A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem, a fundamental problem in the petro-chemical industry. The relaxation targets all nonlinear constraints, resulting in a Lagrangean subproblem with a nonlinear objective function, that is reformulated as a linear mixed integer program. Besides being used to generate lower bounds, the subproblem solutions are exploited within Lagrangean heuristics to find feasible solutions to the pooling problem. Valid cuts, derived from bilinear terms, are added to the subproblem to strengthen the Lagrangean bound, and improve the quality of feasible solutions.The procedure is applied to a benchmark set of fifteen problems from the literature. The Lagrangean lower bound is equal to the global optimum for eight problems, and is on average 8.2% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and are on average 0.043% from the optimum.