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Changes in abundance for emerging populations can develop according to several dynamic scenarios. After rapid biological invasions, the time factor for the development of a reaction from the biotic environment will become important. There are two classic experiments known in history with different endings of the confrontation of biological species. In Gause’s experiments with ciliates, the infused predator, after brief oscillations, completely destroyed its resource, so its $r$-parameter became excessive for new conditions. Its own reproductive activity was not regulated by additional factors and, as a result, became critical for the invader. In the experiments of the entomologist Uchida with parasitic wasps and their prey beetles, all species coexisted. In a situation where a population with a high reproductive potential is regulated by several natural enemies, interesting dynamic effects can occur that have been observed in phytophages in an evergreen forest in Australia. The competing parasitic hymenoptera create a delayed regulation system for rapidly multiplying psyllid pests, where a rapid increase in the psyllid population is allowed until the pest reaches its maximum number. A short maximum is followed by a rapid decline in numbers, but minimization does not become critical for the population. The paper proposes a phenomenological model based on a differential equation with a delay, which describes a scenario of adaptive regulation for a population with a high reproductive potential with an active, but with a delayed reaction with a threshold regulation of exposure. It is shown that the complication of the regulation function of biotic resistance in the model leads to the stabilization of the dynamics after the passage of the minimum number by the rapidly breeding species. For a flexible system, transitional regimes of growth and crisis lead to the search for a new equilibrium in the evolutionary confrontation.

This article is dedicated to a comparison analysis of optimization methods, in order to perform an interval estimation of electrical energy technical losses in distribution networks of voltage 6–20 kV. The issue of interval evaluation is represented as a multi-dimensional conditional minimization/maximization problem with implicit target function. A number of numerical optimization methods of first and zero orders is observed, with the aim of determining the most suitable for the problem of interest. The desired algorithm is BOBYQA, in which the target function is replaced with its quadratic approximation in some trusted region.

Technical diagnosis’ current trends are connected to application of FEM computer simulation, which allows, to some extent, replace real experiments, reduce costs for investigation and minimize risks. Computer simulation, just at the stage of research and development, allows carrying out of diagnostics of equipment to detect permissible fluctuations of parameters of equipment’s work. Peculiarity of diagnosis of rolling equipment is that functioning of rolling equipment is directly tied with manufacturing of product with required quality, including accuracy. At that design of techniques of technical diagnosis and diagnostical modelling is very important. Computer simulation of cold rolling of strip was carried out. At that upper working roll was doing vibrations in horizontal direction according with published data of experiments on continuous 1700 rolling mill. Vibration of working roll in a stand appeared due to gap between roll’s craft and guide in a stand and led to periodical fluctuations of strip’s thickness. After computer simulation with the help of DEFORM software strip with longitudinal and transversal thickness variation was gotten. Visualization of strip’s geometrical parameters, according with simulation data, corresponded to type of inhomogeneity of surface of strip rolled in real. Further analysis of thickness variation was done in order to identify, on the basis of simulation, sources of periodical components of strip’s thickness, whose reasons are malfunctions of equipment. Advantage of computer simulation while searching the sources of forming of thickness variation is that different hypothesis concerning thickness formations may be tested without conducting real experiments and costs of different types may be reduced. Moreover, while simulation, initial strip’s thickness will not have fluctuations as opposed to industrial or laboratorial experiments. On the basis of spectral analysis of random process, it was established that frequency of changing of strip’s thickness after rolling in one stand coincides with frequency of working roll’s vibration. Results of computer simulation correlate with results of the researches for 1700 mill. Therefore, opportunity to apply computer simulation to find reasons of formation of thickness variation of strip on the industrial rolling mill is shown.

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.

This work analyzes the problem of traffic signs recognition in intelligent transport systems. The basic concepts of computer vision and image recognition tasks are considered. The most effective approach for solving the problem of analyzing and recognizing images now is the neural network method. Among all kinds of neural networks, the convolutional neural network has proven itself best. Activation functions such as Relu and SoftMax are used to solve the classification problem when recognizing traffic signs. This article proposes a technology for recognizing traffic signs. The choice of an approach for solving the problem based on a convolutional neural network due to the ability to effectively solve the problem of identifying essential features and classification. The initial data for the neural network model were prepared and a training sample was formed. The Google Colaboratory cloud service with the external libraries for deep learning TensorFlow and Keras was used as a platform for the intelligent system development. The convolutional part of the network is designed to highlight characteristic features in the image. The first layer includes 512 neurons with the Relu activation function. Then there is the Dropout layer, which is used to reduce the effect of overfitting the network. The output fully connected layer includes four neurons, which corresponds to the problem of recognizing four types of traffic signs. An intelligent traffic sign recognition system has been developed and tested. The used convolutional neural network included four stages of convolution and subsampling. Evaluation of the efficiency of the traffic sign recognition system using the three-block cross-validation method showed that the error of the neural network model is minimal, therefore, in most cases, new images will be recognized correctly. In addition, the model has no errors of the first kind, and the error of the second kind has a low value and only when the input image is very noisy.

