All issues

Collective behavior can serve as a mechanism of thermoregulation and play a key role in the joint survival of a group of organisms. In higher animals, such phenomena are usually the subject of study of biology since sudden transitions to collective behavior are difficult to differentiate from the psychological and social adaptation of animals. However, in this paper, we indicate several important examples when a flock of higher animals demonstrates phase transitions similar to known phenomena in liquids and gases. This issue can also be studied experimentally within the framework of synthetic systems consisting of self-propelled robots that act according to a certain given algorithm. Generalizing both of these cases, we consider the problem of phase transitions in a dense group of interacting selfpropelled agents. Within the framework of microscopic theory, we propose a mathematical model of the phenomenon, in which agents are represented as bodies interacting with each other in accordance with an effective potential of a special type, expressing the desire of agents to move in the direction of the gradient of the joint thermal field. We show that the number of agents in the group, the group power, is the control parameter of the problem. A discrete model with individual dynamics of agents reproduces most of the phenomena observed both in natural flocks of higher animals engaged in collective thermoregulation and in synthetic complex systems. A first-order phase transition is observed, which symbolizes a change in the aggregate state in a group of agents. One observes the self-assembly of the initial weakly structured mass of agents into dense quasi-crystalline structures. We demonstrate also that, with an increase in the group power, a second-order phase transition in the form of thermal convection can occur. It manifests in a sudden liquefaction of the group and a transition to vortex motion, which ensures more efficient energy consumption in the case of a synthetic system of interacting robots and the collective survival of all individuals in the case of natural animal flocks.With an increase in the group power, secondary bifurcations occur, the vortex structure in agent medium becomes more complicated.

The article is devoted to first-order adaptive methods for optimization problems with relatively strongly convex functionals. The concept of relatively strong convexity significantly extends the classical concept of convexity by replacing the Euclidean norm in the definition by the distance in a more general sense (more precisely, by Bregman’s divergence). An important feature of the considered classes of problems is the reduced requirements concerting the level of smoothness of objective functionals. More precisely, we consider relatively smooth and relatively Lipschitz-continuous objective functionals, which allows us to apply the proposed techniques for solving many applied problems, such as the intersection of the ellipsoids problem (IEP), the Support Vector Machine (SVM) for a binary classification problem, etc. If the objective functional is convex, the condition of relatively strong convexity can be satisfied using the problem regularization. In this work, we propose adaptive gradient-type methods for optimization problems with relatively strongly convex and relatively Lipschitzcontinuous functionals for the first time. Further, we propose universal methods for relatively strongly convex optimization problems. This technique is based on introducing an artificial inaccuracy into the optimization model, so the proposed methods can be applied both to the case of relatively smooth and relatively Lipschitz-continuous functionals. Additionally, we demonstrate the optimality of the proposed universal gradient-type methods up to the multiplication by a constant for both classes of relatively strongly convex problems. Also, we show how to apply the technique of restarts of the mirror descent algorithm to solve relatively Lipschitz-continuous optimization problems. Moreover, we prove the optimal estimate of the rate of convergence of such a technique. Also, we present the results of numerical experiments to compare the performance of the proposed methods.

Data hiding in digital images is a promising direction of cybersecurity. Digital steganography methods provide imperceptible transmission of secret data over an open communication channel. The information embedding efficiency depends on the embedding imperceptibility, capacity, and robustness. These quality criteria are mutually inverse, and the improvement of one indicator usually leads to the deterioration of the others. A balance between them can be achieved using metaheuristic optimization. Metaheuristics are a class of optimization algorithms that find an optimal, or close to an optimal solution for a variety of problems, including those that are difficult to formalize, by simulating various natural processes, for example, the evolution of species or the behavior of animals. In this study, we propose an approach to data hiding in the hybrid spatial-frequency domain of digital images based on metaheuristic optimization. Changing a block of image pixels according to some change matrix is considered as an embedding operation. We select the change matrix adaptively for each block using metaheuristic optimization algorithms. In this study, we compare the performance of three metaheuristics such as genetic algorithm, particle swarm optimization, and differential evolution to find the best change matrix. Experimental results showed that the proposed approach provides high imperceptibility of embedding, high capacity, and error-free extraction of embedded information. At the same time, storage of change matrices for each block is not required for further data extraction. This improves user experience and reduces the chance of an attacker discovering the steganographic attachment. Metaheuristics provided an increase in imperceptibility indicator, estimated by the PSNR metric, and the capacity of the previous algorithm for embedding information into the coefficients of the discrete cosine transform using the QIM method [Evsutin, Melman, Meshcheryakov, 2021] by 26.02% and 30.18%, respectively, for the genetic algorithm, 26.01% and 19.39% for particle swarm optimization, 27.30% and 28.73% for differential evolution.

