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A new Lagrangean approach to the pooling problem

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Abstract

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.

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References

  • Adhya N., Tawarmalani M., Sahinidis N.: A Lagrangian approach to the pooling problem. Ind. Eng. Chem. Res. 38, 1956–1972 (1999)

    Article  Google Scholar 

  • Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 124–131 (1983)

    Article  Google Scholar 

  • Audet C., Brimberg J., Hansen P., Le Digabel S., Maldenovic N.: Pooling problem: alternate formulations and solution methods. Manage. Sci. 50(6), 761–776 (2004)

    Article  Google Scholar 

  • Baker T.E., Lasdon L.S.: Successive linear programming at Exxon. Manage. Sci. 31, 264–274 (1985)

    Article  Google Scholar 

  • Ben-Tal A., Eiger G., Greshovitz V.: Global minimization by reducing the duality gap. Math. Program. 63, 193–212 (1994)

    Article  Google Scholar 

  • Bodington C.E., Randall W.C.: Nonlinear Programs for Product Blending. Joint National TIMS/ORSA Meeting, New Orleans (1979)

    Google Scholar 

  • Fisher M.L.: The Lagrangian relaxation method for solving integer programming problems. Manage. Sci. 27, 1–18 (1981)

    Article  Google Scholar 

  • Floudas C.A., Aggarwal A.: A Decomposition strategy for the optimum search in the pooling problem. ORSA J. Comput. 2(3), 225–235 (1990)

    Google Scholar 

  • Foulds L.R., Haugland D., Jonsten K.: A bilinear approach to the pooling problem. Optimization 24, 165–180 (1992)

    Article  Google Scholar 

  • GLPK (GNU Linear Programming Kit). http://www.gnu.org/software/glpk/glpk.htmt

  • Griffith R.E., Stewart R.A.: A nonlinear programming technique for the optimization of continuous processing system. Manage. Sci. 7, 379–392 (1961)

    Article  Google Scholar 

  • Haverly C.A.: Studies of the behavior of recursion for the pooling problem. ACM SIGMAP Bull. 25, 29–32 (1978)

    Google Scholar 

  • Haverly C.A.: Behavior of recursion model–more studies. ACM SIGMAP Bull. 26, 22–28 (1979)

    Article  Google Scholar 

  • Lasdon L.S., Waren A.D., Sarkar S., Palacios-Gomez F.: Solving the pooling problem using generalized reduced gradient and successive linear programming algorithm. ACM SIGMAP Bull. 27, 9–15 (1979)

    Article  Google Scholar 

  • Liberti L., Pantelides C.C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J . Glob. Optim. 36, 161–189 (2006)

    Article  Google Scholar 

  • McCormick G.P.: Computability of global solutions to factorable nonconvex programs. Part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  Google Scholar 

  • Meyer C., Floudas C.: Global optimization of a combinatorially complex generalized pooling problem. AIChE J. 52(3), 1027–1037 (2006)

    Article  Google Scholar 

  • Quesada I., Grossmann I.E.: Global optimization of bilinear process networks and multicomponent flows. Comput. Chem. Eng. 19(12), 1219–1242 (1995)

    Article  Google Scholar 

  • Sherali H.D., Alameddine A.: A new reformulation-linearization technique for bilinear programming problems. J . Glob. Optim. 2, 379–410 (1992)

    Article  Google Scholar 

  • Simon J.D., Azma H.M.: Exxon experience with large scale linear and nonlinear programming applications. Comput. Chem. Eng. 7(5), 605–614 (1983)

    Article  Google Scholar 

  • Tawarmalani, M., Sahinidis, N.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications (Kluwer Book Series in Nonconvex Optimization and its Applications, vol. 65). Kluwer Academic, Dordrecht (2002)

  • Visweswaran V., Floudas C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: application of theory and test problems. Comp. Chem. Eng. 14, 1419–1434 (1990)

    Article  Google Scholar 

  • Visweswaran V., Floudas C.A.: New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints. J . Glob. Optim. 3(3), 439–462 (1993)

    Article  Google Scholar 

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Correspondence to Hossa Almutairi.

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Almutairi, H., Elhedhli, S. A new Lagrangean approach to the pooling problem. J Glob Optim 45, 237–257 (2009). https://doi.org/10.1007/s10898-008-9371-1

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  • DOI: https://doi.org/10.1007/s10898-008-9371-1

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