- PII
- S0044466925030068-1
- DOI
- 10.31857/S0044466925030068
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 65 / Issue number 3
- Pages
- 301-324
- Abstract
- The interval interpretation of a first-order divided difference, namely, interval slope is considered. Some properties of interval slopes, including ones for convex (concave) functions are proved. Based on the interval slope, necessary and sufficient conditions for the monotonicity of a function are formulated and proved. These criteria are used to propose an algorithm for the global optimization of a one-variable function taking into account its monotonicity. Numerical experiments are conducted that show that the developed global optimization method is applicable in the nondifferentiable case and significantly accelerates finding an approximate global optimum as compared with the basic version.
- Keywords
- глобальная оптимизация детерминированные методы оптимизации интервальный анализ интервальный наклон критерии монотонности
- Date of publication
- 17.09.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 23
References
- 1. Johnson D.E. Introduction to filter theory. Prentice Hall, 1976.
- 2. Zilinskas A. Optimization of one-dimensional multimodal functions // J. Royal Statistic. Soc., Ser. C (Applied Statistics). 1978. V. 27. № 3.
- 3. Kvasov D.E., Menniti D., Pinnarelli A., et al. Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions // Electric Power Syst. Res. 2008. V. 78. № 7. P. 1217–1229.
- 4. Bedrosian D., Vlach J. Time-domain analysis of networks with internally controlled switches // IEEE Transact. on Circuits and Systems I: Fundament. Theory and Appl. 1992. V. 39. № 3. P. 199–212.
- 5. Femia N., Tucci V. On the modeling of PWM converters for large signal analysis in discontinuous conduction mode // IEEE Transact. on Power Electronics. 1994. V. 9. № 5. P. 487–496.
- 6. Fasano G., Pint’er J.D. Efficient piecewise linearization for a class of nonconvex optimization problems: comparative results and extensions // Model. and Optimizat.: Theory and Appl.: MOPTA, Bethlehem, PA, USA, August 2017, Selected Contributions / Springer. 2019. P. 39–56.
- 7. Lassere J.B. Connecting optimization with spectral analysis of tridiagonal matrices // Math. Program. 2020. https://doi.org/10.1007/s10107-020-01549-3.
- 8. Jensen P.A., Bard J.F., Jensen P. Operations research models and methods. John Wiley & Sons, 2003.
- 9. Pint’er J. Extended univariate algorithms for n-dimensional global optimization // Computing. 1986. V. 36. № 1–2. P. 91–103.
- 10. Strongin R.G., Sergeyev Y.D. Global optimization with non-convex constraints: Sequential and parallel algorithms. Springer Science & Business Media, 2013. V. 45.
- 11. Gergel V., Grishagin V., Gergel A. Adaptive nested optimization scheme for multidimensional global search // J. Global Optimizat. 2016. V. 66. P. 35–51.
- 12. Sergeyev Y.D., Nasso M.C., Lera D. Numerical methods using two different approximations of space-filling curves for black-box global optimization // J. Global Optimizat. 2024. V. 88. № 3. P. 707–722.
- 13. Calvin J.M., Chen Y., Zilinskas A. An adaptive univariate global optimization algorithm and its convergence rate for twice continuously differentiable functions // J. Optimizat. Theory and Appl. 2012. V. 155. P. 628–636.
- 14. Sergeyev Y.D., Candelieri A., Kvasov D.E., Perego R. Safe global optimization of expensive noisy black-box functions in the δLipschitz framework // Soft Comput. 2020. V. 24. № 23. P. 17715–17735.
- 15. Posypkin M., Usov A., Khamisov O. Piecewise linear bounding functions in univariate global optimization // Soft Comput. 2020. V. 24. P. 17631–17647.
- 16. Posypkin M., Khamisov O. Automatic Convexity Deduction for Efficient Function’s Range Bounding // Mathematics. 2021. V. 9. № 2. P. 134.
