Alexei R. Pankov

Robust filtering of process in the stationary difference stochastic system

A technique to construct the robust Kalman filter for process estimation in the difference linear stationary stochastic system with an unknown covariance observation error matrix was developed. Consideration was given to the algorithm of constructing the set of permissible covariance matrices from a priori statistical data. A numerical method for solution of the general minimax optimization problem was proposed; and on its basis an iterative algorithm to calculate the robust filter parameters was developed, and its convergence was proved. Results of the numerical experiment were presented.

Minimax Optimization of Investment Portfolio by Quantile Criterion

Optimization of a single-step investment strategy by a quantile criterion (VaR) is studied. The a priori information on the distribution law for the vector of effective financial instruments is defined by certain constraints on the first- and second-order moments. The concept of a minimax investment strategy is formulated, and the strategy is constructed using the convex programming duality theory. For a set of admissible strategies defined by constraints in the form of linear equalities and inequalities, the minimax strategy is constructed with the help of an analytical dependence on the solution of the dual problem. The existence and uniqueness of the minimax quantile strategy are studied. A computation procedure for solving the dual problem is described. Results of a numerical modeling experiment are given.

Minimax Quadratic Optimization and Its Application to Investment Planning

Minimax optimization with a quadratic criterion and linear equality- and inequality-type constraints is investigated. The minimax solution is expressed in general form. Sufficient conditions for the minimax solution to be uniquely determined by the solution of the dual problem are formulated. The results are applied to construct an investment portfolio having guaranteed characteristics under a priori statistical uncertainty.

On minimax identification: method of dual optimization

The problem of minimax affine identification of a linear uncertain stochastic multivariate model is considered. The minimax optimization problem together with the corresponding dual one are stated and examined. The necessary and sufficient conditions for the minimax affine estimate to exist and to be determined analytically via the dual problem solution are given. The algorithm of minimax stochastic estimation for the infinite-dimensional model given a finite number of observations is also considered. The numerical method for minimax estimation is described, and the results of computer modeling are presented.

Minimax-statistical approach to increasing reliability of measurement information processing

Consideration was given to the methods of guaranteeing estimation of the law of motion of flight vehicles from the results of the trajectory measurements, as well as the methods and algorithms of minimax estimation under fixed sets of uncertainty of the covariance matrices of the observation errors. A minimax-statistical approach to the problem of estimation on the basis of uncertainty sets in the form of confidence regions of given reliability was proposed. The results of numerical modeling were presented.

Minimax estimation by probabilistic criterion

A minimax estimation problem in multidimensional linear regression model containing uncertain parameters and random quantities is considered. Simultaneous distribution of random quantities that are a part of the observation model is not prescribed exactly; however, it has a fixed mean and a covariance matrix from the given set. For estimation algorithm optimization, we applied a minimax approach with the risk measure in the form of the exceedance probability of the estimate of a prescribed level by an error. It was shown that a linear estimation problem is equivalent to the minimax problem with the mean-square criterion. In addition, the corresponding linear estimate will be the best (in the minimax sense) by the probabilistic criterion at the class of all unbiased estimates. The least favorable distribution of random model parameters is also constructed. Several partial cases and a numerical example are considered.

Minimax Estimation in Singular Uncertain Stochastic Models

Minimax parametric identification of a multidimensional uncertain stochastic linear model under incomplete a priori information on the first two moments of the characteristics of the parameters of the model is investigated. The minimax problem is reduced through regularization of the initial mean-square criterion to a dual problem without any additional assumptions on the nondegeneracy of matrices belonging to the uncertainty set. Results are illustrated by concrete examples of singular models.

A solution of the filtering and smoothing problems for uncertain-stochastic linear dynamic systems

The authors present new filtering algorithms for uncertain-stochastic dynamic systems, which are optimal in the sense of a minimax-stochastic criterion. These algorithms allow us to estimate the state vector of dynamic systems given incomplete a priori information about the system characteristics and using observations of the state and input signals. The filtering algorithm is used to construct an optimal two-filter smoothing algorithm for pure uncertain dynamic systems. These algorithms are numerically tested. Results are compared with the results of Kalman filtering and smoothing in the case of complete information about the input signal characteristics.

Minimax control of a process in a linear uncertain-stochastic system with incomplete data

Consideration is given to the control problem in a linear stochastic differential system where constant noise intensities in equations of state and observation are prescribed only accurate within the membership of some known sets. For control optimization, an integral root-mean-square performance criterion is used. The problem is solved by the transition to a dual one, which makes it possible to prove the existence of a saddle point of the criterion and obtain an explicit expression for the minimax control operator as functions of the solution to the dual problem. To solve the latter, an iteration algorithm is proposed; the convergence of the algorithm is proved and investigated by a model example.

Linear stochastic programming with minimax quantile and probability criterions

The problems of linear stochastic model optimization are considered using quantile and probability criterions. The a priori information about the distribution law of the model random coefficients is defined by certain constraints on the first- and second-order moments. The concept of the minimax strategy is formulated and the last one is constructed using the convex programming duality theory. The analytic dependence of the minimax strategy on the solution of the dual problem is derived. A computational procedure for solving the dual problem is examined. The results of computer modeling are presented.