Analysis of MILP Techniques for the Pooling Problem

Abstract

The pq-relaxation for the pooling problem can be constructed by applying McCormick envelopes for each of the bilinear terms appearing in the so-called pq-formulation of the pooling problem. This relaxation can be strengthened by using piecewise-linear functions that over- and under-estimate each bilinear term. Although there is a significant amount of empirical evidence to show that such piecewise-linear relaxations, which can be written as mixed-integer linear programs (MILPs), yield good bounds for the pooling problem, to the best of our knowledge, no formal result regarding the quality of these relaxations is known. In this paper, we prove that the ratio of the upper bound obtained by solving piecewise-linear relaxations (objective function is maximization) to the optimal objective function value of the pooling problem is at most n, where n is the number of output nodes. Furthermore for any ϵ > 0 and for any piecewise-linear relaxation, there exists an instance where the ratio of the relaxation value to the optimal value is at least n − ϵ. This analysis naturally yields a polynomial-time n-approximation algorithm for the pooling problem. We also show that if there exists a polynomial-time approximation algorithm for the pooling problem with guarantee better than n1−ϵ for any ϵ > 0, then NP-complete problems have randomized polynomial-time algorithms. Finally, motivated by the approximation algorithm, we design a heuristic that involves solving an MILP-based restriction of the pooling problem. This heuristic is guaranteed to provide solutions within a factor of n. On large-scale test instances and in significantly lesser time, this heuristic provides solutions that are often orders of magnitude better than those given by commercial local and global optimization solvers.

A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem

“…The GAMS files of the pooling problem instances that we use in this paper, except DeyGupte4, can be found on the website http://www.ii.uib.no/~mohammeda/spooling/. DeyGupte4 is constructed in this paper (Appendix) by using the results of Dey and Gupte (2015). In Table 1, we recall in column "PQ-linear relaxation value" the lower bound proposed in Alfaki and Haugland (2013) value of the PQ-formulation after applying McCormick relaxation for each bilinear term.…”

“…In Table 1, we recall in column "PQ-linear relaxation value" the lower bound proposed in Alfaki and Haugland (2013) value of the PQ-formulation after applying McCormick relaxation for each bilinear term. It is proved in Dey and Gupte (2015) that any optimal value of a piecewise linear approximation of the PQ-formulation (for a precise definition see Appendix) for DeyGupte4, has optimal value in [−4, −3]. Also, columns "# var."…”

“…The optimal value of this problem is −1 with the optimal solution constructed by sending flows from inputs to the first pool, and from it to one of the outputs such that the restriction in the specification in it is satisfied (Dey and Gupte 2015).…”

“…It is proved in Dey and Gupte (2015) that applying this approximation to the PQ-formulation gives an objective value in [−4, −3].…”

“…In this section we use a result in Dey and Gupte (2015) to construct a pooling problem instance (DeyGupte4) for which piecewise linear approximations of the PQ-formulation fail. Consider a standard pooling problem with I = 2 inputs, L = 2 pools and J = 4 outputs.…”