Extremum Of Function Research Articles

MDRM model and its application in hybrid reliability analysis

The high-dimensional model decomposition method is an effective approach for solving the reliability of engineering structures or composite materials with a large number of uncertain parameters. The multiplicative dimensionality reduction method is a high-dimensional model decomposition technique that decomposes a high-dimensional function into a combination of low-dimensional functions and approximates the overall response of the high-dimensional function by solving the responses of these low-dimensional functions. This paper begins by discussing the extrema of functions based on the multiplicative dimensionality reduction method. Then, by combining it with Taylor series expansion, a method for approximating the extremum of a high-dimensional function was proposed. Furthermore, based on the multiplicative dimensionality reduction method, first-order reliability method, and Taylor series expansion, the functional relationship between upper and lower bounds of failure probability and interval variables was derived. Finally, using the algorithm proposed in this paper, the extrema of high-dimensional functions can be calculated to solve the upper and lower bounds of failure probability under random-interval hybrid uncertainty. This paper presents two engineering structures and a numerical example to demonstrate the efficiency and accuracy of the proposed method.

Dimension synthesis of linkage mechanism with approximate poses and velocity constraints

This paper mainly addresses the problem of dimension synthesis for planar open chain two-rod (2R) mechanisms with velocity constraints and position-pose information in agricultural transplanting mechanism (TM). The method proposed in this study extends the dimension synthesis problem of planar 2R mechanisms, which takes position-pose into account, to the dimension synthesis considering both position-pose information and velocity information (PPIAVA) at certain pose points in the field of TM. The geometric constraints of the open chain 2R mechanism are extracted from the prerequisite conditions obtained by a series of given position-pose points and velocity constraints at some pose points to solve the dimension synthesis problem of the planar 2R mechanism. First, the Taylor formula is used to connect the characteristic of the derivative value at a certain point as coefficient with PPIAVA at a certain point. The mathematical model of position-pose and velocity is established by combining the relationship of rod length constraint and velocity direction characteristic. Secondly, the nonlinear equations established according to the position-pose information and velocity constraints are combined with the Lagrange equation and the extremum of the multivariate function to transform the appropriate objective function and then solved the equations. Finally, a transplanting mechanism is designed and tested by combining the solution results with the trajectory characteristics of the TM. The results show that this method is an effective synthesis method for smoothly solving the dimension synthesis problem of planar open chain 2R linkage mechanism with velocity constraints, which can extract geometric constraints from a given series of position-pose and velocity constraints at some poses.

Changes in uroflowmetry indicators with automatic interference elimination

Aim. The aim of this article is to compare the values of measured (QMAX) and calculated (QMCLC) maximal urine flow and the values of measured (TQ) and calculated (TQCLC) time to reach maximal flow in a large group of individual uroflowgrams during home uroflowmetry.Materials and Methods. We analysed 29,110 individual uroflowgrams from 719 patients with prostate adenoma, aged 50 to 73 years (mean age, 60.5 ± 8.2 years), collected between 2004 and 2024. Two algorithms were used: determination of QMAX as the maximum value of the calculated flow (function extremum) and calculation of the actual value of the maximum flow after computer processing to eliminate interference and the WAG-effect (QMCLC).Results. The data showed the following differences between QMAX and QMCLC in the groups: 16.14 % for volumes up to 100 ml; 14.62 % for 100–200 ml; 13.75 % for 200–300 ml; 13.04 % for 300–400 ml; 14.25 % for 400–500 ml; 14.55 % for 500–600 ml; and 12.65 % for 600+ ml. For TQ and TQCLC values, the group differences were as follows: 3.49 % for volumes up to 100 mL; 2.27 % for 100–200 mL; 0.71 % for 200–300 mL; 0.97 % for 300–400 mL; 0.91 % for 400–500 mL; 3.51 % for 500–600 mL; and 1.51 % for 600+ mL.Conclusion. The study demonstrated a statistically significant difference between the maximum urine flow (QMAX) values obtained during measurement and the maximum urine flow value, after removing interference and the WAG effect (QMCLS) characteristic of any recorded volumes. Therefore, the accuracy of the data obtained may differ depending on the algorithm of uroflowgram processing. No statistically significant difference was found between TQ and TQCLC parameters. The algorithm of uroflowgrams processing used in the Sigma and Urovest uroflowmetry hardware and software system ensures high accuracy in determining maximum flow values; Urovest, in particular, has demonstrated reliability in this regard. The average determined difference between the QMAX and QMCLC values at different volumes of urination is 14.14 %.

