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Solving Optimization Problems with Help of the UniCalc Solver

  • Alexander L. Semenov
Part of the Applied Optimization book series (APOP, volume 3)

Abstract

The UniCalc solver [3] was designed at the Russian Institute of Artificial Intelligence to solve systems of nonlinear equations and inequalities with possibly inexact data. UniCalc uses the computation techniques based on the subdefinite computations method [13], which can be regarded as an analogue of constraint propagation with interval labels [4]. To implement this method, algorithms of interval mathematics are used, so UniCalc can also be used to solve interval problems. Another feature of UniCalc is its ability to solve different integer problems and problems with mixed data types (integers and reals). The efficiency of UniCalc is confirmed by the results of its successful applications to many problems, from benchmark (test) problems (see, e.g., [8, 10]) to real-life problems. Some optimization problems [11] have also been successfully solved by these techniques.

Keywords

Constraint Propagation Integer Programming Problem Solve Optimization Problem Unconstrained Optimization Problem USSR Acad 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. Alefeld and Ju. Herzberger, Introduction in Interval Computations, Academic Press, N. Y., 1983.Google Scholar
  2. [2]
    N. S. Asaithambi, Shen Zuhe, and R. E. Moore, “On computing the range of values”, Computing, 1982, Vol. 28, pp. 225–237.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. B. Babichev, O. B. Kadyrova, T. P. Kashevarova, A. S. Leshchenko, and A. L. Semenov, “UniCalc, A Novel Approach to Solving Systems of Algebraic Equations”, Proceedings of the International Conference on Numerical Analysis with Automatic Result Verifications. Lafayette, Louisiana, USA, February–March, 1993. Interval Computations, 1993, No. 2, pp. 29–47.Google Scholar
  4. [4]
    E. Davis, “Constraint propagation with interval labels”, Artificial Intelligence, 1987, Vol. 32, pp. 99–118.CrossRefGoogle Scholar
  5. [5]
    E. Hansen, Global Optimization Using Interval Analysis, Marcel Dekker Inc., N. Y., 1992.zbMATHGoogle Scholar
  6. [6]
    H. K. Jacobsen and H.C. Pedersen, An optimization system for teaching purposes, Eksamenproject, IMSOR, DTH, Lyngby, 1976.Google Scholar
  7. [7]
    T. P. Kashevarova and A. L. Semenov, “Solving Subdefinite Problems for Systems of Ordinary Differential Equations of the First Order in the UniCalc System”, All-Union Scientific-Technical Conference Intelligent Systems in Machine Building, Abstracts, Samara, 1991, pp. 21–24.Google Scholar
  8. [8]
    R. B. Kearfott, “Some Tests of Generalized Bisection”, ACM Transactions on Mathematical Software, 1987, No. 3, pp. 197–220.Google Scholar
  9. [9]
    K. Madsen, Parallel Algorithms for Global Optimization Report 91-07, Institute for Numerical Analysis, The Technical University of Denmark, Lyngby, Denmark, 1991.Google Scholar
  10. [10]
    K. Meintjes and A. P. Morgan, “Chemical Equilibrium Systems as Numerical Test Problems”, ACM Transactions on Mathematical Software, 1990, No. 2, pp. 143–151.CrossRefGoogle Scholar
  11. [11]
    J. J. More, B. S. Garbow, and K. E. Hillstrom, “Testing Unconstrained Optimization Software”, ACM Transactions on Mathematical Software, 1981, No. 1, pp. 17–41.MathSciNetCrossRefGoogle Scholar
  12. [12]
    A. S. Narin’yani, “Active data types for representing and processing of sub-definite information”, In: Actual Problems of the Computer Architecture Development and Computer System Software, Novosibirsk, USSR Acad. of Sciences, Siberian Division, Computer Center, 1983, pp. 128–141.Google Scholar
  13. [13]
    A. S. Narin’yani, “Subdefiniteness in knowledge representation and processing systems”, Transactions of USSR Acad, of Sciences, Technical Cybernetics, 1986, No. 5, pp. 3–28.Google Scholar
  14. [14]
    G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd (eds.), Handbooks in Operational Research and Management Science, Vol. 1. Optimization, North-Holland, 1989.Google Scholar
  15. [15]
    C. C. Petersen, “Computational experience with variants of the Balas algorithm applied to the selection of R and D Projects”, Management Science, 1967, Vol. 13, No. 9, pp. 736–750.CrossRefGoogle Scholar
  16. [16]
    A. L. Semenov, Solving Integer/Real Nonlinear Equations by Constraint Propagation, Technical Report No. 12, Institute of Mathematical Modelling, The Technical University of Denmark, Lyngby, Denmark, 1994.Google Scholar
  17. [17]
    A. L. Semenov and A. S. Leshchenko, “Interval and Symbolic Computations in the UniCalc Solver”, International Conference on Interval and Computer-Algebraic Methods in Science and Engineering INTERVAL-94, Abstracts, St-Petersburg, Russia, March, 1994, pp. 206–208.Google Scholar
  18. [18]
    A. L. Semenov and D. V. Petunin, “The Use of Multiintervals in the UniCalc Solver”, The International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN-95. Wuppertal, Germany, September, 1995 (to appear).Google Scholar
  19. [19]
    V. V. Telerman, Active data types, Preprint 792, USSR Acad. of Sciences, Siberian Division, Computer Center, Novosibirsk, 1988.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Alexander L. Semenov
    • 1
  1. 1.Novosibirsk Division of the Russian Research Institute of Artificial IntelligenceNovosibirskRussia

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