The work solves the problem of establishing the dependence of the potential for spatial selection of useful and interfering signals according to the signal-to-interference ratio criterion on the positioning error of user equipment during beamforming by their location at a base station, equipped with an antenna array. Configurable simulation parameters include planar antenna array with a different number of antenna elements, movement trajectory, as well as the accuracy of user equipment location estimation using root mean square error of coordinate estimates. The model implements three algorithms for controlling the shape of the antenna radiation pattern: 1) controlling the beam direction for one maximum and one zero; 2) controlling the shape and width of the main beam; 3) adaptive beamforming. The simulation results showed, that the first algorithm is most effective, when the number of antenna array elements is no more than 5 and the positioning error is no more than 7 m, and the second algorithm is appropriate to employ, when the number of antenna array elements is more than 15 and the positioning error is more than 5 m. Adaptive beamforming is implemented using a training signal and provides optimal spatial selection of useful and interfering signals without device location data, but is characterized by high complexity of hardware implementation. Scripts of the developed models are available for verification. The results obtained can be used in the development of scientifically based recommendations for beam control in ultra-dense millimeter-wave radio access networks of the fifth and subsequent generations.

The paper is devoted to one approach to constructing subgradient methods for strongly convex programming problems with several functional constraints. More precisely, the strongly convex minimization problem with several strongly convex (inequality-type) constraints is considered, and first-order optimization methods for this class of problems are proposed. The special feature of the proposed methods is the possibility of using the strong convexity parameters of the violated functional constraints at nonproductive iterations, in theoretical estimates of the quality of the produced solution by the methods. The main task, to solve the considered problem, is to propose a subgradient method with adaptive rules for selecting steps and stopping rule of the method. The key idea of the proposed methods in this paper is to combine two approaches: a scheme with switching on productive and nonproductive steps and recently proposed modifications of mirror descent for convex programming problems, allowing to ignore some of the functional constraints on nonproductive steps of the algorithms. In the paper, it was described a subgradient method with switching by productive and nonproductive steps for strongly convex programming problems in the case where the objective function and functional constraints satisfy the Lipschitz condition. An analog of the proposed subgradient method, a mirror descent scheme for problems with relatively Lipschitz and relatively strongly convex objective functions and constraints is also considered. For the proposed methods, it obtained theoretical estimates of the quality of the solution, they indicate the optimality of these methods from the point of view of lower oracle estimates. In addition, since in many problems, the operation of finding the exact subgradient vector is quite expensive, then for the class of problems under consideration, analogs of the mentioned above methods with the replacement of the usual subgradient of the objective function or functional constraints by the $\delta$-subgradient were investigated. The noted approach can save computational costs of the method by refusing to require the availability of the exact value of the subgradient at the current point. It is shown that the quality estimates of the solution change by $O(\delta)$. The results of numerical experiments illustrating the advantages of the proposed methods in comparison with some previously known ones are also presented.

