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This paper presents the results of experimental validation of some structural issues concerning the practical use of methods to overcome catastrophic forgetting of neural networks. A comparison of current effective methods like EWC (Elastic Weight Consolidation) and WVA (Weight Velocity Attenuation) is made and their advantages and disadvantages are considered. It is shown that EWC is better for tasks where full retention of learned skills is required on all the tasks in the training queue, while WVA is more suitable for sequential tasks with very limited computational resources, or when reuse of representations and acceleration of learning from task to task is required rather than exact retention of the skills. The attenuation of the WVA method must be applied to the optimization step, i. e. to the increments of neural network weights, rather than to the loss function gradient itself, and this is true for any gradient optimization method except the simplest stochastic gradient descent (SGD). The choice of the optimal weights attenuation function between the hyperbolic function and the exponent is considered. It is shown that hyperbolic attenuation is preferable because, despite comparable quality at optimal values of the hyperparameter of the WVA method, it is more robust to hyperparameter deviations from the optimal value (this hyperparameter in the WVA method provides a balance between preservation of old skills and learning a new skill). Empirical observations are presented that support the hypothesis that the optimal value of this hyperparameter does not depend on the number of tasks in the sequential learning queue. And, consequently, this hyperparameter can be picked up on a small number of tasks and used on longer sequences.

The work is devoted to the construction of efficient and applicable to real tasks first-order methods of convex optimization, that is, using only values of the target function and its derivatives. Construction uses OGMG, fast gradient method which is optimal by complexity, but requires to know the Lipschitz constant for gradient and the strong convexity constant to determine the number of steps and step length. This requirement makes practical usage very hard. An adaptive on the constant for strong convexity algorithm ACGM is proposed, based on restarts of the OGM-G with update of the strong convexity constant estimate, and an adaptive on the Lipschitz constant for gradient ALGM, in which the use of OGM-G restarts is supplemented by the selection of the Lipschitz constant with verification of the smoothness conditions used in the universal gradient descent method. This eliminates the disadvantages of the original method associated with the need to know these constants, which makes practical usage possible. Optimality of estimates for the complexity of the constructed algorithms is proved. To verify the results obtained, experiments on model functions and real tasks from machine learning are carried out.

Ligand protected metal nanoclusters (NCs) have gained much attention due to their unique physicochemical properties and potential applications in material science. Noble metal NCs protected with thiolate ligands have been of interest because of their long-term stability. The detailed structures of most of the ligandstabilized metal NCs remain unknown due to the absence of crystal structure data for them. Theoretical calculations using quantum chemistry techniques appear as one of the most promising tools for determining the structure and electronic properties of NCs. That is why finding a cost-effective strategy for calculations is such an important and challenging task. In this work, we compare the performance of different theoretical methods of geometry optimization and absorption spectra calculation for silver-thiolate complexes. We show that second order Moller–Plesset perturbation theory reproduces nicely the geometries obtained at a higher level of theory, in particular, with RI-CC2 method. We compare the absorption spectra of silver-thiolate complexes simulated with different methods: EOM-CCSD, RI-CC2, ADC(2) and TDDFT. We show that the absorption spectra calculated with the ADC(2) method are consistent with the spectra obtained with the EOM-CCSD and RI-CC2 methods. CAM-B3LYP functional fails to reproduce the absorption spectra of the silver-thiolate complexes. However, M062X global hybrid meta-GGA functional seems to be a nice compromise regarding its low computational costs. In our previous study, we have already demonstrated that M062X functional shows good accuracy as compared to ADC(2) ab initio method predicting the excitation spectra of silver nanocluster complexes with nucleobases.

