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A numerically stable direct multiplicative method for solving systems of linear equations that takes into account the sparseness of matrices presented in a packed form is considered. The advantage of the method is the calculation of the Cholesky factors for a positive definite matrix of the system of equations and its solution within the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made to the position of the next processed row of the matrix, which allows using static data storage formats. The solution of the system of linear equations by a direct multiplicative algorithm is, like the solution with LU-decomposition, just another scheme for implementing the Gaussian elimination method.

The calculation of the Cholesky factors for a positive definite matrix of the system and its solution underlies the construction of a new mathematical formulation of the unconditional problem of quadratic programming and a new form of specifying necessary and sufficient conditions for optimality that are quite simple and are used in this paper to construct a new mathematical formulation for the problem of quadratic programming on a polyhedral set of constraints, which is the problem of finding the minimum distance between the origin ordinate and polyhedral boundary by means of a set of constraints and linear algebra dimensional geometry.

To determine the distance, it is proposed to apply the known exact method based on solving systems of linear equations whose dimension is not higher than the number of variables of the objective function. The distances are determined by the construction of perpendiculars to the faces of a polyhedron of different dimensions. To reduce the number of faces examined, the proposed method involves a special order of sorting the faces. Only the faces containing the vertex closest to the point of the unconditional extremum and visible from this point are subject to investigation. In the case of the presence of several nearest equidistant vertices, we investigate a face containing all these vertices and faces of smaller dimension that have at least two common nearest vertices with the first face.

The work is dedicated to investigation of possibility to increase the efficiency of solving external aerodynamic problems. Methodical questions of using locally-adaptive grids and wall functions for numerical simulation of turbulent flows past flying vehicles are studied. Reynolds-averaged Navier–Stokes equations are integrated. The equations are closed by standard $k–\varepsilon$ turbulence model. Subsonic turbulent flow of perfect compressible viscous gas past airfoil RAE 2822 is considered. Calculations are performed in CFD software FlowVision. The efficiency of using the technology of smoothing diffusion fluxes and the Bradshaw formula for turbulent viscosity is analyzed. These techniques are regarded as means of increasing the accuracy of solving aerodynamic problems on locally-adaptive grids. The obtained results show that using the technology of smoothing diffusion fluxes essentially decreases the discrepancy between computed and experimental values of the drag coefficient. In addition, the distribution of the skin friction coefficient over the curvilinear surface of the airfoil becomes more regular. These results indicate that the given technology is an effective way to increase the accuracy of calculations on locally-adaptive grids. The Bradshaw formula for the dynamic coefficient of turbulent viscosity is traditionally used in the SST $k–\omega$ turbulence model. The possibility to implement it in the standard $k–\varepsilon$ turbulence model is investigated in the present article. The calculations show that this formula provides good agreement of integral aerodynamic characteristics and the distribution of the pressure coefficient over the airfoil surface with experimental data. Besides that, it essentially augments the accuracy of simulation of the flow in the boundary layer and in the wake. On the other hand, using the Bradshaw formula in the simulation of the air flow past airfoil RAE 2822 leads to under-prediction of the skin friction coefficient. For this reason, the conclusion is made that practical use of the Bradshaw formula requires its preliminary validation and calibration on reliable experimental data available for the considered flows. The results of the work as a whole show that using the technologies discussed in numerical solution of external aerodynamic problems on locally-adaptive grids together with wall functions provides the computational accuracy acceptable for quick assessment of the aerodynamic characteristics of a flying vehicle. So, one can deduce that the FlowVision software is an effective tool for preliminary design studies, for conceptual design, and for aerodynamic shape optimization.