This paper studies a model of a population with non-overlapping generations and density-dependent regulation of birth rate. The population breeds seasonally, and its reproductive potential is determined genetically. The model proposed combines an ecological dynamic model of a limited population with non-overlapping generations and microevolutionary model of its genetic structure dynamics for the case when adaptive trait of birth rate controlled by a single diallelic autosomal locus with allelomorphs A and a. The study showed the genetic composition of the population, namely, will it be polymorphic or monomorphic, is mainly determined by the values of the reproductive potentials of heterozygote and homozygotes. Moreover, the average reproductive potential of mature individuals and intensity of self-regulation processes determine population dynamics. In particularly, increasing the average value of the reproductive potential leads to destabilization of the dynamics of age group sizes. The intensity of self-regulation processes determines the nature of emerging oscillations, since scenario of stability loss of fixed points depends on the values of this parameter. It is shown that patterns of occurrence and evolution of cyclic dynamics regimes are mainly determined by the features of life cycle of individuals in population. The life cycle leading to existence of non-overlapping generation gives isolated subpopulations in different years, which results in the possibility of independent microevolution of these subpopulations and, as a result, the complex dynamics emergence of both stage structure and genetic one. Fixing various adaptive mutations will gradually lead to genetic (and possibly morphological) differentiation and to differences in the average reproductive potentials of subpopulations that give different values of equilibrium subpopulation sizes. Further evolutionary growth of reproductive potentials of limited subpopulations leads to their number fluctuations which can differ in both amplitude and phase.

In the first part of the paper we present convergence analysis of AGMsDR method on a new class of functions — in general non-convex with $M$-Lipschitz-continuous gradients that satisfy Polyak – Lojasiewicz condition. Method does not need the value of $\mu^{PL}>0$ in the condition and converges linearly with a scale factor $\left(1 - \frac{\mu^{PL}}{M}\right)$. It was previously proved that method converges as $O\left(\frac1{k^2}\right)$ if a function is convex and has $M$-Lipschitz-continuous gradient and converges linearly with a~scale factor $\left(1 - \sqrt{\frac{\mu^{SC}}{M}}\right)$ if the value of strong convexity parameter $\mu^{SC}>0$ is known. The novelty is that one can save linear convergence if $\frac{\mu^{PL}}{\mu^{SC}}$ is not known, but without square root in the scale factor.

The second part presents modification of AGMsDR method for solving problems that allow alternating minimization (Alternating AGMsDR). The similar results are proved.

As the result, we present adaptive accelerated methods that converge as $O\left(\min\left\lbrace\frac{M}{k^2},\,\left(1-{\frac{\mu^{PL}}{M}}\right)^{(k-1)}\right\rbrace\right)$ on a class of convex functions with $M$-Lipschitz-continuous gradient that satisfy Polyak – Lojasiewicz condition. Algorithms do not need values of $M$ and $\mu^{PL}$. If Polyak – Lojasiewicz condition does not hold, the convergence is $O\left(\frac1{k^2}\right)$, but no tuning needed.

We also consider the adaptive catalyst envelope of non-accelerated gradient methods. The envelope allows acceleration up to $O\left(\frac1{k^2}\right)$. We present numerical comparison of non-accelerated adaptive gradient descent which is accelerated using adaptive catalyst envelope with AGMsDR, Alternating AGMsDR, APDAGD (Adaptive Primal-Dual Accelerated Gradient Descent) and Sinkhorn's algorithm on the problem dual to the optimal transport problem.

Conducted experiments show faster convergence of alternating AGMsDR in comparison with described catalyst approach and AGMsDR, despite the same asymptotic rate $O\left(\frac1{k^2}\right)$. Such behavior can be explained by linear convergence of AGMsDR method and was tested on quadratic functions. Alternating AGMsDR demonstrated better performance in comparison with AGMsDR.

