Sergey I. Lyashko

A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming

We propose a new iterative two-step proximal algorithm for solving the problem of equilibrium programming in a Hilbert space. This method is a result of extension of L.D. Popov’s modification of Arrow-Hurwicz scheme for approximation of saddle points of convex-concave functions. The convergence of the algorithm is proved under the assumption that the solution exists and the bifunction is pseudo-monotone and Lipschitz-type.

Optimal Factorization of Operators by Operators That Are Consistent with the Graph’s Structure

In this paper we introduce the notion of an operator that is consistent with the structure of a graph and the computational system that is universal in the class of operators. The model fully corresponds to the processes occurring in distributed computing systems. The problem of the factorization of operators by operators consistent with the structure of a graph is formulated. We prove the criterion for the factorization of a linear invertible operator acting in a finite-dimensional linear space. We obtain upper estimations of the factorization depth of the class of linear invertible operators by linear operators compatible with the structure of the graph.

Levitating Orbitron: Grid Computing

Mathematical model of interaction for magnetic symmetric top (i.e. a rigid body and magnetic dipole simultaneously) in external magnetic field under uniform gravitational field is presented. Numerical modeling of the top dynamics, i.e. spinning and rotating around the axis of symmetry in axially-symmetric magnetic field is proposed. Investigation of the dynamics in some neighborhood of a given relative equilibrium for physically reasonable parameters of the system was required to generate a set of random trajectories (Monte-Carlo simulation) with small variations of parameters or initial conditions. More than 1000 of trajectories with 100 turns for each have been tested using grid computing on Grid-clusters of Ukrainian Academic Grid. The motion was limited in certain region for the trajectories with disturbed initial conditions and parameters within 1%. Executed analysis shows the possibility of stable motions and levitation in some neighborhood of a given relative equilibrium. It corresponds to the long trajectories observed in a physical experiment.

Nonparametric Ellipsoidal Approximation of Compact Sets of Random Points

One of the main problems of stochastic control theory is the estimation of attainability sets, or information sets. The most popular and natural approximations of such sets are ellipsoids. B.T. Polyak and his disciples use two kinds of ellipsoids covering a set of points—minimal volume ellipsoids and minimal trace ellipsoids. We propose a way to construct an ellipsoidal approximation of an attainability set using nonparametric estimations. These ellipsoids can be considered as an approximation of minimal volume ellipsoids and minimal trace ellipsoids. Their significance level depends only on the number of points and only one point from the set lays on a bound of such ellipsoid. This unique feature allows to construct a statistical depth function, rank multivariate samples and identify extreme points. Such ellipsoids in combination with traditional methods of estimation allow to increase accuracy of outer ellipsoidal approximations and estimate the probability of attaining a target set of states.

Some Remarks Concerning Supremum Attainment by Complex Functional

we suggested a model of competitiveness of brand product, which takes into account 10 factors. The model is based on 52 fuzzy rules of type. Potential of application of the model for control of competitiveness of brand product is shown. We state a problem of learning of fuzzy model of competitiveness on the basis of experimental data.