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An Improved Successive Linear Programming Algorithm

Author

Listed:
  • Jianzhong Zhang

    (Department of Mathematics, Shanghai Normal University, Shanghai, Peoples Republic of China)

  • Nae-Heon Kim

    (Department of Mechanical Engineering, Ajoy University, San 5 Wonchudon, Suwon, Korea)

  • L. Lasdon

    (Department of General Business and IC 2 Institute, University of Texas, Austin, Texas 78712)

Abstract

Successive Linear Programming (SLP) algorithms solve nonlinear optimization problems via a sequence of linear programs. They have been widely used, particularly in the oil and chemical industries, beginning with their introduction by Griffith and Stewart of Shell Development Company in 1961. Since then, several applications and variants of SLP have appeared, the most recent being the SLPR algorithm described in this journal in 1982 (Palacios-Gomez et al.). SLP procedures are attractive because they are fairly easy to implement if an efficient, flexible LP code is available, can solve nonseparable as well as separable problems, can be applied to as large a problem as the LP code can handle (often thousands of constraints and variables), and have been successful in many practical applications. This paper describes a new SLP algorithm called PSLP (Penalty SLP). PSLP represents a significant strengthening and refinement of the SLPR procedure described in Palacios-Gomez et al. (Palacios-Gomez, F., L. Lasdon, M. Engquist. 1982. Nonlinear optimization by successive linear programming. Management Sci. 28 1106--1120.). We give a convergence proof for PSLP---the first SLP convergence proof for nonlinearly constrained problems of general form. This theory is supported by computational performance---in our tests, PSLP is significantly more robust than SLPR, and at least as efficient. A Fortran computer implementation is described. A simplified version of PSLP has already solved several "real world" NLP problems at Exxon (Baker and Lasdon [Baker, T. E., L. S. Lasdon. 1985. Successive linear programming at Exxon. Management Sci. 31 (March) 264--274.), including nonlinear refinery models of up to 1000 rows. As with other SLP algorithms, PSLP is especially efficient on problems which are highly constrained, i.e., which have nearly as many active constraints as there are variables. For problems with vertex optima (at least as many active constraints as variables), it is quadratically convergent. Nonlinear refinery models often have vertex optima, since they are large and mostly linear, and on line process unit optimization problems are likely to possess highly constrained solutions as well. PSLP has great potential for accurate, efficient solution of such problems.

Suggested Citation

  • Jianzhong Zhang & Nae-Heon Kim & L. Lasdon, 1985. "An Improved Successive Linear Programming Algorithm," Management Science, INFORMS, vol. 31(10), pages 1312-1331, October.
    • Handle: RePEc:inm:ormnsc:v:31:y:1985:i:10:p:1312-1331
    DOI: 10.1287/mnsc.31.10.1312
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    Citations

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    Cited by:

    1. Natashia Boland & Thomas Kalinowski & Fabian Rigterink, 2016. "New multi-commodity flow formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 66(4), pages 669-710, December.
    2. Z. Akbari & R. Yousefpour & M. Reza Peyghami, 2015. "A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 733-754, March.
    3. Ye, Jiao & Jiang, Yu & Chen, Jun & Liu, Zhiyuan & Guo, Renzhong, 2021. "Joint optimisation of transfer location and capacity for a capacitated multimodal transport network with elastic demand: a bi-level programming model and paradoxes," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 156(C).
    4. Chan, Timothy C.Y. & Lee, Taewoo, 2018. "Trade-off preservation in inverse multi-objective convex optimization," European Journal of Operational Research, Elsevier, vol. 270(1), pages 25-39.
    5. Jenny Carolina Saldana Cortés, 2011. "Programación semidefinida aplicada a problemas de cantidad económica de pedido," Documentos CEDE 8735, Universidad de los Andes, Facultad de Economía, CEDE.
    6. Mohammed Alfaki & Dag Haugland, 2014. "A cost minimization heuristic for the pooling problem," Annals of Operations Research, Springer, vol. 222(1), pages 73-87, November.
    7. Charles Audet & Jack Brimberg & Pierre Hansen & Sébastien Le Digabel & Nenad Mladenovi'{c}, 2004. "Pooling Problem: Alternate Formulations and Solution Methods," Management Science, INFORMS, vol. 50(6), pages 761-776, June.
    8. Ted Kutz & Mark Davis & Robert Creek & Nick Kenaston & Craig Stenstrom & Margery Connor, 2014. "Optimizing Chevron’s Refineries," Interfaces, INFORMS, vol. 44(1), pages 39-54, February.
    9. Daniel De Wolf & Yves Smeers, 2000. "The Gas Transmission Problem Solved by an Extension of the Simplex Algorithm," Management Science, INFORMS, vol. 46(11), pages 1454-1465, November.
    10. Elias S. Helou & Sandra A. Santos & Lucas E. A. Simões, 2018. "A fast gradient and function sampling method for finite-max functions," Computational Optimization and Applications, Springer, vol. 71(3), pages 673-717, December.
    11. Volker Maag & Martin Berger & Anton Winterfeld & Karl-Heinz Küfer, 2010. "A novel non-linear approach to minimal area rectangular packing," Annals of Operations Research, Springer, vol. 179(1), pages 243-260, September.
    12. Hong, Sung-Pil & Kim, Taegyoon & Lee, Subin, 2019. "A precision pump schedule optimization for the water supply networks with small buffers," Omega, Elsevier, vol. 82(C), pages 24-37.
    13. Mohammed Alfaki & Dag Haugland, 2013. "Strong formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 56(3), pages 897-916, July.
    14. Dudek, Gregor & Stadtler, Hartmut, 2005. "Negotiation-based collaborative planning between supply chains partners," European Journal of Operational Research, Elsevier, vol. 163(3), pages 668-687, June.
    15. Sarker, Ruhul A. & Gunn, Eldon A., 1997. "A simple SLP algorithm for solving a class of nonlinear programs," European Journal of Operational Research, Elsevier, vol. 101(1), pages 140-154, August.

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    Keywords

    nonlinear programming;

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