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A Deterministic Algorithm for Global Optimization

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Managing Safety of Heterogeneous Systems

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 658))

Abstract

An algorithm for solving global optimization problems is developed. The objective and constraints are required to have gradients satisfying Lipschitz condition. The problem may contain both continuous and integer variables and the objective may be non-convex and multimodal. Improved lower bounds and new techniques to reduce the number of algorithm steps by employing the gradient information are proposed for unconstrained optimization. Computational testing on different test problems demonstrate the efficiency of the proposed method in comparison with the state of the art approaches.

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References

  • Coello Coello, C.A., & Montes, E.M. (2002). Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Advanced Engineering Informatics, 16, 193–203.

    Article  Google Scholar 

  • Ermoliev, Y., & Norkin, V. (2004). Stochastic optimization of risk functions. In K. Marti, Y. Ermoliev, & G. Pflug (Eds.), Dynamic Stochastic Optimization. Lecture Notes in Economics and Mathematical Systems, pp. 225–249. New York: Springer.

    Google Scholar 

  • Evtushenko, Yu. (1971). Numerical methods for finding global extreme (case of a non-uniform mesh). U.S.S.R. Comput. Maths. Math. Phys., Vol. 11, 6, pp. 38–54, Moscow.

    Google Scholar 

  • Evtushenko, Yu., & Potapov., M. (1987). Deterministic global optimization. In E. Spedicato (Eds.), Algorithms for Continuous Optimization. The State of the Art, pp. 481–500. Kluwer.

    Google Scholar 

  • Evtushenko, Yu., Malkova, V., & Stanevichyus, A. (2009). Parallel global optimization of functions of several variables. Computational Mathematics and Mathematical Physics, Vol. 49(2), pp. 246–260. MAIK Nauka. Doi:10.1134/S0965542509020055.

    Google Scholar 

  • Evtushenko, Yu., Posypkin, M., & Sigal, I. (2009). A framework for parallel large-scale global optimization. Computer Science: Research and Development, Vol. 23(3), pp. 211–215. New York: Springer.

    Google Scholar 

  • Gergel, V. P. (1997). A global optimization algorithm for multivariate functions with lipschitzian first derivatives. Journal of Global Optimization, 10, 257–281.

    Article  Google Scholar 

  • Hansen, E. (1992). Global Optimization Using Interval Analysis. New York: Dekker.

    Google Scholar 

  • Horst, R., & Pardalos, P. M. (Eds.) (1995). Handbook of Global Optimization. Dordrecht: Kluwer.

    Google Scholar 

  • Jones, D. R., Perttunen, C. D., & Stuckman, B. E. (1993). Lipschitzian optimization without the Lipschitz constant. Journal of Optimization Theory and Applications, 79(1), 157–181.

    Article  Google Scholar 

  • Jones, D. R. (1999). The DIRECT global optimization algorithm. In A. Floudas, & P. Pardalos (Eds.), Encyclopedia of Optimization, pp. 725–735. New York: Springer.

    Google Scholar 

  • Kalyanmoy, D. (1997). GeneAS: A Robust Optimal Design Technique for Mechanical Component Design. Evolutionary Algorithms in Engineering Applications, pp. 497–514. Berlin: Springer.

    Google Scholar 

  • Kannan, B. K., & Kramer, S. N. (1994). An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. Journal of Mechanical Design, Transactions of the ASME, 116, 318–320.

    Article  Google Scholar 

  • Kearfott, R. B. (1996). Rigorous Global Search: Continuous Problems. Dordrecht: Kluwer.

    Google Scholar 

  • Michalewicz, Z. (1996). Genetic Algorithms + Data Structures = Evolution Programs, 3rd edn. Berlin: Springer.

    Google Scholar 

  • Nesterov, Y. (2003). Introductory Lectures on Convex Optimization: A Basic Course (Applied Optimization). Netherlands: Springer.

    Google Scholar 

  • Pardalos, P. M., Romeijn, E., & Tuy, H. (2000). Recent developments and trends in global optimization. Journal of Computational and Applied Mathematics, 124(1-2), 209–228.