This paper considers various approaches to averaging the generalized travel costs calculated for different modes of travel in the transportation network. The mode of transportation is understood to mean both the mode of transport, for example, a car or public transport, and movement without the use of transport, for example, on foot. The task of calculating the trip matrices includes the task of calculating the total matrices, in other words, estimating the total demand for movements by all modes, as well as the task of splitting the matrices according to the mode, also called modal splitting. To calculate trip matrices, gravitational, entropy and other models are used, in which the probability of movement between zones is estimated based on a certain measure of the distance of these zones from each other. Usually, the generalized cost of moving along the optimal path between zones is used as a distance measure. However, the generalized cost of movement differs for different modes of movement. When calculating the total trip matrices, it becomes necessary to average the generalized costs by modes of movement. The averaging procedure is subject to the natural requirement of monotonicity in all arguments. This requirement is not met by some commonly used averaging methods, for example, averaging with weights. The problem of modal splitting is solved by applying the methods of discrete choice theory. In particular, within the framework of the theory of discrete choice, correct methods have been developed for averaging the utility of alternatives that are monotonic in all arguments. The authors propose some adaptation of the methods of the theory of discrete choice for application to the calculation of the average cost of movements in the gravitational and entropy models. The transfer of averaging formulas from the context of the modal splitting model to the trip matrix calculation model requires the introduction of new parameters and the derivation of conditions for the possible value of these parameters, which was done in this article. The issues of recalibration of the gravitational function, which is necessary when switching to a new averaging method, if the existing function is calibrated taking into account the use of the weighted average cost, were also considered. The proposed methods were implemented on the example of a small fragment of the transport network. The results of calculations are presented, demonstrating the advantage of the proposed methods.

The work is devoted to the description of the author’s formal statements of the clustering problem for a given number of clusters, algorithms for their solution, as well as the results of using this toolkit in medicine.

The solution of the formulated problems by exact algorithms of implementations of even relatively low dimensions before proving optimality is impossible in a finite time due to their belonging to the NP class.

In this regard, we have proposed a hybrid algorithm that combines the advantages of precise methods based on clustering in paired distances at the initial stage with the speed of methods for solving simplified problems of splitting by cluster centers at the final stage. In the development of this direction, a sequential hybrid clustering algorithm using random search in the paradigm of swarm intelligence has been developed. The article describes it and presents the results of calculations of applied clustering problems.

To determine the effectiveness of the developed tools for optimal clustering of multidimensional objects according to a variety of heterogeneous indicators, a number of computational experiments were performed using data sets including socio-demographic, clinical anamnestic, electroencephalographic and psychometric data on the cognitive status of patients of the cardiology clinic. An experimental proof of the effectiveness of using local search algorithms in the paradigm of swarm intelligence within the framework of a hybrid algorithm for solving optimal clustering problems has been obtained.

The results of the calculations indicate the actual resolution of the main problem of using the discrete optimization apparatus — limiting the available dimensions of task implementations. We have shown that this problem is eliminated while maintaining an acceptable proximity of the clustering results to the optimal ones. The applied significance of the obtained clustering results is also due to the fact that the developed optimal clustering toolkit is supplemented by an assessment of the stability of the formed clusters, which allows for known factors (the presence of stenosis or older age) to additionally identify those patients whose cognitive resources are insufficient to overcome the influence of surgical anesthesia, as a result of which there is a unidirectional effect of postoperative deterioration of complex visual-motor reaction, attention and memory. This effect indicates the possibility of differentiating the classification of patients using the proposed tools.

The paper presents a mathematical model that describes the process of obtaining biogas from livestock waste. This model describes the processes occurring in a biogas plant for mesophilic and thermophilic media, as well as for continuous and periodic modes of substrate inflow. The values of the coefficients of this model found earlier for the periodic mode, obtained by solving the problem of model identification from experimental data using a genetic algorithm, are given.

For the model of methanogenesis, an optimal control problem is formulated in the form of a Lagrange problem, whose criterial functionality is the output of biogas over a certain period of time. The controlling parameter of the task is the rate of substrate entry into the biogas plant. An algorithm for solving this problem is proposed, based on the numerical implementation of the Pontryagin maximum principle. In this case, a hybrid genetic algorithm with an additional search in the vicinity of the best solution using the method of conjugate gradients was used as an optimization method. This numerical method for solving an optimal control problem is universal and applicable to a wide class of mathematical models.

