Analysis of MILP Techniques for the Pooling Problem

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 n 1−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.