In this paper we propose two p-th order tensor methods for µstrongly-convex-strongly-concave saddle point problems. The first method is based on the assumption of L p -smoothness of the gradient of the objective and it achieves a convergence rate of O((L p R p /µ) 2 p+1 log(µR 2 /ε)), where R is an estimate of the initial distance to the solution. Under additional assumptions of L 1 -, L 2 and L p -smoothness of the gradient of the objective we connect the first method with a locally superlinear converging algorithm and develop a new method with the complexity of O((L p R p /µ)Since we treat saddle-point problems as a particular case of variational inequalities, we also propose two methods for strongly monotone variational inequalities with the same complexity as the described above.
Ресурсная база Сибири-один из основополагающих элементов конкурентноспособности экономики России. Правильное и рациональное использование имеющихся ресурсов способно трансформировать «сырьевое проклятие» в драйвер значительного экономического и промышленного роста как региональных экономик, так и государства. России принадлежит более 30 % всех природных ресурсов в мире. Огромные топливно-энергетические ресурсы открыты почти во всех экономических районах РФ, но главная, базовая их часть сосредоточена в Сибири. Между тем современный этап в экономическом развитии Сибирских регионов обусловлен внешнеэкономическим фактором-«азиатским» вектором экономического сотрудничества. Российско-китайское сотрудничество способно придать новый импульс социальноэкономическому развитию сибирских регионов.
We address the stability problem for linear switching systems with mode-dependent restrictions on the switching intervals. Their lengths can be bounded as from below (the guaranteed dwell-time) as from above. The upper bounds make this problem quite different from the classical case: a stable system can consist of unstable matrices, it may not possess Lyapunov functions, etc. We introduce the concept of Lyapunov multifunction with discrete monotonicity, which gives upper bounds for the Lyapunov exponent. Its existence as well as the existence of invariant norms are proved. Tight lower bounds are obtained in terms of a modified Berger-Wang formula over periodizable switching laws. Based on those results we develop a method of computation of the Lyapunov exponent with an arbitrary precision and analyse its efficiency in numerical results. The case when some of upper bounds can be cancelled is analysed.
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