In this paper, we examine a mixed integer linear programming (MILP) reformulation for mixed integer bilinear problems where each bilinear term involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resulting product of continuous and binary variables. We present the convex hull of the underlying mixed integer linear set. The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances.

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.

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.

We consider the problem of characterizing the convex hull of the graph of a bilinear function f on the n-dimensional unit cube [0, 1] n . Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of f that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters 44 (2016), 625-629] to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights.