Multilocal Programming and Applications

  • A. I. Pereira
  • O. Ferreira
  • S. P. Pinho
  • Edite M. G. P. Fernandes
Part of the Intelligent Systems Reference Library book series (ISRL, volume 38)


Multilocal programming aims to identify all local maximizers of unconstrained or constrained nonlinear optimization problems. The multilocal programming theory relies on global optimization strategies combined with simple ideas that are inspired in deflection or stretching techniques to avoid convergence to the already detected local maximizers. The most used methods to solve this type of problems are based on stochastic procedures. In general, population-based methods are computationally expensive but rather reliable in identifying all local solutions. Stochastic methods based on point-to-point strategies are faster to identify the global solution, but sometimes are not able to identify all the optimal solutions of the problem. To handle the constraints of the problem, some penalty strategies are proposed. A well-known set of test problems is used to assess the performance of the algorithms. In this chapter, a review on recent techniques for both unconstrained and constrained multilocal programming is presented. Some real-world multilocal programming problems based on chemical engineering process design applications are described.


Particle Swarm Optimization Global Optimization Penalty Function Feasible Region Global Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Afshar, M.H.: Penalty adapting ant algorithm: application to pipe network optimization. Eng. Optim. 40, 969–987 (2008)CrossRefGoogle Scholar
  2. 2.
    Alefeld, G., Mayer, G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ali, M.M., Gabere, M.N.: A simulated annealing driven multi-start algorithm for bound constrained global optimization. J. Comput. Appl. Math. 233, 2661–2674 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baker, L.E., Pierce, A.C., Luks, K.D.: Gibbs energy analysis of phase equilibria. Soc. Petrol. Eng. J. 22, 731–742 (1982)Google Scholar
  5. 5.
    Barbosa, H.J.C., Lemonge, A.C.C.: An adaptive penalty method for genetic algorithms in constrained optimization problems. In: Iba, H. (ed.) Frontiers in Evolutionary Robotics. I-Tech Education Publ., Austria (2008)Google Scholar
  6. 6.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)zbMATHGoogle Scholar
  7. 7.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  8. 8.
    Bonilla-Petriciolet, A., Vásquez-Román, R., Iglesias-Silva, G.A., Hall, K.R.: Performance of stochastic global optimization methods in the calculation of phase analyses for nonreactive and reactive mixtures. Ind. Eng. Chem. Res. 45, 4764–4772 (2006)CrossRefGoogle Scholar
  9. 9.
    Coello, C.A.C.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Technical Report (48 pages), CINVESTAV-IPN, Mexico (2002)Google Scholar
  10. 10.
    Coope, I.D., Watson, G.A.: A projected Lagrangian algorithm for semi-infinite programming. Math. Program. 32, 337–356 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Csendes, T., Pál, L., Sendín, J.O.H., Banga, J.R.: The GLOBAL optimization method revisited. Optim. Lett. 2, 445–454 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fanelli, S.: A new algorithm for box-constrained global optimization. J. Optim. Theory Appl. 149, 175–196 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ferrari, J.C., Nagatani, G., Corazza, F.C., Oliveira, J.V., Corazza, M.L.: Application of stochastic algorithms for parameter estimation in the liquid-liquid phase equilibrium modeling. Fluid Phase Equilib. 280, 110–119 (2009)CrossRefGoogle Scholar
  14. 14.
    Finkel, D.E., Kelley, C.T.: Convergence analysis of the DIRECT algorithm. Optim. Online 14, 1–10 (2004)Google Scholar
  15. 15.
    Floudas, C.A.: Recent advances in global optimization for process synthesis, design and control: enclosure all solutions. Comput. Chem. Eng. 23, S963–S973 (1999)CrossRefGoogle Scholar
  16. 16.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45, 3–38 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gecegormez, H., Demirel, Y.: Phase stability analysis using interval Newton method with NRTL model. Fluid Phase Equilib. 237, 48–58 (2005)CrossRefGoogle Scholar
  18. 18.
    Guo, M., Wang, S., Repke, J.U., Wozny, G.: A simultaneous method for two- and three-liquid-phase stability determination. AIChE J. 50, 2571–2582 (2004)CrossRefGoogle Scholar
  19. 19.
    Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, Inc., New York (2004)zbMATHGoogle Scholar
  20. 20.
    Hendrix, E.M.T., G.-Tóth, B.: Introduction to Nonlinear and Global Optimization. Springer, New York (2010)zbMATHCrossRefGoogle Scholar
  21. 21.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  22. 22.