- 17. Sergeyev Y.D., Nasso M.C., Mukhametzhanov M.S., Kvasov D.E. Novel local tuning techniques for speeding up onedimensional algorithms in expensive global optimization using Lipschitz derivatives // J. Comput. and Appl. Math. 2021. V. 383. P. 113134.
- 18. Posypkin M.A., Sergeyev Y.D. Efficient smooth minorants for global optimization of univariate functions with the first derivative satisfying the interval Lipschitz condition // J. Global Optimizat. 2022. P. 1–29.
- 19. Moore R.E., Kearfott R.B., Cloud M.J. Introduction to interval analysis/Ramon E // Moore, R. Baker Kearfott, Michael J. Cloud. Philadelphia, 2009.
- 20. Шарый С.П. Конечномерный интервальный анализ. Новосибирск: ИВТ СО РАН, 2022.
- 21. Neumaier A. Interval methods for systems of equations. Cambridge Univer. Press, 1990.
- 22. Хансен Э., Уолстер Дж.У. Глобальная оптимизация с помощью методов интервального анализа / Ed. by Шарой, С.П. 2-е изд., перевод с англ. М., Ижевск: Ин-т компьют. исслед.: R&C Dynamics, 2012.
- 23. Ratschek H., Rokne J. New computer methods for global optimization. Halsted Press, 1988.
- 24. Kearfott R.B. Rigorous global search: continuous problems. Springer Science & Business Media, 2013. V. 13.
- 25. Баженов А.Н., Жилин С.И., Кумков С.И., Шарый С.П. Обработка и анализ интервальных данных. М.: НИЦ “Регулярная и хаотическая динамика”, 2024.
- 26. Jafarpour S., Harapanahalli A., Coogan S. Efficient interaction-aware interval analysis of neural network feedback loops // IEEE Transactions on Automatic Control. 2024.
- 27. A branch and prune algorithm for the computation of generalized aspects of parallel robots / Caro S., Chablat D., Goldsztejn A. et al. // Artificial Intelligence. 2014. V. 211. P. 34–50. URL: https://www.sciencedirect.com/science/article/pii/S0004370214000125.
- 28. Евтушенко Ю.Г., Посыпкин М.А. Метод неравномерных покрытий для решения задач многокритериальной оптимизации с гарантированной точностью // Ж. вычисл. матем. и матем. физ. 2013. Т. 53. № 2. С. 209–224.
- 29. Kvasov, D., Sergeev, Y. et al. Lipschitz expensive global optimization // Encyclopedia of Optimization. Springer, 2023.
- 30. Stripinis L., Paulavicius R. Lipschitz-inspired HALRECT algorithm for derivative-free global optimization // J. of Global Optimization. 2024. V. 88. № 1. P. 139–169.
- 31. Евтушенко Ю.Г. Численный метод поиска глобального экстремума функций (перебор на неравномерной сетке) // Ж. вычисл. матем. и матем. физ. 1971. Т. 11. № 6. С. 1390–1403.
- 32. Поляк Б.Т. Введение в оптимизацию. М.: Наука, 1983.
- 33. Ratz D. A nonsmooth global optimization technique using slopes: the one-dimensional case // J. of Global Optimization. 1999. V. 14. P. 365–393.
- 34. Ratz D. An Optimized Interval Slope Arithmetic and its Application // Inst. fur Angewandte Mathematik. 1996.
- 35. Kearfott R.B., Nakao M., Neumaier A. и др. Standardized notation in interval analysis // Вычисл. технологии. 2010. Т. 15. № 1. С. 7–13.
- 36. Бахвалов Н.С., Жидков Н.П., Кобельков Г.М. Численные методы. 1987.
- 37. Ильин В.А., Садовничий В.А., Сендов Бл.Х. Математического анализ: Начальный курс. М.: МГУ, 1985.
- 38. Рокафеллар Р. Выпуклый анализ. М.: МИР, 1973.
- 39. Skelboe Stig. Computation of rational interval functions // BIT Numerical Mathematics. 1974. V. 14. № 1. P. 87–95.
- 40. Grant M., Boyd S., Ye Y. Disciplined convex programming. London: Springer, 2006.
