On the Relation between the Extended Supporting Hyperplane Algorithm and Kelley's Cutting Plane Algorithm

TL;DR

This paper reveals the geometric connection between the supporting hyperplane algorithm and Kelley's cutting plane method, extending its applicability to broader convex and non-convex problems with differentiable functions.

Contribution

It demonstrates the equivalence of the supporting hyperplane algorithm and Kelley's cutting plane algorithm through a geometric perspective, broadening its use to more general problems.

Findings

Supporting hyperplane algorithm is equivalent to Kelley's cutting plane method.

The algorithm applies to non-convex differentiable functions under mild conditions.

Extension improves the algorithm's applicability to a wider class of problems.

Abstract

Recently, Kronqvist et al.~\cite{KronqvistLundellWesterlund2016} rediscovered the supporting hyperplane algorithm of Veinott~\cite{Veinott1967} and demonstrated its computational benefits for solving convex mixed-integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley's cutting plane algorithm~\cite{J.E.Kelley1960} applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by general, not necessarily convex, differentiable functions that satisfy a mild condition.