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Efficient interval partitioning for constrained global optimization

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Abstract

A new efficient interval partitioning approach to solve constrained global optimization problems is proposed. This involves a new parallel subdivision direction selection method as well as an adaptive tree search. The latter explores nodes (intervals in variable domains) using a restricted hybrid depth-first and best-first branching strategy. This hybrid approach is also used for activating local search to identify feasible stationary points. The new tree search management technique results in improved performance across standard solution and computational indicators when compared to previously proposed techniques. On the other hand, the new parallel subdivision direction selection rule detects infeasible and suboptimal boxes earlier than existing rules, and this contributes to performance by enabling earlier reliable deletion of such subintervals from the search space.

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References

  1. Alefeld G. and Herzberger J. (1983). Introduction to Interval Computations. Academic Press, New York

    Google Scholar 

  2. Balogh J. and Tóth B. (2005). Global optimization on Stiefel manifolds: a computational approach. CEJOR 13: 213–232

    Google Scholar 

  3. Benhamou, F., McAllester, D., Van Hentenryck, P.: CLP(intervals) revisited. In: Proc. of ILPS’94, pp. 124–138 (1994)

  4. Casado L.G., García I. and Csendes T. (2000). A new multisection technique in interval methods for global optimization. Computing 65: 263–269

    Article  Google Scholar 

  5. COCONUT, http://www.mat.univie.ac.at/~neum/glopt/coconut/

  6. Csendes T. and Rapcsák T. (1993). Nonlinear coordinate transformations for unconstrained optimization. I. Basic transformations. J. Global Optim. 3: 213–221

    Article  Google Scholar 

  7. Csendes T. and Ratz D. (1996). A review of subdivision direction selection in interval methods for global optimization. ZAMM 76: 319–322

    Google Scholar 

  8. Csendes T. and Ratz D. (1997). Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34: 922–938

    Article  Google Scholar 

  9. Dallwig S., Neumaier A. and Schichl H. (1997). GLOPT—a program for constrained global optimization. In: Bomze, I.M., Csendes, T., Horst, R., and Pardalos, P.M. (eds) Developments in global optimization, pp 19–36. Kluwer, Dordrecht

    Google Scholar 

  10. Drud, A.S.: CONOPT: A System for Large Scale Nonlinear Optimization. Reference Manual for CONOPT Subroutine Library, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1996)

  11. Epperly, T.G.: Global optimization of nonconvex nonlinear programs using parallel branch and bound. PhD dissertation, University of Wisconsin-Madison, USA (1995)

  12. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. Numerical Analysis Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA (1997)

  13. Hansen E.R. (1992). Global Optimization Using Interval Analysis. Marcel Dekker, New York

    Google Scholar 

  14. Hansen E. and Sengupta S. (1980). Global constrained optimization using interval analysis. In: Nickel, K.L. (eds) Interval Mathematics. Academic Press, New York

    Google Scholar 

  15. Kearfott R.B. (1991). Decompostion of arithmetic expressions to improve the behaviour of interval iteration for nonlinear systems. Computing 47: 169–191

    Article  Google Scholar 

  16. Kearfott R.B. (2003). An overview of the GlobSol Package for Verified Global Optimization. Talk given for the Department of Computing and Software. McMaster University, Ontario, Canada

    Google Scholar 

  17. Kearfott R.B. and Manuel N. III (1990). A portable interval Newton/bisection package. ACM Trans. Math. Software 16: 152–157

    Article  Google Scholar 

  18. Knüppel O. (1994). PROFIL/BIAS—a fast interval library. Computing 53: 277–287

    Article  Google Scholar 

  19. Korf R.E. (1985). Depth-first iterative deepening: An optimal admissible tree search. Artif. Intell. 27: 97–109

    Article  Google Scholar 

  20. Markót, M.C.: Reliable global optimization methods for constrained problems and their application for solving circle packing problems. PhD dissertation, University of Szeged, Hungary (2003)

  21. Markót M.C., Fernandez J., Casado L.G. and Csendes T. (2006). New interval methods for constrained global optimization. Math. Program. 106: 278–318

    Article  Google Scholar 

  22. Murtagh, B.A., Saunders, M.A.: MINOS 5.0 User’s Guide. Report SOL 83-20, Department of Operations Research, Stanford University, USA (1987)

