A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem. The relaxation targets all nonlinear constraints, and results in a Lagrangean subproblem with a nonlinear objective function and linear constraints, 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. 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 tested on a benchmark set of fifteen problems from the literature. The proposed bounds are found to outperform or equal earlier bounds from the literature on 14 out of 15 tested problems. Similarly, the Lagrangean heuristics outperform the VNS and MALT heuristics on 4 instances. Furthermore, the Lagrangean lower bound is equal to the global optimum for nine problems, and on average is 2.1% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and on average are 0.043% from the optimum.

[1]  N. Sahinidis,et al.  A Lagrangian Approach to the Pooling Problem , 1999 .

[2]  C. A. Haverly.  Behavior of recursion model - more studies , 1979, SMAP.

[3]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

[4]  V. Visweswaran,et al.  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—II. Application of theory and test problems , 1990 .

[5]  Jerome D. Simon,et al.  Exxon experience with large scale linear and nonlinear programming applications , 1983 .

[6]  Christodoulos A. Floudas,et al.  A Decomposition Strategy for Global Optimum Search in the Pooling Problem , 1990, INFORMS J. Comput..

[7]  C. Floudas,et al.  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory , 1990 .

[8]  Hanif D. Sherali,et al.  A new reformulation-linearization technique for bilinear programming problems , 1992, J. Glob. Optim..

[9]  Aharon Ben-Tal,et al.  Global minimization by reducing the duality gap , 1994, Math. Program..

[10]  L. Foulds,et al.  A bilinear approach to the pooling problem , 1992 .

[11]  L. S. Lasdon,et al.  Solving the pooling problem using generalized reduced gradient and successive linear programming algorithms , 1979, SMAP.

[12]  Christodoulos A. Floudas,et al.  New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints , 1993, J. Glob. Optim..

[13]  R. E. Griffith,et al.  A Nonlinear Programming Technique for the Optimization of Continuous Processing Systems , 1961 .

[14]  C. A. Haverly.  Studies of the behavior of recursion for the pooling problem , 1978, SMAP.

[15]  I. Grossmann,et al.  Global optimization of bilinear process networks with multicomponent flows , 1995 .

[16]  Marshall L. Fisher,et al.  The Lagrangian Relaxation Method for Solving Integer Programming Problems , 2004, Manag. Sci..

[17]  T. E. Baker,et al.  Successive Linear Programming at Exxon , 1985 .

[18]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[19]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[20]  Pierre Hansen,et al.  Pooling Problem: Alternate Formulations and Solution Methods , 2000, Manag. Sci..

[21]  Christodoulos A. Floudas,et al.  Global optimization of a combinatorially complex generalized pooling problem , 2006 .

[22]  Leo Liberti,et al.  An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms , 2006, J. Glob. Optim..