We consider a control problem over an infinite time horizon with a linear stochastic system with an unstable asymptotically unbounded state matrix. We extend the notion of anti-stability of a matrix to the case of non-exponential anti-stability, and introduce an antistability rate function as a characteristic of the rate of growth for the norm of the corresponding fundamental matrix. We show that the linear stable feedback control law is optimal with respect to the criterion of the adjusted extended long-run average. The designed criterion explicitly includes information about the rate of anti-stability and the parameters of the disturbances. We also analyze optimality conditions.
Existence, weak uniqueness, and Markov - Dobrushin's conditions are established for Markov solutions of highly degenerate stochastic differential equations.
Consideration was given to optimization of the queue control strategy in the MlGl1l queuing system where decision about continuing or stopping admission of customers is made at the service completion instants of each customer in compliance with the distribution on the set of decisions depending on the number of customers remaining in the system. The mean specific income in the stationary mode was used as the efficiency criterion and the set of permissible strategies coincided with the set of homogeneous randomised Markov strategies. It was proved that if there exists an optimal strategy, then it is degenerate and threshold with one point on control switching, that is, if the number of customers in the system exceeds a certain level, then admission of customers must be stooped or, otherwise, it must be continued.
Two optimal stopping problems for geometric random walks with the observer’s power payoff function, on the finite and infinite horizons, are solved. For these problems, an explicit form of the cut value and also optimal stopping rules are established. It is proved that the optimal stopping rules are nonrandomized thresholds and describe the corresponding free boundary. An explicit form of the free boundary is presented.
We consider the problem of stochastic linear regulator over an infinite time horizon with superexponentially stable matrix in the equation of state dynamics. The form of the optimal control based on the criterion taking into account the information about the parameters of disturbances and the matrix stability rate was determined. The results obtained were used to analyze the model of a system with extremely impatient agents where the objective functional includes discounting by the asymptotically unbounded rate.
We find optimal (from the insurer’s point of view) strategies for insurance and reinsurance
in a controllable Cramer–Lundberg risk process that describes the capital dynamics of
an insurance company over an infinite time interval. As the optimality criterion being minimized,
we use the stationary variation coefficient, taking into account additional constraints
on residual risks for both insurers and reinsurer. We establish that it is best to use stop-loss
reinsurance with an upper limit and insurance which is a combination of a stop-loss strategy
and deductible. Equations that define optimal strategies parameters are derived.
This paper is devoted to the study of multicriteria cooperative games with vector payoffs and coalition partition. An imputation based on the concept of the Owen value is proposed. The definition of a stable coalition partition for bi-criteria games is formulated. A three-player cooperative game with the 0-1 characteristic function is considered and stability conditions of a coalition partition are established.
We state an optimal control problem for a class of dynamical systems whose nonlinear objects can be represented as objects with linear structure and state-dependent parameters. The linearity of the structure of the transformed nonlinear system and the quadratic performance functional allow one to move from the need to search for solutions of the Hamilton–Jacobi equation to an equation of the Riccati type with state-dependent parameters when synthesizing the optimal control, i.e., the controller parameters. The main problem of implementing the optimal control is related to the problem of being capable of finding solutions of such an equation online, at the object operation rate. An algorithmic method for the parametric optimization of the controller is proposed. The method is based on using the necessary optimality conditions for the control system in question. Our algorithms can be used both to optimize the time-varying objects themselves given an appropriate choice of the parameters for this purpose and to optimize the entire control system using an appropriate parametric adjustment of the controllers. The efficiency of the algorithms is demonstrated by the example of drug treatment of patients with HIV.
We present the basic properties of the a new pattern analysis method in parallel coordinates; results of the method do not depend on the ordering of data in the original sample of objects being analyzed. We prove that clusters obtained with this method do not overlap. We also show the possibility of representing objects of one cluster in the form of monotonically increasing/decreasing functions.
The prototype of the isolated words recognition software based on the phonetic decoding method with the Kullback-Leibler divergence is presented. The architecture and basic algorithms of the software are described. Finally, an example of application to the problem of isolated words recognition is provided.
This paper considers a voting problem in which the individual preferences of electors are defined by the ranked lists of candidates. For single-winner elections, we apply the criterion of weak positional dominance (WPD, PD), which is closely related to the positional scoring rules. Also we formulate the criterion of weak mutual majority (WMM), which is stronger than the majority criterion but weaker than the criterion of mutual majority (MM). Then we construct two modifications for the median voting rule that satisfy the Condorcet loser criterion. As shown below, WPD and WMM are satisfied for the first modification while PD and MM for the second modification. We prove that there is no rule satisfying WPD and MM simultaneously. Finally, we check a list of 37 criteria for the constructed rules.
An axiomatics of power indices in voting with quota was proposed. It relies on the additivity and dictator axioms. Established was an important property that the player’s power index is representable as the sum of contributions of the coalitions in which it is a pivot member. The coalition contributions are independent of the players’ weights or the quota. The general theorem of power index representation and the theorem of representation for a power index of anonymous players were formulated and proved.
We show the results of a statistical study of the complexity of the asymmetric traveling salesman problem (ATSP) obtained by processing a specially generated pool of matrices. We show that the normal distribution can serve as an approximation to the distribution of the logarithm of complexity for a fixed problem dimension. We construct a family of probability distributions that represent satisfactory approximations of the complexity distribution with a dimension of the cost matrix from 20 to 49. Our main objective is to make probabilistic predictions of the complexity of individual problems for larger values of the dimension of the cost matrix. We propose a representation of the complexity distribution that makes it possible to predict the complexity. We formulate the unification hypothesis and show directions for further study, in particular proposals on the task of clustering “complex” and “simple” ATSP problems and proposals on the task of directly predicting the complexity of a specific problem instance based on the initial cost matrix.
Consideration was given to a special problem of controlling a formation of mobile agents, that of uniform deployment of several identical agents on a segment of the straight line. For the case of agents obeying the first-order dynamic model, this problem seems to be first formulated in 1997 by I.A. Wagner and A.M. Bruckstein as “row straightening.” In the present paper, the straightening algorithm was generalized to a more interesting case where the agent dynamics obeys second-order differential equations or, stated differently, it is the agent’s acceleration (or the force applied to it) that is the control.
In Russia, chain stores have achieved considerable market power. In this work, we combine a Dixit–Stiglitz industry model with a monopolistic retailer in order to address the following questions: does the retailer always impair prices, variety of goods, and ultimately welfare? Which market structure is worse: Nash or Stackelberg? What should be the public policy in this area?