  23. Pedamallu, C.S., Özdamar, L., Csendes, T.: An interval partitioning approach for continuous constrained optimization. In: Models and Algorithms in Global Optimization, pp. 73–96. Springer, Berlin (2006)

  24. Pedamallu C.S., Özdamar L. and Csendes T. (2007). Symbolic interval inference approach for subdivision direction selection in interval partitioning algorithms. J. Global Optim. 37: 177–194

    Article  Google Scholar 

  25. Pedamallu, C.S., Pósfai, J., Csendes, T.: Interval partitioning algorithm for constraint satisfaction problems. Int. J. of Model. Identif. Control. (accepted for publication)

  26. Pintér J.D. (1997). LGO—a program system for continuous and Lipschitz global optimization. In: Bomze, I.M., Csendes, T., Horst, R. and Pardalos, P.M. (eds) Developments in Global Optimization, pp 183–197. Kluwer, Boston

    Google Scholar 

  27. PrincetonLib.: Princeton Library of Nonlinear Programming Models. http://www.gamsworld.org/performance/princetonlib/princetonlib.htm

  28. Rapcsák T. and Csendes T. (1993). Nonlinear coordinate transformations for unconstrained optimization. II. Theoretical background. J. Global Optim. 3: 359–375

    Article  Google Scholar 

  29. Ratschek H. and Rokne J. (1988). New computer Methods for Global Optimization. Ellis Horwood, Chichester

    Google Scholar 

  30. Ratz D. and Csendes T. (1995). On the selection of subdivision directions in interval branch-and-bound methods for global optimization. J. Global Optim. 7: 183–207

    Article  Google Scholar 

  31. Robinson S.M. (1973). Computable error bounds for nonlinear programming. Math. Program. 5: 235–242

    Article  Google Scholar 

  32. Sahinidis, N.V.: Global optimization and constraint satisfaction: the branch-and-reduce approach. In: Bliek, C., Jermann, C., Neumaier, A. (eds.): COCOS 2002, LNCS, pp. 1–16 vol. 2861 (2003)

  33. Shcherbina O., Neumaier A., Sam-Haroud D., Vu X.-H. and Nguyen T.-V. (2003). Benchmarking global optimization and constraint satisfaction codes. LNCS 2861: 211–222

    Google Scholar 

  34. Smith E.M.B. and Pantelides C.C. (1999). A symbolic reformulation/spatial branch and bound algorithm for the global optimization of nonconvex MINLP’s. Comp. Chem. Eng. 23: 457–478

    Article  Google Scholar 

  35. Tawarmalani M. and Sahinidis N.V. (2004). Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99: 563–591

    Article  Google Scholar 

  36. Wolfe M.A. (1994). An interval algorithm for constrained global optimization. J. Comput. Appl. Math. 50: 605–612

    Article  Google Scholar 

  37. Yeniay O. (2004). Penalty function methods for constrained optimization with genetic algorithms. Math. Comp. Appl. 10: 45–56

    Google Scholar 

  38. Zhou J.L. and Tits A.L. (1996). An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions. SIAM J. Optim. 6: 461–487

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, Singapore

    Chandra Sekhar Pedamallu

  2. New England Biolabs Inc., 240 County Road, Ipswich, MA, USA

    Chandra Sekhar Pedamallu

  3. Department of Systems Engineering, Yeditepe University, Istanbul, Turkey

    Linet Özdamar

  4. Institute of Informatics, University of Szeged, Szeged, Hungary

    Tibor Csendes

  5. ESA/ESTEC, Advanced Concepts Team, Noordwijk, The Netherlands

    Tamás Vinkó

Authors
  1. Chandra Sekhar Pedamallu
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  2. Linet Özdamar
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  3. Tibor Csendes
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  4. Tamás Vinkó
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Correspondence to Tibor Csendes.

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Pedamallu, C.S., Özdamar, L., Csendes, T. et al. Efficient interval partitioning for constrained global optimization. J Glob Optim 42, 369–384 (2008). https://doi.org/10.1007/s10898-008-9297-7

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  • DOI: https://doi.org/10.1007/s10898-008-9297-7

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