Piecewise parametric structure in the pooling problem: from sparse strongly-polynomial solutions to NP-hardness

Abstract

The standard pooling problem is a NP-hard subclass of non-convex quadraticallyconstrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity, focusing on the single quality 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 under relaxed flow availability 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/N P boundary. We formally present the results obtained and their derivations for various specialized single quality pooling problem subclasses.

A study of rank-one sets with linear side constraints and application to the pooling problem

“…Proposition 2. Consider the set U row (n 1 ,n 2 ) (l, u) described in (7) where we assume u i is finite for every i ∈ [n 1 ]. Then, the following hold:…”

“…While the standard pooling problem is NP-hard to solve [6,27], there are also some positive results. For instance, a polynomial-time n-approximation algorithm (n is the number of output nodes) is presented in [22], convex hull of special substructure (with one pool node) has been studied [34], and recently some very special cases of the pooling problem has been shown to be polynomially solvable [27,28,13,7]. However, none of these positive results apply for the generalized pooling problem in which the corresponding graph has arcs from the set (I × (I ∪ T )).…”

Strong Convex Nonlinear Relaxations of the Pooling Problem

“…Next, (30) u − γx < xt − γx = x(t − γ) < t − γ as x < 1 and t > γ, and so ( 13) is satisfied by p with strict inequality. To show that ( 15) is satisfied strictly, we aggregate (7) with weight 1, (8) with weight −β, and (10) with weight 1 and get…”

“…Although the pooling problem has been studied for decades, it was only proved to be strongly NP-hard in 2013 by Alfaki and Haugland [3]. Further complexity results on special cases of the pooling problem can be found in [7,9,20].…”

Solving the pooling problem at scale with extensible solver GALINI

“…We focus on the pooling problem because of its many industrial applications, including (Misener and Floudas, 2009): crude-oil scheduling (Lee et al, 1996;Li et al, 2007Li et al, , 2012a, water networks (Galan and Grossmann, 1998;Castro et al, 2007), natural gas production (Selot et al, 2008;Li et al, 2011), fixedcharge transportation with product blending (Papageorgiou et al, 2012), hybrid energy systems (Baliban et al, 2012), multi-period blend scheduling (Kolodziej et al, 2013), and mining (Boland et al, 2015). Solving the pooling problem is NP-hard (Alfaki and Haugland, 2013b;Baltean-Lugojan and Misener, 2018;Letsios et al, 2020), so deterministic global optimization solvers use algorithms such as branch & bound to solve the problem. Our goal with explicitly using special structure information is practically solving larger problem sizes.…”