Optimization Design of Pt/C Multilayer Supermirrorsfor Hard X-rays

Aperiodic multilayers have important applications in hard X-rays astronomical observations due to their broadband responses. This paper presents an optimization model of Pt/C power-law graded stack multilayers using the IMD software. The reflectivity expression of multilayers was derived from the reflection theory of X-ray multilayers and Fresnel recursion formulas. Then, the effects of maximum bi-layer thickness ([Formula: see text]), ratio of high-density material thickness to bi-layer thickness ([Formula: see text]), power-law index ([Formula: see text]), number of bi-layers ([Formula: see text]), grazing incidence angle ([Formula: see text]), interfacial roughness ([Formula: see text]), and photon energy ([Formula: see text]) on the reflectivity were analyzed. The calculation results showed that the [Formula: see text], [Formula: see text] and [Formula: see text] control the shift of response range, absorption peak and Bragg peaks, respectively. And the decrease of [Formula: see text] and increase of [Formula: see text] and [Formula: see text] can result in the reduction of reflectivity at different energies. Additionally, an optimized multilayer structure was proposed according to the extremum of merit function, which can achieve a broadband reflectivity of no less than 0.6 from 1[Formula: see text]keV up to 80[Formula: see text]keV. These results can provide important references for the response characteristic analysis and optimization design of multilayers.

Probability analysis of compressor blade vibration based on radial basis function extremum neural network method

To enhance the transient response and modeling accuracy of compressor blade vibration characteristics in time domain dynamic reliability analysis, a method called the radial basis function extremum neural network (RBFENN) is proposed. This method combines the radial basis function neural network (RBFNN) with the extreme response surface method (ERSM). By considering the time domain and frequency domain characteristics of unsteady aerodynamic excitation on blades affected by multi-row stator/rotor interference, the stress response distribution of the blades is determined through deterministic analysis. The RBFENN mathematical model is established using RBFNN to address the high nonlinearity in surrogate modeling. The ERSM is used to process the transient problem of motion reliability analysis, taking extreme values into account throughout the response process. In evaluating the vibrational reliability of the compressor blades, the maximum stress is considered as the primary focus of the study, taking into account the stochastic nature of aerodynamic excitation frequency, rotational velocity, and material parameters. It is observed that the unsteady aerodynamic excitation induces higher-order harmonic resonances in the blade velocity margin, resulting in increased stress at the leading edge of the blade root. This stress distribution follows a normal distribution pattern. A comparison among Monte Carlo methods, ERSM, and RBFENN demonstrates the superiority of the proposed RBFENN method in terms of computational accuracy and efficiency. This study introduces the RBFENN method as a learning-based approach for blade vibration reliability design, enriching the theory of mechanical reliability.

Analysis and evaluation of the effectiveness of program codes for particle swarm and ant colony methods for lower semi-continuous functions

In this paper, a comparative analysis of two stochastic methods of global optimization is carried out: the Particle Swarm Optimization method and the Ant Colony Optimization method. The analysis and evaluation of efficiency were carried out using parameters such as the number of calls to the objective function, the rate of convergence, resources, computational efficiency, sensitivity to parameters, and resistance to initial conditions. The article presents the program codes of these methods in the Python environment, designed to calculate the global extremum of semi-continuous functions from below. Well- known test functions were used to construct (by gluing) semi-continuous functions at different values of input parameters.

MODIFICATION OF A QUANTUM-INSPIRED GENETIC ALGORITHM FOR NUMERICAL OPTIMIZATION USING QUDIT UNDER CONDITIONS OF SIMULATING QUANTUM DECOHERENCE

The genetic algorithm for numerical optimization (GA) of the metaheuristic class is a method for finding optimal solutions based on the biological principles of natural selection and variability. GA is characterized by high operating speed, resistance to noise in the data, low probability of hitting the local extremum of the multimodal objective function, as well as the simultaneous application of probabilistic and deterministic rules for generating search space points. An alternative to the classical GA is the quantum-inspired genetic algorithm for numerical optimization (QIGA), which has advantages that are unattainable for GA by using the concepts and principles of quantum computing. The article proposes a new approach to the implementation of a quantum-inspired genetic numerical optimization algorithm for searching for the global maximum of the objective function, based on modeling the functioning of the GA by simulating the execution of quantum calculations based on qudit in the conditions of the existence of quantum decoherence in the era of noisy medium-scale quantum algorithms. For this purpose, to carry out quantum operations of rotating the states of multilevel quantum systems, the paper presents a density matrix based on Heisenberg-Weyl operators as an analogue of the Bloch sphere for qudits. The simulation of quantum decoherence is interpreted from the point of view of the influence of extraneous noise emanating from the environment on the qudit and is presented as the use of a normal random variable with zero mathematical expectation and unit variance in quantum gates. At the same time, the work presents detailed pseudocodes of the functioning of both the most modified quantum-inspired genetic algorithm for numerical optimization and its individual operations. Testing is carried out by conducting computational experiments with the implementation of a modified algorithm on two-dimensional and multidimensional functions of test optimization problems, as well as when solving an applied optimization problem of planning hybrid flow production in the manufacturing industry based on financial costs and solving the problem of increasing forecasting accuracy based on compact extreme learning machines. The experimental results demonstrate the superiority of the new algorithm over QIGA and classical optimization algorithms in the accuracy of the solution, the speed of convergence with the target value of the global maximum and the execution time of the algorithm.

