Vol 80, No 10 (2019)

The maximum output deviation of a linear time-varying system is defined as the worst-case measure of the maximum value of the output Euclidean norm over a finite horizon provided that the sum of the squared energy of an external disturbance and a quadratic form of the initial state is 1. Maximum deviation is characterized in terms of solutions to differential matrix equations or inequalities. A modified concept of the boundedness of the system on a finite horizon under an external and initial disturbances is introduced and its connection with the concept of maximum deviation is established. Necessary and sufficient conditions for the boundedness of the system on a finite horizon are obtained. It is demonstrated that optimal controllers (including the multiobjective ones that minimize the maximum deviations of several outputs) as well as controllers ensuring the boundedness of the system can be designed using linear matrix inequalities.

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the l2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form \(\sqrt {\left( {s/n} \right)\log \left( {\sqrt p /s} \right)}\). We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.

The classical linear quadratic regulation problem is considered in the robust formulations where the matrices of the system and/or initial conditions are not know precisely. Several approaches are proposed where the quadratic cost is minimized against the worst-case uncertainties. Finding such controllers is performed via reducing the matrix Riccati equation with uncertainty to a single linear matrix inequality. The properties of the solutions are discussed and the comparison with previously known approaches is performed.

We discuss an approach to signal recovery in Generalized Linear Models (GLM) in which the signal estimation problem is reduced to the problem of solving a stochastic monotone Variational Inequality (VI). The solution to the stochastic VI can be found in a computationally efficient way, and in the case when the VI is strongly monotone we derive finite-time upper bounds on the expected ‖ · ‖²₂ error converging to 0 at the rate O(1/K) as the number K of observation grows. Our structural assumptions are essentially weaker than those necessary to ensure convexity of the optimization problem resulting from Maximum Likelihood estimation. In hindsight, the approach we promote can be traced back directly to the ideas behind the Rosenblatt’s perceptron algorithm.

In this paper, new approaches to obtain optimal behavior in dynamic multicriteria games are suggested. The multicriteria Nash equilibrium is designed via the Nash bargaining scheme (Nash products), and the cooperative equilibrium is determined by the Nash bargaining solution for the entire planning horizon. The process of coalition formation in dynamic mul-ticriteria games is studied. For constructing the characteristic function the Nash bargaining scheme is applied, where the multicriteria Nash equilibrium plays the role of the status quo points. Two modifications of the characteristic function are presented that take into account the information structure of the game (the models without and with information). The dynamic multicriteria bioresource management problem is considered. The players’ strategies and the quantities of resource are compared under the cooperative and noncooperative behavior for the above modifications of the characteristic function.