Extremum Of Function Research Articles

Development and application of combined methods to expand the potential for predicting the defining characteristics of composites

The issue of promising ways to expand potential opportunities for improving the effectiveness of methods for predicting the remaining life, durability, and reliability of composite materials and structures under the influence of extreme environmental factors is considered. Modifications of methods for searching for the global extremum of multiparametric functions have been developed and studied, taking into account the structural peculiarities of the efficiency criteria used in forecasting the remaining life and durability of structures made of composite and polymer materials. One of the promising ways to increase the potential opportunities has been analyzed. It is connected with the improvement of methods for constructing optimal directions for searching for the absolute minimum of efficiency indicators related to solving the problems of predicting the determining characteristics of polymer composites under the influence of extreme environmental factors and operational loads.

Amplitude electro-optical modulation of radiation by sequences of Fabry-Perot resonators

In this paper, we considered the method of amplitude electro-optical modulation of radiation using sequences of Fabry-Perot resonators based on the transverse electro-optical effect on the example of lithium niobate LiNbO3. With this method, it is possible to significantly reduce the voltage of the control electromagnetic field of the electro-optical amplitude modulator operating in the transmission mode of the light beam while maintaining its high efficiency. The reduction of the control voltage is achieved by increasing the number of Fabry-Perot resonators installed in series and the phase shift relative to the extremum of the transmittance function. This method allows to diminish the duration of the received light signals which leads to an increase in the clock frequency while maintaining a high efficiency of the radiation modulation. Diminishing the duration of light signals is achieved by using separate modulation channels of two sequences of electro-optical Fabry-Perot resonators, the first of which works on the transmission and the second one on the reflection. Increasing the clock frequency at the output of the modulator is achieved by summing the signals coming from several modulation channels. It is shown that the value of the control voltage for an amplitude electro-optical modulator based on a sequence of Fabry-Perot resonators made of lithium niobate LiNbO3, with an operating wavelength of 1.307 microns, can be 4 V in the case when its initial operating point corresponds to the maximum transmittance. The control voltage is 2 V if the initial operating point is shifted in phase relative to the extremum of the transmittance function.

FRAGMENT SELECTION METHOD FOR BINDING MULTI-CHANNEL IMAGES

The method of selecting reference fragments for coordinate matching of images from different observation channels, including images from different spectral ranges, is considered. The proposed method is based on calculating the Fourier spectrum in the local vicinity of fragment points and analyzing the anisotropy of the spectrum. Image fragments with the maximum summary anisotropy value are assigned as reference fragments. The procedure is performed automatically and does not require entering threshold values. A reference fragment is detected by searching for the extremum of the target function. It is shown that the error of image coordinate binding depends on the total value of the anisotropy of reference fragments.

ANALYSIS OF ROTATION AND SCALE INVARIANCE FOR PROXIMITY MEASURES IN TARGET DETECTION

The problem of stability of object detection in images using proximity measures is considered. The purpose of the work is to determine the degree of invariance of various proximity measures for detecting objects by reference when rotating and zooming the scanned image. The proximity measure that is most resistant to these geometric transformations of the image is found out. The proximity measures are analyzed: correlation, comparison, Chamfer Distance. The target location is based on the coordinates of the extremum of the target function. Modeling is performed in the Matlab software package. A database of thirty television images was created to test the proximity measures. Test images contain the required objects and imitations of both complex and simple backgrounds. It was determined that all considered proximity measures steadily determine the target with small turns and scaling factors.

Impact of soil moisture on wear intensity of the actuating elements of soil processing machines