The simplest and most desirable method of traffic signal control is precalculated regulation, when the parameters of the traffic light object operation are calculated in advance and activated in accordance to a schedule. This work proposes a method of forming a signal plan that allows one to calculate the control programs and set the period of their activity. Preparation of initial data for the calculation includes the formation of a time series of daily traffic intensity with an interval of 15 minutes. When carrying out field studies, it is possible that part of the traffic intensity measurements is missing. To fill up the missing traffic intensity measurements, the spline interpolation method is used. The next step of the method is to calculate the daily set of signal plans. The work presents the interdependencies, which allow one to calculate the optimal durations of the control cycle and the permitting phase movement and to set the period of their activity. The present movement control systems have a limit on the number of control programs. To reduce the signal plans' number and to determine their activity period, the clusterization using the $k$-means method in the transport phase space is introduced In the new daily signal plan, the duration of the phases is determined by the coordinates of the received cluster centers, and the activity periods are set by the elements included in the cluster. Testing on a numerical illustration showed that, when the number of clusters is 10, the deviation of the optimal phase duration from the cluster centers does not exceed 2 seconds. To evaluate the effectiveness of the developed methodology, a real intersection with traffic light regulation was considered as an example. Based on field studies of traffic patterns and traffic demand, a microscopic model for the SUMO (Simulation of Urban Mobility) program was developed. The efficiency assessment is based on the transport losses estimated by the time spent on movement. Simulation modeling of the multiprogram control of traffic lights showed a 20% reduction in the delay time at the traffic light object in comparison with the single-program control. The proposed method allows automation of the process of calculating daily signal plans and setting the time of their activity.

Recently, saddle point problems have received much attention due to their powerful modeling capability for a lot of problems from diverse domains. Applications of these problems occur in many applied areas, such as robust optimization, distributed optimization, game theory, and many applications in machine learning such as empirical risk minimization and generative adversarial networks training. Therefore, many researchers have actively worked on developing numerical methods for solving saddle point problems in many different settings. This paper is devoted to developing a numerical method for solving saddle point problems in the nonconvex uniformly-concave setting. We study a general class of saddle point problems with composite structure and H\"older-continuous higher-order derivatives. To solve the problem under consideration, we propose an approach in which we reduce the problem to a combination of two auxiliary optimization problems separately for each group of variables, the outer minimization problem w.r.t. primal variables, and the inner maximization problem w.r.t the dual variables. For solving the outer minimization problem, we use the Adaptive Gradient Method, which is applicable for nonconvex problems and also works with an inexact oracle that is generated by approximately solving the inner problem. For solving the inner maximization problem, we use the Restarted Unified Acceleration Framework, which is a framework that unifies the high-order acceleration methods for minimizing a convex function that has H\"older-continuous higher-order derivatives. Separate complexity bounds are provided for the number of calls to the first-order oracles for the outer minimization problem and higher-order oracles for the inner maximization problem. Moreover, the complexity of the whole proposed approach is then estimated.

In this study we discuss methods to solve optimal control problems based on neural network techniques. We study hierarchical dynamical two-level system for surface water quality control. The system consists of a supervisor (government) and a few agents (enterprises). We consider this problem from the point of agents. In this case we solve optimal control problem with constraints. To solve this problem, we use Pontryagin’s maximum principle, with which we obtain optimality conditions. To solve emerging ODEs, we use feedforward neural network. We provide a review of existing techniques to study such problems and a review of neural network’s training methods. To estimate the error of numerical solution, we propose to use defect analysis method, adapted for neural networks. This allows one to get quantitative error estimations of numerical solution. We provide examples of our method’s usage for solving synthetic problem and a surface water quality control model. We compare the results of this examples with known solution (when provided) and the results of shooting method. In all cases the errors, estimated by our method are of the same order as the errors compared with known solution. Moreover, we study surface water quality control problem when no solutions is provided by other methods. This happens because of relatively large time interval and/or the case of several agents. In the latter case we seek Nash equilibrium between agents. Thus, in this study we show the ability of neural networks to solve various problems including optimal control problems and differential games and we show the ability of quantitative estimation of an error. From the numerical results we conclude that the presence of the supervisor is necessary for achieving the sustainable development.

In this paper we describe an algorithm of numerical solving of an inverse problem on a hyperbolic heat equation with additional second time derivative with a small parameter. The problem in this case is finding an initial distribution with given final distribution. This algorithm allows finding a solution to the problem for any admissible given precision. Algorithm allows evading difficulties analogous to the case of heat equation with inverted time. Furthermore, it allows finding an optimal grid size by learning on a relatively big grid size and small amount of iterations of a gradient method and later extrapolates to the required grid size using Richardson’s method. This algorithm allows finding an adequate estimate of Lipschitz constant for the gradient of the target functional. Finally, this algorithm may easily be applied to the problems with similar structure, for example in solving equations for plasma, social processes and various biological problems. The theoretical novelty of the paper consists in the developing of an optimal procedure of finding of the required grid size using Richardson extrapolations for optimization problems with inexact gradient in ill-posed problems.

