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Vol 214, No 3 (2023)
Solving strongly convex-concave composite saddle point problems with a small dimension of one of the variables
Abstract
Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional.
3-53
Topological analysis of the pseudo-Euclidean Euler top for special values of parameters
Abstract
An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations.
54-70
Diophantine exponents of lattices and growth of multidimensional analogues of partial quotients
Abstract
A three-dimensional analogue of the connection between the exponent of the irrationality of a real number and the growth of the partial quotients of its expansion in a simple continued fraction is investigated. As a multidimensional generalization of continued fractions, Klein polyhedra are considered.
71-84
Geometric progression stabilizer in common metric
Abstract
So-called normalized metrics are considered on the set of elements of a geometric progression. A full description of normalized metrics with maximal stabilizer, which is the group of integer degrees of the common ratio of the progression, is presented. Previously, it was known that this group is the stabilizer for the minimal normalized metric (inherited from the real line) and the maximal normalized metric (an intrinsic metric all paths in which pass through zero). The stabilizer of a metric space is understood as the set of positive numbers such that multiplying the metric by this number produces a metric space lying at a finite Gromov-Hausdorff distance from the original space.
85-105
Degeneration of a graph describing conformal structure
Abstract
We consider the cell decomposition of the moduli space of real genus
106-119
On a class of interpolation inequalities on the 2D sphere
Abstract
We prove estimates for the
120-134
Bernstein–Szegö inequality for Riesz derivative of trigonometric polynomials in the spaces $L_p$, $0\le p\le\infty$, with the classical value of the best constant
Abstract
The Bernstein-Szegő inequality for the Weyl derivative of real order
135-152
An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fiber bundle
Abstract
We present an elementary approach to local combinatorial formulae for the Euler class of a fibre-oriented triangulated spherical fibre bundle. This approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those due to Mnëv.
153-168