The standard pooling problem is a NP-hard sub-class of non-convex quadratically-constrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity of the standard pooling problem in its p-formulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas' intuition that pooling problems are rooted in piecewise-defined functions. We introduce dominant active topologies to explicitly identify pooling problem sparsity and show that the sparse patterns of active topological structure are associated with a piecewise objective function. Finally, the paper explains the conditions under which sparsity vanishes and where the combinatorial complexity emerges to cross over the P/NP boundary. We formally present the results obtained and their derivations for various specialized pooling problem instances.
Baltean-Lugojan, R. & Misener, R. J Glob Optim (2018) 71: 655. https://doi.org/10.1007/s10898-017-0577-y
View Piecewise Parametric Structure in the Pooling Problem - from Sparse Strongly-Polynomial Solutions to NP-Hardness