    Ingber, L.: Very fast simulated re-annealing. Math. Comput. Model. 12, 967–973 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ingber, L.: Simulated annealing: practice versus theory. Math. Comput. Model. 18, 29–57 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ingber, L.: Adaptive simulated annealing (ASA): lessons learned. Control Cybern. 25, 33–54 (1996)zbMATHGoogle Scholar
  25. 25.
    Jones, D.R., Perttunen, C.C., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Jones, D.R.: Direct global optimization algorithm. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 725–735. Springer (2009)Google Scholar
  27. 27.
    Kiseleva, E., Stepanchuk, T.: On the efficiency of a global non-differentiable optimization algorithm based on the method of optimal set partitioning. J. Glob. Optim. 25, 209–235 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    León, T., Sanmatias, S., Vercher, E.: A multilocal optimization algorithm. TOP 6, 1–18 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello, C.A.C., Deb, K.: Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization. Technical Report (2006)Google Scholar
  30. 30.
    Liu, J.L., Lin, J.H.: Evolutionary computation of unconstrained and constrained problems using a novel momentum-type particle swarm optimization. Eng. Optim. 39, 287–305 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Glob. Optim. 48, 113–128 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    McDonald, C.M., Floudas, C.A.: Global optimization for the phase stability problem. AIChE J. 41, 1798–1814 (1994)CrossRefGoogle Scholar
  33. 33.
    McDonald, C.M., Floudas, C.A.: Global optimization for the phase and chemical equilibrium problem: application to the NRTL equation. Comput. Chem. Eng. 19, 1111–1139 (1995)CrossRefGoogle Scholar
  34. 34.
    Michalewicz, Z.: A survey of constraint handling techniques in evolutionary computation methods. In: Proceedings of the 4th Annual Conference on Evolutionary Programming, pp. 135–155 (1995)Google Scholar
  35. 35.
    Michelsen, M.L.: The isothermal flash problem. Part I. Stability. Fluid Phase Equilib. 9, 1–19 (1982)CrossRefGoogle Scholar
  36. 36.
    Mieltinen, K., Mäkelä, M.M., Toivanen, J.: Numerical comparison of some penalty-based constraint handling techniques in genetic algorithms. J. Glob. Optim. 27, 427–446 (2003)CrossRefGoogle Scholar
  37. 37.
    Nagatani, G., Ferrari, J., Cardozo Filho, L., Rossi, C.C.R.S., Guirardello, R., Oliveira, J.V., Corazza, M.L.: Phase stability analysis of liquid-liquid equilibrium with stochastic methods. Braz. J. Chem. Eng. 25, 571–583 (2008)CrossRefGoogle Scholar
  38. 38.
    Parsopoulos, K.E., Plagianakos, V., Magoulas, G., Vrahatis, M.N.: Objective function stretching to alleviate convergence to local minima. Nonlinear Anal. 47, 3419–3424 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Parsopoulos, K.E., Vrahatis, M.N.: Recent approaches to global optimization problems through particle swarm optimization. Nat. Comput. 1, 235–306 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Parsopoulos, K.E., Vrahatis, M.N.: On the computation of all global minimizers through particle swarm optimization. IEEE Transaction on Evolutionary Computation 8, 211–224 (2004)CrossRefGoogle Scholar
  41. 41.
    Petalas, Y.G., Parsopoulos, K.E., Vrahatis, M.N.: Memetic particle swarm optimization. Ann. Oper. Res. 156, 99–127 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: On a reduction line search filter method for nonlinear semi-infinite programming problems. In: Sakalauskas, L., Weber, G.W., Zavadskas, E.K. (eds.) Euro Mini Conference Continuous Optimization and Knowledge-Based Technologies, pp. 174–179 (2008)Google Scholar
  43. 43.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: Numerical experiments with a continuous L2-exponential merit function for semi-infinite programming. In: Simos, T.E., Psihoyios, G. (eds.) International Electronic Conference on Computer Science, AIP, vol. 1060(1), pp. 1354–1357. Springer (2008)Google Scholar
  44. 44.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: A reduction method for semi-infinite programming by means of a global stochastic approach. Optim. 58, 713–726 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: Constrained multi-global optimization using a penalty stretched simulated annealing framework. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics, AIP, vol. 1168, pp. 1354–1357. Springer (2009)Google Scholar
  46. 46.