The aim of the study consisted in the determination of the impact of soil moisture on wear intensity of bipod and tined actuating elements of soil processing machines made of different materials. We conducted the operation study according to the current regulatory documents on three types of soil processing machines: cultivators, plows, and common disk assemblies. Movement speed varied between 10...13 km/h for plows, and between 11...15 km/h for disk assemblies. During operation, we observed the change of linear dimensions and weight of the actuating elements of soil processing machines. Determination of the change of weight and linear dimensions was done after 10 ha of running time per one arrowheaded tine, 30 ha of running time per one disk, and 5 ha of running time per one trail. As a result of the studies, we have determined that the increase of soil moisture in sandy loam and clay loam soils leads to the increased wear intensity of the actuating elements of soil processing machines. Obtained mathematical dependencies allow determining the value of moisture where the most intensive abrasive wear of actuating elements takes place, after which the wear intensity decreases. Depending on the type of actuating elements, the moisture of sandy loam soils where the process of abrasive wear is the most intensive is 8...12 %, and for clay loam soils this figure is 9...13 %. The study of the impact of soil moisture on wear intensity of the actuating elements in clay soils was conducted at the moisture before and after the extremum of function, therefore the value of moisture where the most intensive wear takes place was determined experimentally. Consequently, when working in clay loam soils, the wear process for the actuating elements of the soil processing machine is the most intensive at the moisture of 13...16 %. The material of the actuating element of the soil processing machine does not influence the overall tendency of the impact of soil moisture on wear intensity

Automated Search for Locations of Detention Groups to Reduce Security Activity Risk

Property security management is a crucial problem in keeping tangible and intangible assets safe. Its solution using technical means and mobile detention groups ensures assets preservation and reduces security activities risk. Classical definition of a risk, which relates the probability of damage to the protected object to the amount of damage, is proposed to use. For determining the probability, an assumption, where only arrival time of the detention group at secured facility is recorded by the alarm signal, is introduced.
 In order to minimize the total risk of security activities the task of finding the location of the detention group on the ground with dispersed objects is formulated. As a solution to the formulated problem, a search for the location of the detention groups which takes into account the magnitude of the damage and the current coordinates of the object on the plane is proposed. Examples of calculating locations of detention groups on the plane are presented.
 An objective function Ф of the effectiveness of security activities implementation, linking the financial and economic indicators of security organization with the value of security activities risk is introduced and substantiated. The analysis of function Φ behavior with a change in the number of detention groups is shown, the presence of function extremum is shown, the interval of finding the extremum is determined.
 It is proposed to calculate the distances between objects on a map and to use it to determine the coordinates in a new auxiliary plane. Coordinates are calculated using Gram matrices. A computational example is presented.
 A step-by-step algorithm for secured facility allocation between detention groups with minimization of the total risk of security activities is developed; an example of its use is presented.
 A search procedure for the location of the detention group on the ground by the determined coordinates on the plane is defined; a search process is illustrated. A general location search algorithm is proposed and the results are presented.

Multi-Objective Function for SWIPT System by SADDE

Simultaneous wireless information and power transfer (SWIPT) optimization with multiple objective function optimization is presented in the millimeter band in this paper. Three different objective functions that are used for harvest power (HP), capacity, and bit error rate (BER) were studied. There are three different nodes in real environment for wireless power transfer (WPT) and SWIPT. The channel estimation calculated by shooting and bouncing ray/image techniques includes multi-path, fading effect, and path-loss in the real environment. We applied beamforming techniques at the transmitter to focus the transmitter energy in order to reduce the multi-path effect and adjust the length of the feed line on each array element in order to find the extremum of the objective functions by the self-adaptive dynamic differential evolution (SADDE) method. Numerical results showed that SWIPT node cannot achieve good performance by single objective function, but wireless power transfer (WPT) can. Nevertheless, both WPT and SWIPT nodes can meet the criteria by the multiple objective function. The harvesting power ratio as well as the BER and capacity can be improved by the multiple objective function to an acceptable level by only reducing a little harvesting energy compared to the best harvesting energy for the single objective function. Finally, the multiple optimization function cannot merely provide good information quality for SWIPT node but achieve good total harvesting power for WPT and SWIPT node as well.

