Abstract
A general class of derivative-free optimization procedures is presented including the corresponding convergence theory. This theory turns out to be very constructive, in the sense that the convergence conditions not only can be verified easily for many existing algorithms, but also allow one to construct new procedures. It is shown that popular methods such as branch-and-bound concepts, Pintér's general class of procedures, the algorithms of Pijavskii, Shubert, and Mladineo, and the approach of Zheng and Galperin can not only be subsumed under this class of methods, but also partly be improved by regarding them within the framework presented.
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References
Horst, R.,A General Class of Branch-and-Bound Methods in Global Optimization with Some New Approaches for Concave Minimization, Journal of Optimization Theory and Applications, Vol. 51, pp. 271–291, 1986.
Horst, R., andTuy, H.,Remarks on Convergence and Finiteness of Branch-and-Bound Algorithms in Global Optimization, Discussion Paper, Universität Oldenburg, 1985.
Pinter, J.,A Unified Approach to Globally Convergent One-Dimensional Optimization Algorithms, Technical Report No. IAMI-83.5, Istituto per le Applicazioni della Matematica e dell'Informatica, CNR, Milan, Italy, 1983.
Pinter, J.,Globally Convergent Methods for N-Dimensional Multiextremal Optimization, Optimization, Vol. 17, pp. 1–16, 1986.
Zheng, Q., Jiang, B., andZhuang, S.,A Method for Finding the Global Extremum, Acta Mathematicae Applagatea Sinica, Vol. 2, pp. 164–174, 1978 (in Chinese).
Galperin, E. A., andZheng, Q.,Nonlinear Observation via Global Optimization Methods, Proceedings of the 1983 Conference on Information Sciences and Systems, Baltimore, Maryland, pp. 620–628, 1983.
Galperin, E. A.,The Cubic Algorithm, Journal of Mathematical Analysis and Applications, Vol. 112, pp. 635–640, 1985.
Pijavskii, S. A.,An Algorithm for Finding the Absolute Extremum of a Function, USSR Computational Mathematics and Mathematical Physics, Vol. 12, pp. 57–67, 1972.
Shubert, B. O.,A Sequential Method Seeking the Global Maximum of a Function, SIAM Journal of Numerical Analysis, Vol. 9, pp. 379–388, 1972.
Mladineo, R. H.,An Algorithm for Finding the Global Maximum of a Multimodal, Multivariate function, presented at the 12th International Symposium on Mathematical Programming, Cambridge, Massachusetts, 1985.
Horst, R.,An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312–321, 1976.
Thoai, N. V., andTuy, H.,Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556–566, 1980.
Strongin, R. G.,Numerical Methods for Multiextremal Problems, Nauka, Moscow, USSR, 1978 (in Russian).
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Communicated by G. Leitmann
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Horst, R., Tuy, H. On the convergence of global methods in multiextremal optimization. J Optim Theory Appl 54, 253–271 (1987). https://doi.org/10.1007/BF00939434
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DOI: https://doi.org/10.1007/BF00939434