A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem

2023/03/06 by Taherinassaj, Ali, Chen, Yiling
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2303.05423

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.

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