Abstract
A modification of Polyak’s special method of convex optimization is proposed. The properties of the corresponding algorithm are studied by computational experiments for convex separable and nonseparable optimization problems, nonconvex optimization problems for the potentials of atomic-molecular clusters, and a model optimal control problem. Sequential and parallel versions of the algorithm have been implemented, which made it possible to solve problems with dimensions of up to one hundred billion variables.
Similar content being viewed by others
REFERENCES
B. T. Polyak, “Minimization of unsmooth functionals,” USSR Comput. Math. Math. Phys. 9 (3), 14–29 (1969).
B. T. Polyak, Introduction to Optimization (Nauka, Moscow, 1983) [in Russian].
Yu. E. Nesterov, Introduction to Convex Optimization (MTsNMO, Moscow, 2010) [in Russian].
Yu. Nesterov, “Subgradient methods for huge-scale optimization problems,” Math. Program. 146 (1–2), 275–297 (2014).
J. M. C. Marques, A. A. C. C. Pais, and P. E. Abreu, “On the use of big-bang method to generate low-energy structures of atomic clusters modeled with pair potentials of different ranges,” J. Comput. Chem. 33 (4), 442–452 (2012).
P. N. Keating, “Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure,” Phys. Rev. 145, 637–645 (1966).
Yu. G. Evtushenko, Preprint, VTs im. A.A. Dorodnitsyna RAN (Dorodnitsyn Computing Center, Russian Academy of Sciences, Moscow, 2013).
A. Yu. Gornov, Computational Techniques for Solving Optimal Control Problems (Nauka, Novosibirsk, 2009) [in Russian].
A. Yu. Gornov, T. S. Zarodnyuk, and A. S. Anikin, “The computational technique for nonlinear nonconvex optimal control problems based on modification gully method,” DEStech Transactions on Computer Science and Engineering (2018), pp. 152–162.
A. Yu. Gornov, A. I. Tyatyushkin, and E. A. Finkelstein, “Numerical methods for solving terminal optimal control problems,” Comput. Math. Math. Phys. 56 (2), 221–234 (2016).
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 18-07-00587.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by E. Chernokozhin
Rights and permissions
About this article
Cite this article
Gornov, A.Y., Anikin, A.S. & Andrianov, A.N. Numerical Study of High-Dimensional Optimization Problems Using a Modification of Polyak’s Method. Comput. Math. and Math. Phys. 61, 1053–1062 (2021). https://doi.org/10.1134/S0965542521070034
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542521070034