Strong Convex Nonlinear Relaxations of the Pooling Problem

TL;DR

This paper develops new convex relaxations for the pooling problem, improving the bounds of existing formulations by deriving valid inequalities and analyzing their convex hulls, with demonstrated computational improvements.

Contribution

It introduces novel convex nonlinear inequalities for the pooling problem's relaxation, enhancing the strength of existing bounds and providing a detailed analysis of their convex hulls.

Findings

Inequalities significantly strengthen the convex relaxation.

Convex hulls are characterized for specific parameter cases.

Computational results show improved bounds on test instances.

Abstract

We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull, and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. Computational results on literature instances and newly created larger test instances demonstrate that the inequalities can significantly strengthen the convex relaxation of the pq-formulation of the pooling problem, which is the relaxation known to have the strongest bound.