The problem of developing efficient numerical methods for non-convex (including non-smooth) problems is relevant due to their widespread use of such problems in applications. This paper is devoted to subgradient methods for minimizing Lipschitz $\mu$-weakly convex functions, which are not necessarily smooth. It is well known that subgradient methods have low convergence rates in high-dimensional spaces even for convex functions. However, if we consider a subclass of functions that satisfies sharp minimum condition and also use the Polyak step, we can guarantee a linear convergence rate of the subgradient method. In some cases, the values of the function or it’s subgradient may be available to the numerical method with some error. The accuracy of the solution provided by the numerical method depends on the magnitude of this error. In this paper, we investigate the behavior of the subgradient method with a Polyak step when inaccurate information about the objective function value or subgradient is used in iterations. We prove that with a specific choice of starting point, the subgradient method with some analogue of the Polyak step-size converges at a geometric progression rate on a class of $\mu$-weakly convex functions with a sharp minimum, provided that there is additive inaccuracy in the subgradient values. In the case when both the value of the function and the value of its subgradient at the current point are known with error, convergence to some neighborhood of the set of exact solutions is shown and the quality estimates of the output solution by the subgradient method with the corresponding analogue of the Polyak step are obtained. The article also proposes a subgradient method with a clipped step, and an assessment of the quality of the solution obtained by this method for the class of $\mu$-weakly convex functions with a sharp minimum is presented. Numerical experiments were conducted for the problem of low-rank matrix recovery. They showed that the efficiency of the studied algorithms may not depend on the accuracy of localization of the initial approximation within the required region, and the inaccuracy in the values of the function and subgradient may affect the number of iterations required to achieve an acceptable quality of the solution, but has almost no effect on the quality of the solution itself.

Studies of the crisis socio-demographic situation in the Russian Far East require not only the use of traditional statistical methods, but also a conceptual analysis of possible development scenarios based on the synergy principles. The article is devoted to the analysis and modeling of the number of employed, unemployed and economically inactive population using nonlinear autonomous differential equations. We studied a basic mathematical model that takes into account the principle of pair interactions, which is a special case of the model for the struggle between conditional information of D. S. Chernavsky. The point estimates for the parameters are found using least squares method adapted for this model. The average approximation error was no more than 5.17%. The calculated parameter values correspond to the unstable focus and the oscillations with increasing amplitude of population number in the asymptotic case, which indicates a gradual increase in disparities between the employed, unemployed and economically inactive population and a collapse of their dynamics. We found that in the parametric space, not far from the inertial scenario, there are domains of blow-up and chaotic regimes complicating the ability to effectively manage. The numerical study showed that a change in only one model parameter (e.g. migration) without complex structural socio-economic changes can only delay the collapse of the dynamics in the long term or leads to the emergence of unpredictable chaotic regimes. We found an additional set of the model parameters corresponding to sustainable dynamics (stable focus) which approximates well the time series of the considered population groups. In the mathematical model, the bifurcation parameters are the outflow rate of the able-bodied population, the fertility (“rejuvenation of the population”), as well as the migration inflow rate of the unemployed. We found that the transition to stable regimes is possible with the simultaneous impact on several parameters which requires a comprehensive set of measures to consolidate the population in the Russian Far East and increase the level of income in terms of compensation for infrastructure sparseness. Further economic and sociological research is required to develop specific state policy measures.

High frequency algorithmic trading became is a subclass of trading which is focused on gaining basis-point like profitability on sub-second time frames. Such trading strategies do not depend on most of the factors eligible for the longer-term trading and require specific approach. There were many attempts to utilize machine learning techniques to both high and low frequency trading. However, it is still having limited application in the real world trading due to high exposure to overfitting, requirements for rapid adaptation to new market regimes and overall instability of the results. We conducted a comprehensive research on combination of known quantitative theory and reinforcement learning methods in order derive more effective and robust approach at construction of automated trading system in an attempt to create a support for a known algorithmic trading techniques. Using classical price behavior theories as well as modern application cases in sub-millisecond trading, we utilized the Reinforcement Learning models in order to improve quality of the algorithms. As a result, we derived a robust model which utilize Deep Reinforcement learning in order to optimise static market making trading algorithms’ parameters capable of online learning on live data. More specifically, we explored the system in the derivatives cryptocurrency market which mostly not dependent on external factors in short terms. Our research was implemented in high-frequency environment and the final models showed capability to operate within accepted high-frequency trading time-frames. We compared various combinations of Deep Reinforcement Learning approaches and the classic algorithms and evaluated robustness and effectiveness of improvements for each combination.