Keywords: global optimization, black-box functions, derivative-free methods, Lipschitz condition, applied problems.

Decision-making problems stated as problems of global optimization of an objective function subject to a set of constraints arise in various fields of human activity such as engineering design, economic models, geophysical studies, etc. [1,2,3,4,5]. Optimization problems characterized by functions with several local optima (typically, their number is unknown and can be very high) have a great importance for practical applications. These problems are usually referred to as global optimization ones. Both the objective function and constraints can be black-box and hard to evaluate functions with unknown analytical representations. Such a type of function is frequently met in real-life applications, especially in engineering, but the problems related to them often cannot be solved by traditional optimization techniques. This explains the growing interest of researchers in developing numerical global optimization methods able to tackle this difficult class of problem.

Because of the enormous computational cost involved, a researcher is typically willing to perform only a small number of functions evaluations when optimizing such costly functions. Thus, the main goal is to develop fast global optimization algorithms that produce reasonably good solutions with a limited number of function (and constraint) evaluations. In this work, various deterministic approaches proposed by the authors [1,2,3] for constructing efficient and reliable numerical methods for solving these problems based on the Lipschitz continuity assumption are discussed. The application of the proposed techniques to studying important practical optimization problems (actually intractable numerically or solved roughly) from different applied areas (such as from electrical engineering and telecommunications and geological mechanics [1,4,5] is shown.

1

Ya.D. Sergeyev, D.E. Kvasov, "Diagonal Global Optimization Methods", FizMatLit, Moscow, 2008. (In Russian)

2

R.G. Strongin, Ya.D. Sergeyev, "Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms", Kluwer, Dordrecht, 2000.

3

Ya.D. Sergeyev, D.E. Kvasov, "Lipschitz global optimization", in J.J. Cochran, (Editor), "Wiley Encyclopedia of Operations Research and Management Science", 4, 2812-2828, Wiley, New York, 2011.

4

D.E. Kvasov, D. Menniti, A. Pinnarelli, Ya.D. Sergeyev, N. Sorrentino, "Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions", Electric Power Systems Research, 78(7), 1217-1229, 2008. doi:10.1016/j.epsr.2007.10.009

5

V.I. Golubev, D.E. Kvasov, I.E. Kvasov, "Identification of seismogeological cracks location by using numerical global optimization methods", in I.B. Petrov, (Editor), Proc. of the 53rd MIPT Scientific Conference "Recent Advances in Basic and Applied Sciences", Moscow-Dolgoprudny, MIPT Press, Moscow, VII(2), 20-22, November 24-29, 2010.

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