Ж.-Л. ЛАГРАНЖ КАК ОДИН ИЗ ОСНОВОПОЛОЖНИКОВ ТЕОРИИ ЭКСТРЕМУМОВ ФУНКЦИЙ МНОГИХ ПЕРЕМЕННЫХ

In the article we carried out a detailed analysis of the results obtained by J.-L. Lagrange in his first. The theory of extrema of functions of many variables, as part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems on the conditional extremum of the function of many variables. The relevance of this topic is determined by the fact that the methods for solving problems on the extremum of the function of many variables obtained in the mid 18th - early 20th centuries are used in solving modern problems. A special place here is occupied by L. Euler and J.-L. Lagrange. The aim of the article is to study the conditions for the maximum and minimum functions of many variables obtained by J.-L. Lagrange, and a comparison of its results with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis. It was established that in his first printed work he first formulated and proved sufficient conditions for the existence of an extremum of the function of many variables by actually establishing a criterion for the positive (negative) definiteness of quadratic forms, long before it appeared in J. Sylvester in the mid-19th century. A comparative analysis of the results of L. Euler and J.-L. Lagrange. It was found that sufficient conditions for the existence of an extremum of functions of many variables obtained in the first printed work of the young Lagrange are included in all modern textbooks in virtually the same form. The examples shown illustrate his theory. These are tasks of geometric and physical content. Special cases are considered in detail: functions of two and three variables. It is noted that this article became programmatic for the young Lagrange, although it remained unnoticed by his contemporaries. Subsequently, based on the method he obtained, he created the variational calculus, using the principle of least action and the theory of extrema, derived the basic laws of mechanics, the rule of factors for finding the conditional extremum of functions of many variables, which is named after him.

Continuum Sensitivity Analysis for Electrode Shape Optimization in Electrostatic System with State Variable Constraint

This article proposes an electrode shape optimization strategy with state variable constraints. The continuum sensitivity formula is used for the electrostatic system to determine the extremum of objective functions with state variable constraints. We use the gradient projection method and a restoration process to treat geometric and state variable constraints. The geometric changes of the design parameter are determined by the velocity field calculated from the constrained sensitivity analysis and are expressed by the level set method. The application models are tested to demonstrate the feasibility and utility of the proposed method.

Оптимизация состава оборудования гибридных энергетических систем с возобновляемыми источниками энергии

The article develops of a methodology and a software application to optimize the equipment composition of hybrid energy systems with renewable energy sources. The paper describes the proposed methodology and the mathematical models of the main components of hybrid systems: a photovoltaic station, a wind power plant, a battery and a diesel generator set. A distinctive feature of the proposed method is an original model used to forecast the solar radiation based on the data of the National Aeronautics and Space Administration. This allows it to be used for predicting the main characteristics of solar radiation in any location of Russia, including the territories with no regular actinometric observations data. The reverse Weibull distribution function is used to predict the daily course of the wind speed, which provides an increase in the reliability of predicting the generation of electricity by the wind power plant at daily time intervals. An evolutionary particle swarm algorithm is utilized to solve the optimization problem, which provides for a reliable and efficient definition of the global extremum of the objective function under various optimization criteria and constraints. A practical example of the proposed methodology use in selecting the optimal composition of the equipment of a hybrid energy system with a different equipment configuration is considered for an area located in the Vladivostok city. The proposed technique is implemented as a software application in the popular mathematical complex MATLAB, which ensures the convenience of its practical application.

Asymptotics of Control Problem for the Diffusion Process in Markov Environment

A stochastic optimization procedure and a limit generator of the original problem are constructed for a system of stochastic differential equations with Markov switching and diffusion perturbation with control, which is determined by the condition for the extremum of the quality function. The complexity of the studied evolutionary model lies primarily in the fact that the system is in conditions of external random influence, which is modeled using a switching process. The main assumption is the condition for uniform ergodicity of the Markov switching process, that is, the existence of a stationary distribution for the switching process over large time intervals. This allows one to construct explicit algorithms for the analysis of the asymptotic behavior of a controlled process. An important property of the generator of the Markov switching process is that the space in which it is defined splits into the direct sum of its zero-subspace and a subspace of values, followed by the introduction of a projector that acts on the subspace of zeros. Another complexity of the model under study is the presence of an approximation scheme determined by normalization. The question of how the behavior of the limit process depends on the prelimit normalization by a small parameter of the stochastic system in an ergodic Markovian environment is studied. A stochastic differential equation is written for determining the limiting processes of transfer and control. For the first time, a model of the control problem for the diffusion transfer process using the stochastic optimization procedure for control is proposed. A singular expansion in the small parameter of the generator of the three-component Markov process is obtained, and the problem of a singular perturbation with the representation of the limiting generator of this process is solved.

Разработка модифицированного самоорганизующегося миграционного алгоритма оптимизации (MSOMA)

The modified self-organizing migration optimization algorithm (MSOMA) based on a self-organizing migration algorithm (SOMA) is suggested. An algorithm for solving the problem of finding the global conditional extremum of the objective function on a given set is developed. Examples illustrating the application of the algorithm and created software are given.

