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Methods for solving multi-extremal problems (global search)

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Abstract

This work considers some methods for searching for the absolute (global) minimum of functions having concave minorants on bounded convex sets. Special methods for minimizing Lipschitz and concave functions are proposed. The methods are based on deeper cuts that use second-order information. The possibility of applying some cutting schemes that solve convex programming problems for the global minimization of concave functions on bounded convex sets is analyzed. A new original class of cuts, and a new method for minimizing concave functions, are presented. The work also considers a new approach for the solution of multi-extremal problems that is different from cutting methods. It is based on the approximation of convex sets by the convex envelopes over the convex hull of a finite number of points. This work comes close to the studies by Tuy [1,8].

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References

  1. H. Tuy, Concave programming with linear constraints, Dokl. AN SSSR 159 (1964) 32–35.

    Google Scholar 

  2. V. P. Bulatov and L.I. Kasinskaya, Some methods for concave function minimization and their applications,Metody Optimizatsii i ih Prilozheniya (Nauka. Sib. Otd-nie, Novosibirsk, 1982) pp. 71–80.

    Google Scholar 

  3. V.P. Bulatov,The Imbedding Methods in Optimization Problems (Nauka. Sib. otd-nie, Novosibirsk, 1977).

    Google Scholar 

  4. L.T. Ashchepkov et al.,Methods for Solving Problems of Mathematical Programming and Optimal Control (Nauka. Sib otd-nie, Novosibirsk, 1984) p. 7–43.

    Google Scholar 

  5. J.E. Kelley, The cutting-plane method for solving convex programming, SIAM J. 8 (1960) 703–712.

    Google Scholar 

  6. J.B. Rosen, Iterative solution of nonlinear optimal control problems, SIAM J. Control (1966) 223–244.

  7. V.P. Bulatov, The methods for solving multiextremal problems,Metody Chislennogo Analiza i Optimizatsii (Nauka, Novosibirsk, 1987).

    Google Scholar 

  8. H. Tuy and R. Horst, Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and d.c. optimization problems, Math. Progr. 41 (1988) pp. 161–185.

    Google Scholar 

  9. N.I. Merenkova,Mathematical Models for Optimizing the Laying and Structure of Pipeline Systems (SEI, Irkutsk, 1977) p. 145.

    Google Scholar 

  10. V.P. Bulatov,Approximation Methods for Solving Some Problems of Mathematical Programming (Prikladnaya matematika, Irkutsk, 1969).

    Google Scholar 

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Authors and Affiliations

  1. Department of Operations Research, Siberian Energy Institute, USSR Academy of Sciences, Lermontov Street 130, 664033, Irkutsk, USSR

    V. P. Bulatov

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  1. V. P. Bulatov
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Editors' note: The original manuscript was edited by Faiz Al-Khayyal and Panos Pardalos. In this process, we endeavoured to focus only on corrections (both technical and grammatical) and on the insertion of clarifying phrases and passages. Major insertions are flagged by footnotes. The revised manuscript was typed in TEX by Carmella Bell of Georgia Tech.

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Bulatov, V.P. Methods for solving multi-extremal problems (global search). Ann Oper Res 25, 253–277 (1990). https://doi.org/10.1007/BF02283699

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