A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem
Optimization and Control
2023-10-10 v2
Abstract
We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The insight is that two sets are separable by a hyperplane if there is a point from whose perspective the two sets appear to be convex. This global topological to local vector space relationship anticipates generalizations to manifolds as a possible future direction. The result might also be of pedagogical interest as it re-wires the structure in which the supporting hyperplane theorem and its dependencies are usually presented in convex optimization and analysis textbooks.
Cite
@article{arxiv.2303.05423,
title = {A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem},
author = {Ali Taherinassaj and Yiling Chen},
journal= {arXiv preprint arXiv:2303.05423},
year = {2023}
}