Two modifications of extension of piyavskii’s global optimization algorithm to a function continuous on a compact interval and its convergence

В работах R.J. Vanderbei доказано, что непрерывная на выпуклом компактном множестве функция обладает свойством $\varepsilon $-липшицевости, обобщающим классическое понятие липшицевости. На основе этого свойства R.J. Vanderbei предложено одно обобщение метода Пиявского поиска глобального минимума непрерывной на отрезке функции. В данной работе предлагаются одна модификация этого метода для положительной $\varepsilon $-константы и одна модификация для положительной $\varepsilon $-константы и условия останова, не зависящего от выбора $\varepsilon $. Доказана сходимость предлагаемых алгоритмов, приведены результаты численных экспериментов на основе применения разработанной программы. Данные методы могут быть применены для оптимизации любых непрерывных на отрезке функций, например, при решении некоторых обратных задачах баллистики и в экономике в прямых задачах потребительского выбора маршаллианского типа с переменными ценами благ и с непрерывной функцией полезности. R.J. Vanderbei in his works proves that any continuous on a compact set function has the $\varepsilon $-Lipschitz property which extends conventional Lipschitz continuity. Based on this feature Vanderbei proposed one extension of Piyavskii’s global optimization algorithm to the continuous function case. In this paper we propose one modification of the Vanderbei’s algorithm for a positive $\varepsilon $-constant and another modification for a positive $\varepsilon $-constant and $\varepsilon $ value independent termination condition. We prove proposed methods convergence and perform several computational experiments with designed software for known test functions.

Halpern-type relaxed inertial algorithms with Bregman divergence for solving variational inequalities

It is well-known that the use of Bregman divergence is an elegant and effective technique for solving many problems in applied sciences. In this paper, we introduce and analyze two new inertial-like algorithms with Bregman divergence for solving pseudomonotone variational inequalities in a real Hilbert space. The first algorithm is inspired by Halpern -type iteration and subgradient extragradient method and the second algorithm is inspired by Halpern -type iteration and Tseng's extragradient method. Under suitable conditions, the strong convergence theorems of the algorithms are established without assuming the Lipschitz continuity and the sequential weak continuity of any mapping. Finally, several numerical experiments with various types of Bregman divergence are also performed to illustrate the theoretical analysis. The results presented in this paper improve and generalize the related works in the literature.

On Lipschitz continuity with respect to the Poincaré metric of linear contractions between Möbius gyrovector spaces

For every linear operator between inner product spaces whose operator norm is less than or equal to one, we show that the restriction to the Möbius gyrovector space is Lipschitz continuous with respect to the Poincaré metric. Moreover, the Lipschitz constant is precisely the operator norm.

Lipschitz regularity for degenerate elliptic integrals with 𝑝, 𝑞-growth

We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the 𝑥-variable.