In the course of the study, various modes of submission of the substrate to the digesters, temperature environments and types of raw materials were analyzed. It is shown that the rate of biogas production in the continuous feed mode is 1.4–1.9 times higher in the mesophilic medium (1.9–3.2 in the thermophilic medium) than in the periodic mode over the period of complete fermentation, which is associated with a higher feed rate of the substrate and a greater concentration of nutrients in the substrate. However, the yield of biogas during the period of complete fermentation with a periodic mode is twice as high as the output over the period of a complete change of the substrate in the methane tank at a continuous mode, which means incomplete processing of the substrate in the second case. The rate of biogas formation for a thermophilic medium in continuous mode and the optimal rate of supply of raw materials is three times higher than for a mesophilic medium. Comparison of biogas output for various types of raw materials shows that the highest biogas output is observed for waste poultry farms, the least — for cattle farms waste, which is associated with the nutrient content in a unit of substrate of each type.

The paper proposes a mathematical model for the therapy of glioma, taking into account the blood-brain barrier, radiotherapy and antibody therapy. The parameters were estimated from experimental data and the evaluation of the effect of parameter values on the effectiveness of treatment and the prognosis of the disease were obtained. The possible variants of sequential use of radiotherapy and the effect of antibodies have been explored. The combined use of radiotherapy with intravenous administration of $mab$ $Cx43$ leads to a potentiation of the therapeutic effect in glioma.

Radiotherapy must precede chemotherapy, as radio exposure reduces the barrier function of endothelial cells. Endothelial cells of the brain vessels fit tightly to each other. Between their walls are formed so-called tight contacts, whose role in the provision of BBB is that they prevent the penetration into the brain tissue of various undesirable substances from the bloodstream. Dense contacts between endothelial cells block the intercellular passive transport.

The mathematical model consists of a continuous part and a discrete one. Experimental data on the volume of glioma show the following interesting dynamics: after cessation of radio exposure, tumor growth does not resume immediately, but there is some time interval during which glioma does not grow. Glioma cells are divided into two groups. The first group is living cells that divide as fast as possible. The second group is cells affected by radiation. As a measure of the health of the blood-brain barrier system, the ratios of the number of BBB cells at the current moment to the number of cells at rest, that is, on average healthy state, are chosen.

The continuous part of the model includes a description of the division of both types of glioma cells, the recovery of BBB cells, and the dynamics of the drug. Reducing the number of well-functioning BBB cells facilitates the penetration of the drug to brain cells, that is, enhances the action of the drug. At the same time, the rate of division of glioma cells does not increase, since it is limited not by the deficiency of nutrients available to cells, but by the internal mechanisms of the cell. The discrete part of the mathematical model includes the operator of radio interaction, which is applied to the indicator of BBB and to glial cells.

Within the framework of the mathematical model of treatment of a cancer tumor (glioma), the problem of optimal control with phase constraints is solved. The patient’s condition is described by two variables: the volume of the tumor and the condition of the BBB. The phase constraints delineate a certain area in the space of these indicators, which we call the survival area. Our task is to find such treatment strategies that minimize the time of treatment, maximize the patient’s rest time, and at the same time allow state indicators not to exceed the permitted limits. Since the task of survival is to maximize the patient’s lifespan, it is precisely such treatment strategies that return the indicators to their original position (and we see periodic trajectories on the graphs). Periodic trajectories indicate that the deadly disease is translated into a chronic one.

