On the relation between the extended supporting hyperplane algorithm and Kelley’s cutting plane algorithm

Abstract

Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) 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 (J Soc Ind Appl Math 8(4):703–712, 1960) 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 a class of general, not necessarily convex nor differentiable, functions.

The SCIP Optimization Suite 8.0

“…In the last decades, solvers for convex MINLPs have demonstrated that the choice of the reference point in which to linearize convex nonlinear constraints is essential. While using the solution of the LP relaxation still leads to a convergent algorithm [54], better performance is achieved by using a reference point that is close to or at the boundary of the feasible region [26,97]. Therefore, also the new implementation of cons nonlinear includes a feature where feasible solutions are used as reference points to generate cutting planes.…”

Using projected cutting planes in the extended cutting plane method

“…In a recent paper of Serrano et al [22] it was shown that the supporting hyperplane algorithm [23] is equivalent to Kelley's cutting plane algorithm when reformulating the feasible region of the problem using its (sublinear) gauge function. This is theoretically interesting and in principle the case (neclecting the numerical differences) when comparing the smooth convex NLP algorithms of Kelley [6] and Veinott [24].…”