  The pooling problem is a classical NP-hard problem in the chemical process
and petroleum industries. This problem is modeled as a nonlinear, nonconvex
network flow problem in which raw materials with different specifications are
blended in some intermediate tanks, and mixed again to obtain the final
products with desired specifications. The analysis of the pooling problem is
quite an active research area, and different exact formulations, relaxations
and restrictions are proposed. In this paper, we focus on a recently proposed
rank-one-based formulation of the pooling problem. In particular, we study a
recurring substructure in this formulation defined by the set of nonnegative,
rank-one matrices with bounded row sums, column sums, and the overall sum. We
show that the convex hull of this set is second-order cone representable. In
addition, we propose an improved compact-size polyhedral outer-approximation
and families of valid inequalities for this set. We further strengthen these
convexification approaches with the help of various bound tightening techniques
specialized to the instances of the pooling problem. Our computational
experiments show that the newly proposed polyhedral outer-approximation can
improve upon the traditional linear programming relaxations of the pooling
problem in terms of the dual bound. Furthermore, bound tightening techniques
reduce the computational time spent on both the exact, linear programming and
mixed-integer linear programming relaxations.
