The Pooling Problem
Convexification techniques based on disjunctive programming have been extensively employed to solve many hard combinatorial optimization problems. In this chapter, we demonstrate the potential of convexification techniques in the context of continuous nonlinear programming by applying them to the pooling problem. The pooling problem is a nonlinear network flow problem that models the operation of the supply chain of petrochemicals where crude oils are mixed to produce intermediate quality streams that are blended at retail locations to produce the final products. The main challenge in finding optimal solutions to pooling problems is that the nonlinearities result in many local optima. We address the question of finding a tight polyhedral outer-approximation of the convex hull of solutions of the pooling problem. In particular, we consider a formulation of the pooling problem that is derived after the addition of one class of nonlinear equalities to the classical q-formulation of the problem. These constraints are obtained using the reformulation-linearization technique (Sherali & Adams 1990) Using convexification and disjunctive programming arguments, we prove that the linear programming relaxation of this formulation dominates the linear programming and Lagrangian relaxations of the classical p- and q-formulations of the problem. We finally present extensive eomputational results demonstrating that the use of this tight formulation in the eontext of a spatial braneh-and-bound algorithm makes it trivial to solve to global optimality all pooling problems from the open literature.
KeywordsConvex Hull Lagrangian Relaxation Dual Solution Linear Programming Relaxation Global Optimization Algorithm
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