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Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

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Abstract

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.

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References

  1. Hedar, A-R., Ahmed, A.: Studies on Metaheuristics for continuous global optimization problems. Ph.D. dissertation, Kyoto University, Kyoto, Japan (2004)

  2. Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.: Revising hull and box consistency. In: Proceedings of the sixteenth international conference on logic programming (ICLP’99), The MIT Press, Las Cruces, USA, pp. 230–244 (1999)

  3. Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global optimization algorithms. Springer, London (1990)

  4. Floudas C.A., Pardalos P.M.: Handbook of Test problems in Local and Global Optimization. Kluwer, The Netherlands (1999)

    Google Scholar 

  5. Garloff J.: Convergent bounds for range of multivariate polynomials. Interval mathematics. In: Nickel, K (eds) Lecture Notes in Computer Science, vol. 212, pp. 37–56. Springer, Berlin (1985)

    Google Scholar 

  6. Garloff J.: The Bernstein algorithm. Interval Comput. 2, 154–168 (1993)

    Google Scholar 

  7. Garloff J.: Solving strict polynomial inequalities by Bernstein expansion. The use of symbolic methods in control system analysis and design. In: Munro, N. (eds) IEE Control Engineering Series, vol. 56, pp. 339–352. Springer, Berlin (1999)

    Google Scholar 

  8. Global Library, Avaiable from http://www.gamsworld.org/global/globallib.

  9. Granvilliers L.: On the combination of interval constraint solvers. Reliab. Comput. 7, 467–483 (2001)

    Article  Google Scholar 

  10. Granvilliers L., Benhamou F.: Algorithm 852: real paver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. 32(1), 138–156 (2006)

    Article  Google Scholar 

  11. Hansen E., Walster G.W.: Global Optimization Using Interval Analysis. Marcel Dekker Inc, New York (2004)

    Google Scholar 

  12. Henrion, D., Lasserre J.B.: GloptiPoly: global optimization with polynomials with MATLAB and SeDuMi, version 2.3.0 (2006)

  13. Horst R., Pardalos P.M.: Handbook of Global Optimization. Kluwer, The Netherlands (1995)

    Google Scholar 

  14. Jaulin L., Kieffer M., Didrit O.: Applied Interval Analysis. Springer, London (2001)

    Google Scholar 

  15. Kearfott R.B.: Rigorous Global Search: Continuous problems. Kluwer, Dordrecht (1996)

    Google Scholar 

  16. Lebbah Y., Michel C., Rueher M.: An efficient and safe framework for solving optimization problems. J. Comput. Appl. Math. 199, 372–377 (2007)

    Article  Google Scholar 

  17. Li H.L., Chang C.T.: An approximate approach of global optimization for polynomial programming problems. Eur J Oper Res 107, 625–632 (1998)

    Article  Google Scholar 

  18. Malan, S., Milanese, M., Tarangna, M., Garloff, J.: B3 algorithm for robust performances analysis in presence of mixed parametric and dynamic perturbations. In: Proceedings of 31st Conference on Decision and Control, Tucson, Arizona, pp. 128–133 (1992)

  19. MATLAB Reference Guide.: MATLAB version 6.1, The Mathworks Inc., Natick, MA (2001)

  20. Nataraj P.S.V., Kotecha K.: An improved interval global optimization algorithm using higher order inclusion function forms. J. Global Optim. 32(1), 35–63 (2005)

    Article  Google Scholar 

  21. Nataraj P.S.V., Kotecha K.: Global optimization with higher order inclusion function forms part 1: a combined Taylor–Bernstein form. Reliab. Comput. 10(1), 27–44 (2004)

    Article  Google Scholar 

  22. Nataraj P.S.V., Arounassalame M.: A new subdivision algorithm for bernstein polynomial approach to global optimization. Int. J. Automat. Comput. 4(4), 342–352 (2007)

    Article  Google Scholar 

  23. Ratschek H., Rokne J.: Computer Methods for the Range of Functions. Ellis Horwood Limited Publishers, Chichester (1984)

    Google Scholar 

  24. Ratz D., Csendes T.: On the selection of subdivision directions in interval Branch-and-Bound methods for global optimization. J. Global Optim. 7(2), 183–207 (1995)

    Article  Google Scholar 

  25. Rump, S.M.: INTLAB-Interval Laboratory, Developments in Reliable Computing. Kluwer, Dordrecht, pp. 77–104 (1999)

  26. Shaswathi Ray, S.: A new approach to range computation of polynomials using the Bernstein form. Ph.D. dissertation, Indian Institute of Technology Bombay, Mumbai, India (2006)

  27. Stahl. V.: Interval methods for bounding the range of polynomials and solving systems of nonlinear equations. Ph.D. dissertation, University of linz (2005)

  28. Vaidyanathan R., Ed-halwagi M.: Global optimization of nonconvex nonlinear programs via interval analysis. Comput. Chem. Eng. 18(10), 889–897 (1994)

    Article  Google Scholar 

  29. Venkataraman R.: Applied Optimization with MATLAB Programming. Wiley, London (2001)

    Google Scholar 

  30. Zettler M., Garloff J.: Robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion. IEEE Trans. Autom. Contr. 43(5), 425–431 (1998)

    Article  Google Scholar 

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  1. Systems and Control Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India

    P. S. V. Nataraj & M. Arounassalame

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  1. P. S. V. Nataraj
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  2. M. Arounassalame
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Correspondence to P. S. V. Nataraj.

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Nataraj, P.S.V., Arounassalame, M. Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm. J Glob Optim 49, 185–212 (2011). https://doi.org/10.1007/s10898-009-9485-0

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