In this paper we propose high-order (tensor) methods for two types of saddle point problems. Firstly, we consider the classic min-max saddle point problem. Secondly, we consider the search for a stationary point of the saddle point problem objective by its gradient norm minimization. Obviously, the stationary point does not always coincide with the optimal point. However, if we have a linear optimization problem with linear constraints, the algorithm for gradient norm minimization becomes useful. In this case we can reconstruct the solution of the optimization problem of a primal function from the solution of gradient norm minimization of dual function. In this paper we consider both types of problems with no constraints. Additionally, we assume that the objective function is $\mu$-strongly convex by the first argument, $\mu$-strongly concave by the second argument, and that the $p$-th derivative of the objective is Lipschitz-continous.

For min-max problems we propose two algorithms. Since we consider strongly convex a strongly concave problem, the first algorithm uses the existing tensor method for regular convex concave saddle point problems and accelerates it with the restarts technique. The complexity of such an algorithm is linear. If we additionally assume that our objective is first and second order Lipschitz, we can improve its performance even more. To do this, we can switch to another existing algorithm in its area of quadratic convergence. Thus, we get the second algorithm, which has a global linear convergence rate and a local quadratic convergence rate.

Finally, in convex optimization there exists a special methodology to solve gradient norm minimization problems by tensor methods. Its main idea is to use existing (near-)optimal algorithms inside a special framework. I want to emphasize that inside this framework we do not necessarily need the assumptions of strong convexity, because we can regularize the convex objective in a special way to make it strongly convex. In our article we transfer this framework on convex-concave objective functions and use it with our aforementioned algorithm with a global linear convergence and a local quadratic convergence rate.

Since the saddle point problem is a particular case of the monotone variation inequality problem, the proposed methods will also work in solving strongly monotone variational inequality problems.

The principle of minimal differentiation, based on the Hotelling model, is well known in the economy. It is applicable to horizontal differentiated goods of almost any nature. The Hotelling approach to modeling competition of oligopolies corresponds to a modern description of monopolistic competition with increasing returns to scale and imperfect competition. We develop a modification of the Hotelling model that endows a firm with a non-market advantage, which is introduced alike the valence advantage known in problems of political economy. The nonmarket (valence) advantage can be interpreted as advertisement (brand awareness of firms). Problem statement. Consider two firms competing with prices and location. Homogeneous consumers vary with its location on a segment. They minimize their costs, which additively includes the price of the product and the distance from them to the product. The utility function is linear with respect to the price and quadratic with respect to the distance. It is also expected that one of the firms (for certainty, firm № 1) has a market advantage d. The consumers are assumed to take into account the sum of the distance to the product and the market advantage of firm 1. Thus, the strategy of the firms and the consumers depend on two parameters: the unit t of the transport costs and the non-market advantage d. I explore characteristics of the equilibrium in the model as a function of the non-market advantage for different fixed t. The aim of the research is to assess the impact of the non-market advantage on the equlibrium. We prove that the Nash equilibrium exists and it is unique under additive consumers' preferences de-pending on the square of the distance between consumers and firms. This equilibrium is ‘richer’ than that in the original Hotelling model. In particular, non-market advantage can be excessive and inefficient to use.