Використання задачі про максимальний потік для бізнес-процесів

The article is devoted to the use of optimization methods for managing business processes. Problems of synthesizing the structure and choosing parameters of systems, as well as problems of optimal control of them, are most often solved by reducing them to problems of finding a number of parameters so that the extremum of the quality function is achieved under a set of restrictions. A separate group of problems related to the choice of topology can be solved using the apparatus of graph theory. Problems of choosing channel capacity can be reduced to classical optimization problems, such as linear programming, transport problem, nonlinear programming, etc. The minimum flow problem is a special case of the transport problem, which is related to linear programming problems. The paper analyzes the use of the maximum flow problem in economic systems, namely for planning business processes. The use of tasks to optimize business processes in various industries, such as logistics, production and finance, and personnel management, is described. A review of the developments and research of the maximum flow problem is carried out, a number of algorithms for solving this problem and their main points are given. The problems, the mathematical formulation of the problem, the algorithm for placing marks and the calculation of arc flows for each iteration are described in detail. An example of the flagging algorithm shows how maximum flow models can be used to analyze and optimize production processes. Maximum flow problems can identify bottlenecks in a production chain, optimize resource utilization, and plan production. Using the maximum flow problem when planning business processes will improve efficiency and increase production, help reduce costs and improve profitability, and ensure that products are in stock to meet demand.

Mathematical modeling of minimization of electricity consumption by industrial enterprises with continuous production

Relevance. Determined by the importance of minimizing electrical power consumption in industrial enterprises with continuous production, considering the specific characteristics of their technological processes and the requirements to maintain the output volume of their products. Aim. To solve the task of minimizing electrical power consumption based on a mathematical model and gradient method under optimal planning of the production volume of the industrial enterprises with continuous production; to develop a mathematical model for optimal distribution of production over a time cycle (month, quarter, year) across departments, taking into account both simple and functional constraints, derived from the condition of ensuring minimal electrical power consumption in industrial enterprises with continuous production. Methods. When developing the mathematical model for ensuring minimal electrical power consumption while preserving the production volume, classic Lagrange optimization methods were used. To ensure sufficient calculation accuracy, iterative methods were also applied. For the task under consideration, a calculation error margin of ε=0,1 was assumed and established. It is known that the choice of calculation error margin depends on the specifics of the problem at hand and the decision-maker. To verify the adequacy of the developed model, the method of finding the relative extremum of a function of several variables was used. Results. The use of the mathematical model, which takes into account the nature of the technological process and boundary conditions in both simple and integral forms, demonstrated the feasibility of optimal planning of electrical power consumption by the enterprise. The effectiveness of the developed approaches was verified using a metallurgical enterprise as an example of an industrial enterprise with continuous production, in solving the task of minimizing electrical power consumption for products produced during the reporting period. The use of the proposed model allowed for a reduction in annual electrical power consumption by 2.5% while maintaining the same production volume. One of the classic optimization methods – the method of finding the relative extremum of functions of several variables – showed almost identical results upon verification. This serves as further evidence of the adequacy of the proposed model.

About the Kyiv school mathematicians contribution to the extrema of functions of many variables theory

The subject of this article is the study of Kyiv school mathematicians’ contribution to the many variable extrema function theory. The purpose of this article is to study the work of Kyiv mathematician Professor M. Ye. Vashchenko-Zakharchenko. Task: for the first time, to carry out a detailed analysis of the results obtained in the article by M. Ye. Vashchenko-Zakharchenko «Signs of the highest and lowest value of functions». The research method is a historical and scientific analysis of the original source, which allows the scientific results obtained in the late 19th and early 20th centuries to be estimated from the viewpoint of modern mathematical analysis. The following results were obtained. Thanks to the analysis of this article, it was found out that here M. Ye. Vashchenko-Zakharchenko first used the methods of linear algebra and applied D. Sylvester’s criterion of positive (negative) definiteness of quadratic form to obtain sufficient conditions for the existence of an extremum of functions of many variables. The problem is considered in general for functions of many variables and for special cases of functions of two and three variables. Conclusions. Comparing the results obtained from the Kyiv mathematician, Professor M. Ye. Vashchenko-Zakharchenko with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis, we can conclude that they are included in these textbooks in virtually the same form. This is what determines the relevance of this topic: methods for solving problems for the extremum of functions of many variables, obtained at the turn of the 19th and 20th centuries, are used in modern optimization problems. The breadth of the scientific erudition of M. Ye. Vashchenko-Zakharchenko allowed him to immediately perceive new methods obtained by foreign scientists and immediately find their application to obtain sufficient conditions for the existence of an extremum of functions of many variables. This shows the high scientific level of the state of mathematics at Kyiv University in the second half of the 19th century. Further research should focus on the contribution of other mathematicians of the Kyiv school to the development of the theory of extrema of functions of many variables.