Since 1950s the field of city transport modelling has progressed rapidly. The first equilibrium distribution models of traffic flow appeared. The most popular model (which is still being widely used) was the Beckmann model, based on the two Wardrop principles. The core of the model could be briefly described as the search for the Nash equilibrium in a population demand game, in which losses of agents (drivers) are calculated based on the chosen path and demands of this path with correspondences being fixed. The demands (costs) of a path are calculated as the sum of the demands of different path segments (graph edges), that are included in the path. The costs of an edge (edge travel time) are determined by the amount of traffic on this edge (more traffic means larger travel time). The flow on a graph edge is determined by the sum of flows over all paths passing through the given edge. Thus, the cost of traveling along a path is determined not only by the choice of the path, but also by the paths other drivers have chosen. Thus, it is a standard game theory task. The way cost functions are constructed allows us to narrow the search for equilibrium to solving an optimization problem (game is potential in this case). If the cost functions are monotone and non-decreasing, the optimization problem is convex. Actually, different assumptions about the cost functions form different models. The most popular model is based on the BPR cost function. Such functions are massively used in calculations of real cities. However, in the beginning of the XXI century, Yu. E. Nesterov and A. de Palma showed that Beckmann-type models have serious weak points. Those could be fixed using the stable dynamics model, as it was called by the authors. The search for equilibrium here could be also reduced to an optimization problem, moreover, the problem of linear programming. In 2013, A.V.Gasnikov discovered that the stable dynamics model can be obtained by a passage to the limit in the Beckmann model. However, it was made only for several practically important, but still special cases. Generally, the question if this passage to the limit is possible remains open. In this paper, we provide the justification of the possibility of the above-mentioned passage to the limit in the general case, when the cost function for traveling along the edge as a function of the flow along the edge degenerates into a function equal to fixed costs until the capacity is reached and it is equal to plus infinity when the capacity is exceeded.

A previously developed model that describes the dynamics of social tension in a society divided into two groups: the elite and the people was considered. This model took into account the impact of economic situation changes and the elite–people interaction. The model has been modified by including in the equation describing the tension of the people, a term that takes into account the adaptation of the people to the current situation.

The model coefficients estimation is an important task, the solution of which allows obtaining information about the nature of the interaction between elite and people. We believe that the solution of the system of model equations with optimal coefficients is closest to the values of the indicator characterizing social tension. We used the normalized level of homicide rate as an indicator of social tension.

The model contains seven coefficients. Two coefficients characterizing the influence of economic situation changes on elite and people are taken equal to each other and the same for all countries. We obtained their estimations using a simplified model that takes into account only the change in the economic situation and allows an analytical solution.

The Bayesian approach was used to estimate the remaining five coefficients of model for post-Soviet countries. The prior probability densities of the four coefficients for all countries under consideration were taken to be the same. The prior probability density of fifth coefficient was considered to depend on the regime of government (authoritarian or «transitional»). We assumed that the calculated tension matches with the corresponding indicator of tension in cases where the difference between them does not exceed 5%.

The calculations showed that for the post-Soviet countries, a good coincidence was obtained between the calculated values of the people tension and the normalized level of homicide rate. The coincidence is satisfactory only on average.

The following main results was obtained at the work: under the influence of some «significant» events in 40% of post-Soviet countries, there was a rapid change in the nature of interaction between the elite and the people; regional feature have some influence on the elite–people interaction; the type of government does not significantly affect the elite–people interaction; the method for assessing the stability of the country by the value of the model coefficients is proposed.

The most important principle of military science and border security is the principle of concentrating the main efforts on the main directions and tasks. At the tactical level, there are many mathematical models for computing the optimal resource allocation by directions and objects, whereas at the state level there are no corresponding models. Using the statistical data on the results of the protection of the US border, an exponential type border production function parameter is calculated that reflects the organizational and technological capabilities of the border guard. The production function determines the dependence of the probability of detaining offenders from the density of border guards per kilometer of the border. Financial indicators in the production function are not taken into account, as the border maintenance budget and border equipment correlate with the number of border agents. The objective function of the border guards is defined — the total prevented damage from detained violators taking into account their expected danger for the state and society, which is to be maximized. Using Slater's condition, the solution of the problem was found — optimal density of border guard was calculated for the regions of the state. Having a model of resource allocation, the example of the three border regions of the United States has also solved the reverse problem — threats in the regions have been assessed based on the known allocation of resources. The expected danger from an individual offender on the US-Canada border is 2–5 times higher than from an offender on the US-Mexican border. The results of the calculations are consistent with the views of US security experts: illegal migrants are mostly detained on the US-Mexican border, while potential terrorists prefer to use other channels of penetration into the US (including the US-Canadian border), where the risks of being detained are minimal. Also, the results of the calculations are consistent with the established practice of border protection: in 2013 the number of border guards outside the checkpoints on the US-Mexican border increased by 2 times compared with 2001, while on the American-Canadian border — 4 times. The practice of border protection and the views of specialists give grounds for approval of the verification of the model.