We consider a finite-dimensional optimization problem, the formulation of which in addition to the required variables contains parameters. The solution to this problem is a dependence of optimal values of variables on parameters. In general, these dependencies are not functions because they can have ambiguous meanings and in the functional case be nondifferentiable. In addition, their domain of definition may be narrower than the domains of definition of functions in the condition of the original problem. All these properties make it difficult to solve both the original parametric problem and other tasks, the statement of which includes these dependencies. To overcome these difficulties, usually methods such as non-differentiable optimization are used.

This article proposes an alternative approach that makes it possible to obtain solutions to parametric problems in a form devoid of the specified properties. It is shown that such representations can be explored using standard algorithms, based on the Taylor formula. This form is a function smoothly approximating the solution of the original problem for any parameter values, specified in its statement. In this case, the value of the approximation error is controlled by a special parameter. Construction of proposed approximations is performed using special functions that establish feedback (within optimality conditions for the original problem) between variables and Lagrange multipliers. This method is described for linear problems with subsequent generalization to the nonlinear case.

From a computational point of view the construction of the approximation consists in finding the saddle point of the modified Lagrange function of the original problem. Moreover, this modification is performed in a special way using feedback functions. It is shown that the necessary conditions for the existence of such a saddle point are similar to the conditions of the Karush – Kuhn – Tucker theorem, but do not contain constraints such as inequalities and conditions of complementary slackness. Necessary conditions for the existence of a saddle point determine this approximation implicitly. Therefore, to calculate its differential characteristics, the implicit function theorem is used. The same theorem is used to reduce the approximation error to an acceptable level.

Features of the practical implementation feedback function method, including estimates of the rate of convergence to the exact solution are demonstrated for several specific classes of parametric optimization problems. Specifically, tasks searching for the global extremum of functions of many variables and the problem of multiple extremum (maximin-minimax) are considered. Optimization problems that arise when using multicriteria mathematical models are also considered. For each of these classes, there are demo examples.

The article presents an approach to configuration of an artificial neural network architecture and a training set size. Configuration is based on parameter minimization with constraints specifying neural network model quality criteria. The algorithm of artificial neural network architecture and training set size configuration is applied to dynamic object artificial neural network approximation.
Series of computational experiments were performed. The method is applicable to construction of dynamic object models based on non-linear autocorrelation neural networks.

An approach to the decrease of norm of the correction in Newton’s methods of optimization, based on the Cholesky’s factorization is presented, which is based on the integration with the technique of the choice of leading element of algorithm of linear programming as a method of solving the system of equations. We investigate the issues of increasing of the numerical stability of the Cholesky’s decomposition and the Gauss’ method of exception.

Efficiency of production directly depends on quality of the management of technology which, in turn, relies on the accuracy and efficiency of the processing of control and measuring information. Development of the mathematical methods of research of the system communications and regularities of functioning and creation of the mathematical models taking into account structural features of object of researches, and also writing of the software products for realization of these methods are an actual task. Practice has shown that the list of parameters that take place in the study of complex object of modern production, ranging from a few dozen to several hundred names, and the degree of influence of each factor in the initial time is not clear. Before working for the direct determination of the model in these circumstances, it is impossible — the amount of the required information may be too great, and most of the work on the collection of this information will be done in vain due to the fact that the degree of influence on the optimization of most factors of the original list would be negligible. Therefore, a necessary step in determining a model of a complex object is to work to reduce the dimension of the factor space. Most industrial plants are hierarchical group processes and mass volume production, characterized by hundreds of factors. (For an example of realization of the mathematical methods and the approbation of the constructed models data of the Moldavian steel works were taken in a basis.) To investigate the systemic linkages and patterns of functioning of such complex objects are usually chosen several informative parameters, and carried out their sampling. In this article the sequence of coercion of the initial indices of the technological process of the smelting of steel to the look suitable for creation of a mathematical model for the purpose of prediction is described. The implementations of new types became also creation of a basis for development of the system of automated management of quality of the production. In the course of weak correlation the following stages are selected: collection and the analysis of the basic data, creation of the table the correlated of the parameters, abbreviation of factor space by means of the correlative pleiads and a method of weight factors. The received results allow to optimize process of creation of the model of multiple-factor process.