The Lipschitz continuous property has been used for a long time to solve the global optimization problem and continues to be used. Here we can mention the work of Piyavskii, Yevtushenko, Strongin, Shubert, Sergeyev, Kvasov and others. Most papers assume a priori knowledge of the Lipschitz constant, but the derivation of this constant is a separate problem. Further still, we must prove that an objective function is really Lipschitz, and it is a complicated problem too. In the case where the Lipschitz continuity is established, Strongin proposed an algorithm for global optimization of a satisfying Lipschitz condition on a compact interval function without any a priori knowledge of the Lipschitz estimate. The algorithm not only finds a global extremum, but it determines the Lipschitz estimate too. It is known that every function that satisfies the Lipchitz condition on a compact convex set is uniformly continuous, but the reverse is not always true. However, there exist models (Arutyunova, Dulliev, Zabotin) whose study requires a minimization of the continuous but definitely not Lipschitz function. One of the algorithms for solving such a problem was proposed by R. J. Vanderbei. In his work he introduced some generalization of the Lipchitz property named $\varepsilon$-Lipchitz and proved that a function defined on a compact convex set is uniformly continuous if and only if it satisfies the $\varepsilon$-Lipchitz condition. The above-mentioned property allowed him to extend Piyavskii’s method. However, Vanderbei assumed that for a given value of $\varepsilon$ it is possible to obtain an associate Lipschitz $\varepsilon$-constant, which is a very difficult problem. Thus, there is a need to construct, for a function continuous on a compact convex domain, a global optimization algorithm which works in some way like Strongin’s algorithm, i.e., without any a priori knowledge of the Lipschitz $\varepsilon$-constant. In this paper we propose an extension of Strongin’s global optimization algorithm to a function continuous on a compact interval using the $\varepsilon$-Lipchitz conception, prove its convergence and solve some numerical examples using the software that implements the developed method.

In this article we consider min-min type of problems or minimization by two groups of variables. In some way it is similar to classic min-max saddle point problem. Although, saddle point problems are usually more difficult in some way. Min-min problems may occur in case if some groups of variables in convex optimization have different dimensions or if these groups have different domains. Such problem structure gives us an ability to split the main task to subproblems, and allows to tackle it with mixed oracles. However existing articles on this topic cover only zeroth and first order oracles, in our work we consider high-order tensor methods to solve inner problem and fast gradient method to solve outer problem.

We assume, that outer problem is constrained to some convex compact set, and for the inner problem we consider both unconstrained case and being constrained to some convex compact set. By definition, tensor methods use high-order derivatives, so the time per single iteration of the method depends a lot on the dimensionality of the problem it solves. Therefore, we suggest, that the dimension of the inner problem variable is not greater than 1000. Additionally, we need some specific assumptions to be able to use mixed oracles. Firstly, we assume, that the objective is convex in both groups of variables and its gradient by both variables is Lipschitz continuous. Secondly, we assume the inner problem is strongly convex and its gradient is Lipschitz continuous. Also, since we are going to use tensor methods for inner problem, we need it to be p-th order Lipschitz continuous ($p > 1$). Finally, we assume strong convexity of the outer problem to be able to use fast gradient method for strongly convex functions.

We need to emphasize, that we use superfast tensor method to tackle inner subproblem in unconstrained case. And when we solve inner problem on compact set, we use accelerated high-order composite proximal method.

Additionally, in the end of the article we compare the theoretical complexity of obtained methods with regular gradient method, which solves the mentioned problem as regular convex optimization problem and doesn’t take into account its structure (Remarks 1 and 2).

The work is devoted to the study of subgradient methods with different variations of the Polyak stepsize for minimization functions from the class of weakly convex and relatively weakly convex functions that have the corresponding analogue of a sharp minimum. It turns out that, under certain assumptions about the starting point, such an approach can make it possible to justify the convergence of the subgradient method with the speed of a geometric progression. For the subgradient method with the Polyak stepsize, a refined estimate for the rate of convergence is proved for minimization problems for weakly convex functions with a sharp minimum. The feature of this estimate is an additional consideration of the decrease of the distance from the current point of the method to the set of solutions with the increase in the number of iterations. The results of numerical experiments for the phase reconstruction problem (which is weakly convex and has a sharp minimum) are presented, demonstrating the effectiveness of the proposed approach to estimating the rate of convergence compared to the known one. Next, we propose a variation of the subgradient method with switching over productive and non-productive steps for weakly convex problems with inequality constraints and obtain the corresponding analog of the result on convergence with the rate of geometric progression. For the subgradient method with the corresponding variation of the Polyak stepsize on the class of relatively Lipschitz and relatively weakly convex functions with a relative analogue of a sharp minimum, it was obtained conditions that guarantee the convergence of such a subgradient method at the rate of a geometric progression. Finally, a theoretical result is obtained that describes the influence of the error of the information about the (sub)gradient available by the subgradient method and the objective function on the estimation of the quality of the obtained approximate solution. It is proved that for a sufficiently small error $\delta > 0$, one can guarantee that the accuracy of the solution is comparable to $\delta$.