An Algorithmic Separating Hyperplane Theorem and Its Applications

TL;DR

This paper introduces a new separating hyperplane theorem and a generalized Triangle Algorithm to efficiently solve convex set intersection, distance, and margin problems with applications in machine learning and optimization.

Contribution

It presents a novel separating hyperplane theorem and extends the Triangle Algorithm for broader, more powerful applications in convex analysis and computational geometry.

Findings

Developed a new hyperplane separation theorem for convex sets.

Created a stronger, generalized Triangle Algorithm for various convex problems.

Achieved a fully polynomial-time approximation scheme for key convex set tasks.

Abstract

We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets $K, K'$ of the Euclidean space intersect, and when they are disjoint. The theorem is distinct from classical separation theorems. It generalizes the {\it distance duality} proved in our earlier work for testing the membership of a distinguished point in the convex hull of a finite point set. Next by utilizing the theorem, we develop a substantially generalized and stronger version of the {\it Triangle Algorithm} introduced in the previous work to perform any of the following three tasks: (1) To compute a pair $(p,p') \in K \times K'$, where either the Euclidean distance $d(p,p')$ is to within a prescribed tolerance, or the orthogonal bisecting hyperplane of the line segment $pp'$ separates the two sets; (2) When $K$ and $K'$ are disjoint, to compute $(p,p') \in K \times K'$ so that $d(p,p')$ approximates $d(K,K')$ to within a prescribed tolerance; (3) When $K$ and $K'$ are disjoint, to compute a pair of parallel supporting hyperplanes $H,H'$ so that $d(H,H')$ is to within a prescribed tolerance of the optimal margin. The worst-case complexity of each iteration is solving a linear objective over $K$ or $K'$. The resulting algorithm is a fully polynomial-time approximation scheme for such important special cases as when $K$ and $K'$ are convex hulls of finite points sets, or the intersection of a finite number of halfspaces. The results find many theoretical and practical applications, such as in machine learning, statistics, linear, quadratic and convex programming. In particular, in a separate article we report on a comparison of the Triangle Algorithm and SMO for solving the hard margin problem. In future work we extend the applications to combinatorial and NP-complete problems.