After a series of publications of T.E. O’Neil et al. (e.g. in 2010), dynamic program- ming seems to be the most promising way to solve knapsack problems. Some techniques are known to make dynamic programming algorithms (DPA) faster. One of them is the graphical method that deals with piecewise linear Bellman functions. For some problems, it was previ- ously shown that the graphical algorithm has a smaller running time in comparison with the classical DPA and also some other advantages. In this paper, an exact graphical algorithm (GrA) and a fully polynomial-time approximation scheme based on it are presented for an in- vestment optimization problem having the best known running time. The algorithms are based on new Bellman functional equations and a new way of implementing the GrA.

For the nonlinear minimum phase single-input single-output (SISO) systems, this paper formalizes the structure of the equivalent “input–output” representation taking into account the smooth perturbations that do not belong to the control space. Using this representation, we design a basic combined control law ensuring the asymptotic stability of the tracking error. In contrast to the traditional approach with generation of a reference signal and a perturbation, the complex estimation of the external signals and their derivatives employs a sliding mode observer. And finally, we formalize the physical implementability conditions of the invariant tracking system in the case when only the output (controlled variable) and the reference signal are directly measured.

A problem of minimization of integral losses on given horizon is considered for stochastic system in continuous time. The losses occur in jump times of a Poisson process, and represent continuous convex function of control parameter on convex compact finite-dimensional set. At the jump times an oracle provide stochastically perturbed sub-gradient of the loss function, bounded in mean squares; the noise is additive and centered. Control strategy generated by Mirror Descent algorithm is suggested. For the strategy an explicit upper bound for integral loss discrepancy over its minimum is proved. Example of such strategy application to queueing model is examined.

In the nonparametric kernel estimation of the unknown probability densities and their derivatives there exist several methods for estimation of the kernel function bandwidth of which the CV and SCV methods of cross-validation are most simple and suitable. The former method was developed both for the density itself and its derivatives; the latter one, for density only. Yet it generates estimates with a higher rate of convergence and substantially smaller scatter. For the problem of nonparametric restoration of the density derivative from an independent sample, a data-based estimate of the kernel function bandwidth was constructed

This paper introduces the concept of a truncated order statistic. We derive analytical expressions for the mean value and the variance of a truncated order statistic with a threshold determined by a given number of the variational series term. The algorithm of noisy signal detection is developed for detection systems, which employ the method of statistical hypotheses testing and the energy detection criterion. The algorithm utilizes statistical characteristics of order statistics.