints satisfy \(\t A_j > 0\), meaning that the point at
\(-\infty \cdot \t\) is a solution.
- This actually feels quite limiting, but I’ll keep it to keep things simple.
- Without this constraint, the most-orthogonal face could be on the wrong/opposite side of the polygon.
Idea for an algorithm
Suppose \(n=2\), and we are given the polytope, which is an unbounded convex polygon in this case. The boundary of this polygon is given by a series of segments of increasing slope. The optimal solution happens around the segment that is most perpendicular to \(\t\), exactly where the slope of transitions from less than the slope of the perpendicular to \(\t\) to more than the slope of the perpendicular to \(\t\).
For general \(n\), we first find the face \(j_1\) that minimizes the angle between \(A_j\) and \(t\). If \(A_j\) is exactly orthogonal to \(t\), we found an optimal solution and we are done. Otherwise, we must find the ’next most-orthogonal’ face \(j_2\), with the restriction that it must not be ‘behind’ the previous face: In the \(2\) dimensional cases it could be that there are many lines very close to orthogonal to \(\t\) on one side of the optimal solution, and none on the other side. Those should be excluded.
To filter out near-orthogonal faces that are behind \(j_1\), I see a few options: (I am not sure how to formalize them at this point.)
- Only consider the part of the angle orthogonal to the angle between \(t\) and \(A_{j_1}\).
- This removes a bit too much information, since in the same plane there could be faces with a small angle in the opposite direction.
- Remove from the angle any component in the same direction as the angle between \(t\) and \(A_{j_1}\).
- Do some change of basis so that face \(j_1\) is not nearly orthogonal anymore (maybe by making \(\t\) and \(A_{j_1}\) basis vectors?) and find the most-orthogonal face after the transformation.
Then ideally we can repeatedly find the most-orthogonal face and once we find \(n\) of them (or once \(\t\) is a linear combination of \(A_{j_1}\) to \(A_{j_k}\)) we know that the optimal solution is at the intersection of those \(n\) faces.