Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1588610.31857/S0869-56524875493-495Research ArticleConstructive generalization of classical sufficient second-order optimality conditionsEvtushenkoYu. G.<p>Academicaian of the Russian Academy of Sciences</p>yuri-evtushenko@yandex.ruTret’yakovA. A.tret@ap.siedlce.plDorodnitsyn Computing Centre, Federal Research Center Computer Science and Control of the Russian Academy of SciencesMoscow Institute of Physics and TechnologyMoscow Aviation Institute (National Research University)System Research Institute of the Polish Academy of SciencesSiedlce University0209201948754934952908201929082019Copyright © 2019, Russian academy of sciences2019<p class="a"><span lang="EN-US">In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.</span></p>optimization problemequality constraintsinequitiesslack variablessufficient conditionsequivalenceзадача оптимизацииограничения равенстванеравенстваискусственные переменныедостаточные условияэквивалентность[Евтушенко Ю. Г. Методы решения экстремальных задач и их применение в системах оптимизации. М.: Наука, 1982. 432 с.][Поляк Б. Т. Введение в оптимизацию. М.: Наука. Гл. ред. физ.-мат. лит., 1983.][Брежнева О. А., Евтушенко Ю. Г., Третьяков А. А. 2-фактор-метод модифицированных функций Лагранжа для решения вырожденных задач условной оптимизации // ДАН. 2006. Т. 408. № 4. С. 439-442.][Bertsekas D. P. Nonlinear Programming. Belmont: Athena Scientific, 1999. P. 191-276.][Brezhneva O. A., Tret’yakov A. A. The p-Factor Lagrange Methods for Degenerate Nonlinear Programming // Numerical Functional Analysis and Optimization. 2007. V. 28. № 9/10. P. 1051-1086.]