    Rangaiah, G.P.: Evaluation of genetic algorithms and simulated annealing for phase equilibrium and stability problems. Fluid Phase Equilib. 187-188, 83–109 (2001)CrossRefGoogle Scholar
  47. 47.
    Renon, H., Prausnitz, J.M.: Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J. 14, 135–144 (1968)CrossRefGoogle Scholar
  48. 48.
    Sepulveda, A.E., Epstein, L.: The repulsion algorithm, a new multistart method for global optimization. Struct. Multidiscip. Optim. 11, 145–152 (1996)Google Scholar
  49. 49.
    Tessier, S.R., Brennecke, J.F., Stadtherr, M.A.: Reliable phase stability analysis for excess Gibbs energy models. Chem. Eng. Sci. 55, 1785–1796 (2000)CrossRefGoogle Scholar
  50. 50.
    Tsoulos, L.G., Lagaris, I.E.: MinFinder: locating all the local minima of a function. Comput. Phys. Commun. 174, 166–179 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Tu, W., Mayne, R.W.: Studies of multi-start clustering for global optimization. Int. J. Numer. Methods Eng. 53, 2239–2252 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Voglis, C., Lagaris, I.E.: Towards ”Ideal Multistart”. A stochastic approach for locating the minima of a continuous function inside a bounded domain. Appl. Math. Comput. 213, 216–229 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Wang, Y.J.: Derivative-free simulated annealing and deflecting function technique for global optimization. J. Appl. Math. Comput. 1-2, 49–66 (2008)CrossRefGoogle Scholar
  54. 54.
    Wang, Y., Cai, Z., Zhou, Y., Fan, Z.: Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique. Struct. Multidiscip. Optim. 37, 395–413 (2008)CrossRefGoogle Scholar
  55. 55.
    Wu, Z.Y., Bai, F.S., Lee, H.W.J., Yang, Y.J.: A filled function method for constrained global optimization. J. Glob. Optim. 39, 495–507 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Xavier, A.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. Int. Trans. Oper. Res. 8, 659–671 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Yeniay, Ö.: Penalty function methods for constrained optimization with genetic algorithms. Math. Comput. Appl. 10, 45–56 (2005)Google Scholar
  58. 58.
    Yushan, Z., Zhihong, X.: A reliable method for liquid-liquid phase equilibrium calculation and global stability analysis. Chem. Eng. Commun. 176, 113–160 (1999)CrossRefGoogle Scholar
  59. 59.
    Yushan, Z., Zhihong, X.: Calculation of liquid-liquid equilibrium based on the global stability analysis for ternary mixtures by using a novel branch and bound algorithm: application to Uniquac Equation. Ind. Eng. Chem. Res. 38, 3549–3556 (1999)CrossRefGoogle Scholar
  60. 60.
    Zahara, E., Hu, C.H.: Solving constrained optimization problems with hybrid particle swarm optimization. Eng. Optim. 40, 1031–1049 (2008)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Zhang, X., Liu, S.: Interval algorithm for global numerical optimization. Eng. Optim. 40, 849–868 (2008)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Optimization and Its Applications. Springer (2007)Google Scholar
  63. 63.
    Zhu, W.: A class of filled functions for box constrained continuous global optimization. Appl. Math. Comput. 169, 129–145 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Zhu, W., Ali, M.M.: Solving nonlinearly constrained global optimization problem via an auxiliary function method. J. Comput. Appl. Math. 230, 491–503 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Zhu, Y., Inoue, K.: Calculation of chemical and phase equilibrium based on stability analysis by QBB algorithm: application to NRTL equation. Chem. Eng. Sci. 56, 6915–6931 (2001)CrossRefGoogle Scholar
  66. 66.
    Zhu, Y., Xu, Z.: A reliable prediction of the global phase stability for liquid-liquid equilibrium through the simulated annealing algorithm: application to NRTL and UNIQUAC equations. Fluid Phase Equilib. 154, 55–69 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • A. I. Pereira
    • 1
  • O. Ferreira
    • 2
  • S. P. Pinho
    • 2
  • Edite M. G. P. Fernandes
    • 3
  1. 1.Polytechnic Institute of Bragança, Bragança, and Algoritmi R&D Centre, University of MinhoBragaPortugal
  2. 2.LSRE/LCM Laboratory of Separation and Reaction EngineeringPolytechnic Institute of BragançaBragançaPortugal
  3. 3.Algoritmi R&D CentreUniversity of MinhoBragaPortugal

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