A note on the turning point for the quadratic trend

Abstract The quadratic trend is a statistical model described by the quadratic function. Finding its extremum (also called the vertex or the turning point) using differential calculus or completing the square method is very well known in the literature. In this paper, a new method for finding the extremum of the quadratic function, based on a simple mathematical inequality is proposed. In comparison with the other two known methods, our method does not require the differentiability assumption and it takes fewer steps than completing the square method. Also, it is shown how the turning point for the quadratic trend can be applied in forecasting the unemployment rate in Croatia in the first quarter of 2019. The obtained conclusions are equal to the conclusions obtained in the usual way by using forecasting software.

Two Subpopulations Cuckoo Search Algorithm Based on Mean Evaluation Method for Function Optimization Problems

In order to better apply the cuckoo search (CS) algorithm in solving the problem of function extremum optimization, and further improve the phenomenon of low precision and slow convergence in the optimization process of algorithm, the two subpopulations CS algorithm based on mean value evaluation is proposed. On the one hand, the algorithm introduces dynamic inertia weight to adjust the lévy flight mechanism, thus dynamically constraining the moving step-size of each generation of population, so that the algorithm has certain self-adaptability. On the other hand, the algorithm changes the way of mutation in the preference random walk. First, the average fitness evaluation mechanism is used to divide the current population into two subpopulations: good and bad. Then, it adopts a directional mutation strategy for the better population, so that the individual can search purposefully. The worse population uses differential mutation mechanism of the disturbance items with the [Formula: see text]-distribution characteristics, and makes the individual to search in the best orientation of current, so as to enhance the local search performance and accelerate the convergence rate of the algorithm. Theoretical analysis proves the convergence and time complexity of the algorithm in this paper. The simulation results show that the improved algorithm has good applicability in solving the function optimization problem, and the optimization results and convergence speed have been significantly improved in the algorithm.

Calculation of sealing joints with elastic edge in PTC MathCAD

The article touches upon the issues of the development of engineering methods of calculating a thin-walled shell-plate seat for valve and flange sealing joints, followed by the formulation of tasks using modern automated calculation software. To achieve the purpose, the authors used Parametric Technology Corporation MathCAD (PTC MathCAD) with subroutines for the search for extremum of functions and the computational block ‘Given - minimize’. The calculation is presented in the form of algorithms consisting of three logically interrelated parts: 1) stiffness calculation; 2) strength calculation; 3) the calculation of rational dimensions. The obtained results of the calculation of geometric dimensions are tested, which eliminates a probable error in the calculated block ‘Given’ of PTC MathCAD. Among the developed engineering methodologies, the article describes the following ones: a method for calculating the rational geometrical dimensions of a shell-plate valve seat, operating under shock loading at an uncertain maximum dynamic load. It ensures that the allowable stresses in a thin-walled shell-plate seat are not exceeded. Another is a method for testing the stress-strain state of a shell-plate valve seat under static loading with a force of sealing and pressure of the working medium. The last one is the method of calculating rational geometrical dimensions of the shell-plate seat of a flange connection, which minimizes the unit stiffness (minimizing sealing force), provided that the shell-plate seat strength is ensured.

Computer simulation of sublayer temperature mode during deposition of diamond coatings using gas-plasma method

The research presents a computer simulation of the temperature field on the sublayer surface, performed in a stationary mode during the deposition of diamond coatings using the gas-plasma method with different parameters of the cooling system. The optimization problem of mathematical programming with the search for the extremum of the performance function by varying the controlled parameters within the permissible range is considered.

On the program implementation of a Markov inhomogeneous random search algorithm with normal distributions

A program that implements a Markov inhomogeneous monotonous random search algorithm of an extremum with normal distributions is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

On the program implementation of a Markov homogeneous random search algorithm of an extremum with normal distributions

A program that implements a Markov homogeneous monotonous random search algorithm of an extremum with normal distributions is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

On the program implementation of one inhomogeneous Markov algorithm of search for extremum

A program that implements a Markov inhomogeneous monotonous random search algorithm of an extremum is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

On the program implementation of a Markov homogeneous random search algorithm of an extremum with Ingber’s distribution

A program that implements a Markov homogeneous monotonous random search algorithm of an extremum with Ingber’s distribution is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.