    Article  Google Scholar 

  • Pardalos, P. M., & Resende, M. (Eds.) (2002). Handbook of Applied Optimization. Oxford: Oxford University Press.

    Google Scholar 

  • Pinter, J. D. (1996). Global Optimization in Action. Dordrecht: Kluwer.

    Google Scholar 

  • Pinter, J. (2005). Nonlinear optimization in modeling environments: software implementations for compilers, spreadsheets, modeling languages, and integrated computing systems. In V. Jeyakumar, & A. M. Rubinov (Eds.), Continuous Optimization: Current Trends and Applications, pp. 147–173. New York: Springer Science + Business Media.

    Google Scholar 

  • Piyavskii, S. A. (1972). An algorithm for finding the absolute extremum of a function. USSR Computational Mathematics and Mathematical Physics, 12, 57–67.

    Article  Google Scholar 

  • Snir, M., Otto, S., Huss-Lederman, S., Walker, D., & Dongarra, J. (1996). MPI: The Complete Reference. Boston: MIT Press.

    Google Scholar 

  • Sandgren, E. (1988). Nonlinear integer and discrete programming in mechanical design. Proceedings of the ASME Design Technology Conference, pp. 95–105. Florida: Kissimine.

    Google Scholar 

  • Sergeyev, Ya. D. (1998). Global one-dimensional optimization using smooth auxiliary functions. Mathematical Programming, 81(1), 127–146.

    Google Scholar 

  • Spall, J. C. (2003). Introduction to Stochastic Search and Optimization. Wiley.

    Google Scholar 

  • Strongin, R. G., & Sergeyev, Ya. D. (2000). Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Dordrecht: Kluwer.

    Google Scholar 

  • Shubert, B. (1972). A sequential method seeking the global maximum of a function. SIAM Journal on Numerical Analysis, 9, 379–388.

    Article  Google Scholar 

  • Tahera, K., Ibrahim, R. N., Lochert, P. B. (2008). GADYM – A novel genetic algorithm in mechanical design problems. Journal for Universal Computer Science, 14(15), 2566–2581.

    Google Scholar 

  • Takahama, T., & Sakai, S. (2006). Solving Constrained Optimization Problems by the ε-Constrained Particle Swarm Optimizer with Adaptive Velocity Limit Control, pp. 1–7, CIS.

    Google Scholar 

  • Tawarmalani, M., & Sahinidis, N. V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical Programming, 103(2), 225–249.

    Article  Google Scholar 

  • Tawarmalani, M., Sahinidis N. V. (2002). Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Dordrecht: Kluwer.

    Google Scholar 

  • Zhigljavsky, A. A. (1991). Theory of Global Random Search. Dordrecht: Kluwer.

    Google Scholar 

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (Projects 08-01-00619-a, 09-01-12098-ofi-m) and by the Program P-14 of the President of the Russian Academy of Sciences.

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Authors and Affiliations

  1. Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Moscow, Russia

    Yury Evtushenko & Mikhail Posypkin

Authors
  1. Yury Evtushenko
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  2. Mikhail Posypkin
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Corresponding author

Correspondence to Mikhail Posypkin .

Editor information

Editors and Affiliations

  1. Applied Systems Analysis (IIASA), International Institute for, Schloßplatz 1, Laxenburg, 2361, Austria

    Yuri Ermoliev

  2. Applied Systems Analysis (IIASA), International Institute for, Schloßplatz 1, Laxenburg, 2361, Austria

    Marek Makowski

  3. Aerospace Engineering and, Technology, Federal Armed Forces University Munich, Werner-Heisenberg-Weg 39, Neubiberg/München, 85577, Germany

    Kurt Marti

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Evtushenko, Y., Posypkin, M. (2012). A Deterministic Algorithm for Global Optimization. In: Ermoliev, Y., Makowski, M., Marti, K. (eds) Managing Safety of Heterogeneous Systems. Lecture Notes in Economics and Mathematical Systems, vol 658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22884-1_10

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