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On proving existence of feasible points in equality constrained optimization problems

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Abstract

Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an actual feasible point exists within these bounds. Hansen and others have proposed techniques for proving the existence of feasible points within given bounds, but practical implementations have not, to our knowledge, previously been described. Various alternatives are possible in such an implementation, and details must be carefully considered. Also, in addition to Hansen’s technique for handling the underdetermined case, it is important to handle the overdetermined case, when the approximate feasible point corresponds to a point with many active bound constraints. The basic ideas, along with experimental results from an actual implementation, are summarized here.

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References

  1. R.B. Kearfott, A review of techniques in the verified solution of constrained global optimization problems, in: R.B. Kearfott, V. Kreinovich (Eds.), Applications of Interval Computations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996, pp. 23–60.

    Google Scholar 

  2. E.R. Hansen, Global Optimization using Interval Analysis, Marcel Dekker, New York, 1992.

    MATH  Google Scholar 

  3. R.B. Kearfott, Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

    MATH  Google Scholar 

  4. H. Ratschek, J. Rokne, New Computer Methods for Global Optimization, Wiley, New York, 1988.

    MATH  Google Scholar 

  5. M.A. Wolfe, An interval algorithm for constrained global optimization, Journal of Computational and Applied Mathematics 50 (1994) 605–612.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Jansson, O. Knüppel, A global minimization method: The multi-dimensional case, Technical Report 92.1, Informationstechnik, Technische Uni. Hamburg-Harburg, Hamburg, Germany, 1992.

    Google Scholar 

  7. O. Caprani, B. Godthaab, K. Madsen, Use of a real-valued local minimum in parallel interval global optimization, Interval Computations 3 (1993) 71–82.

    MathSciNet  Google Scholar 

  8. R.B. Kearfott, On verifying feasibility in equality constrained optimization problems, Technical report, University of Southwestern, Louisiana, Lafayette, LA, 1994.

    Google Scholar 

  9. A. Neumaier, Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, UK, 1990.

    MATH  Google Scholar 

  10. G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983.

    MATH  Google Scholar 

  11. R.E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979.

    MATH  Google Scholar 

  12. R.B. Kearfott, Preconditioners for the interval Gauss-Seidel method, SIAM Journal on Numerical Analysis 27 (1977) 804–822.

    Article  MathSciNet  Google Scholar 

  13. G. Mayer, Epsilon-inflation in verification algorithms, Journal of Computational and Applied Mathematics 60 (1994) 147–169.

    Article  Google Scholar 

  14. S.M. Rump Kleine Fehlerschranken bei Matrixproblemen, Ph.D. Thesis., Universität Karlsruhe, Karlsruhe, Germany, 1980.

    Google Scholar 

  15. S.M. Rump, Verification methods for dense and sparse systems of equations, in: J. Herzberger (Ed.), Topics in Validated Computations, Elsevier, Amsterdam, The Netherlands, 1994, pp. 63–135.

    Google Scholar 

  16. C.A. Floudas, P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer, New York, 1990.

    MATH  Google Scholar 

  17. R.B. Kearfott, A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization. ACM Transactions on Mathematical Software 21 (1995) 63–78.

    Article  MATH  Google Scholar 

  18. A.R. Conn, N. Gould, Ph.L. Toint, LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization, Springer, New York, 1992.

    MATH  Google Scholar 

  19. B.P. Kristinsdottir, Z.B. Zabinsky, T. Csendes, M.E. Tuttle, Methodologies for tolerance intervals, Interval Computations 3 (1993) 133–147.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, University of Southwestern Louisiana, P.O. Box 41010, 70504-1010, Layfayette, LA, USA

    R. Baker Kearfott

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  1. R. Baker Kearfott
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This work was supported in part by National Science Foundation grant CCR-9203730.

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Kearfott, R.B. On proving existence of feasible points in equality constrained optimization problems. Mathematical Programming 83, 89–100 (1998). https://doi.org/10.1007/BF02680551

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