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A Derivative-Free Algorithm for Bound Constrained Optimization

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Abstract

In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a “suitable” descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.

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References

  1. E.J. Anderson and M.C. Ferris, “A direct search algorithm for optimization with noisy functions evaluations,” Mathematical Programming Technical Report 96-11, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 1996.

    Google Scholar 

  2. A.R. Conn, K. Scheinberg, and Ph.L. Toint, “Recent progress in unconstrained nonlinear optimization without derivatives,” Mathematical Programming, vol. 79, pp. 397–415, 1997.

    Google Scholar 

  3. C. Elster and A. Neumaier, “A grid algorithm for bound constrained optimization of noisy functions,” IMA Journal of Numerical Analysis, vol. 15, pp. 585–608, 1995.

    Google Scholar 

  4. L. Grippo, F. Lampariello, and S. Lucidi, “Global convergence and stabilization of unconstrained minimization methods without derivatives,” Journal of Optimization Theory and Applications, vol. 56, no. 3, pp. 385–406, 1988.

    Google Scholar 

  5. W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Springer-Verlag: Berlin, 1981. Lectures Notes in Economics and Mathematical Systems, vol. 187.

    Google Scholar 

  6. C.T. Kelley, Iterative Methods for Optimization. Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics: Philadelphia, 1999.

    Google Scholar 

  7. R.M. Lewis and V. Torczon, “Pattern search methods for bound constrained minimization,” SIAM Journal on Optimization, vol. 9, pp. 1082–1099, 1999.

    Google Scholar 

  8. S. Lucidi and M. Sciandrone, “On the global convergence of derivative free methods for unconstrained optimization,” Technical Report 18-96, DIS, Università di Roma “La Sapienza”.

  9. S. Lucidi and M. Sciandrone, “A derivative-free algorithm for bound constrained optimization,” R. 498 1999, IASI, Consiglio Nazionale delle Ricerche.

  10. J.J. Moré, B.S. Garbow, and K.E. Hillstrom, “Testing unconstrained optimization software; ACM Trans. on Math. Software, vol. 7, pp. 17–41, 1981.

    Google Scholar 

  11. M.J.D. Powell, “Direct search algorithms for optimization calculations,” Acta Numerica, vol. 7, pp. 287–336, 1998.

    Google Scholar 

  12. K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag: Berlin, 1987. Lectures Notes in Economics and Mathematical Systems, vol. 282.

    Google Scholar 

  13. M. Sciandrone, G. Placidi, L. Testa, and A. Sotgiu, “Compact low field magnetic resonance imaging magnet: Design and optimization,” Review of Scientific Instruments, vol. 71, no. 3, pp. 1534–1538, 2000.

    Google Scholar 

  14. V. Torczon, “On the convergence of pattern search methods,” SIAM Journal on Optimization, vol. 7, pp. 1–25, 1997.

    Google Scholar 

  15. P. Tseng, “Fortified-descent simplicial search method: A general approach,” SIAM Journal on Optimization, vol. 10, pp. 269–288, 2000.

    Google Scholar 

  16. M.H. Wright, “Direct search methods: Once scorned, now respectable,” Numerical Analysis 1995, “Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis, D.F. Griffiths and G.A. Watson (Eds.) Addison-Wesley Longman: Harlow, UK, 1996, pp. 191–208.

    Google Scholar 

  17. S.K. Zavriev, “On the global optimization properties of finite-differences local descent algorithms,” Journal of Global Optimization, vol. 3, pp. 67–78, 1993.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Roma, Italy

    Stefano Lucidi

  2. Istituto di Analisi dei Sistemi ed Informatica, CNR, Roma, Italy

    Marco Sciandrone

Authors
  1. Stefano Lucidi
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  2. Marco Sciandrone
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Lucidi, S., Sciandrone, M. A Derivative-Free Algorithm for Bound Constrained Optimization. Computational Optimization and Applications 21, 119–142 (2002). https://doi.org/10.1023/A:1013735414984

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