GRID-STEP OPTIMIZATION OF THE PARAMETERS OF THERMAL IMPACT ON THE TECHNICAL SYSTEM

The article solves applied problems of optimizing the parameters of technical systems under the influence of concentrated, discrete sources of thermal load. To improve the accuracy and speed of solving applied optimization problems, the authors propose a generalized procedure for finding the optimal thermal load parameters. This procedure consists of well-known computational methods for solving boundary value problems and ensuring the optimization of the objective function and its parameters. To calculate the values of the temperature field, it is necessary to solve a nonlocal boundary value problem of a system of nonlinear differential heat transfer equations. In order to find the conditions for the correctness of boundary value problems, the authors used the theory of pseudo-differential operators in the space of generalized slow power functions. This made it possible to guarantee the correctness of computational and applied optimization mathematical models. The procedure for grid-step optimization of the parameters of thermal action on a multilayer material proposed by the authors is based on a grid approach with discretization of the main parameters of thermal load and grinding of a large uniform grid in the vicinity of the selected node of the large grid. The directed search for local extremes of the mesh model is carried out by the nodes of the crushed mesh. The search for local extrema is performed by selecting the next node of a large grid. This approach will increase the accuracy of solving applied optimization problems by allowing further mesh refinement and refinement of the values of local extremes, and, consequently, refinement of the values of the global extremum of the objective function. The optimization of the thermal impact parameters was carried out according to the criterion of reducing damage to the test material. This makes it possible to increase the accuracy of the technological process of thermal action on a multilayer material.

On the properties of one auxiliary function for calculating the global extremum

In this paper, we investigate the properties of one Auxiliary Function (AF) for calculating the global optimum of a function of many variables. The considered AF is constructed by transforming the objective function using the Lebesgue integral and is a function of one variable. Despite the fact that in previous works this AF was used to find the global minimum of smooth functions on convex closed sets using the algorithm of dividing a segment in half, its properties have not been studied. Here, this AF is used to calculate the global extremum of continuous functions defined on bounded closed sets of a multidimensional Euclidean space. The basic properties of the AF for any continuous objective function are established, such as non-negativity, positive uniformity, uniform continuity, differentiability and strict convexity, moreover, derivatives up to a certain order are found. The optimality criterion is formulated. The main criterion of optimality is that the value of a free variable, at which the AF and its derivatives are equal to zero up to some order, coincides with the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the AF or its derivative up to some order.

An in-depth analysis and derivation of extremum conditions based on gradient information

This paper explores the necessary conditions for extremum in both constrained and unconstrained problems by extracting fundamental principles of constraint conditions. We provide a precise geometric understanding of the Lagrange Multiplier, prioritizing analytical insight. Beginning with a geometric interpretation of the gradient, we leverage the expansion of functions and their images to comprehend extremum and detail the Lagrangian derivation process.We expand the base vectors of the constraint surface into those of the full space and use a transition matrix to assess the function's extremum. This demonstrates how the second derivative matrix is transformed into its full-space representation to discern extrema in optimization problems. Additionally, we introduce incremental variables to optimize the second-order derivative matrix in full space, providing a novel perspective to solve extremal necessary conditions.

Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

In this research work, a new derived optimal quadrature formula is discussed, which includes the sum of the values of the function and its first and second order derivatives at the points located at the same distance on the interval [0,1] in the L2(m)(0,1) space. we first obtain an analytical representation of the error function norm, and a system of equations of the Wiener-Hopf type construct using the method of Lagrange unknown multipliers for finding the conditional extremum of multivariable functions. Optimal coefficients found by solving the system. Using the exact form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula for m=3 and m=4 calculate and the order of approximation was shown to be O(hm). The obtain theoretical conclusions confirmed by numerical experiments.

On Problems of Extremum and Estimates of Control Function for Parabolic Equation

On Problems of Extremum and Estimates of Control Function for Parabolic Equation

Optimal recovery of the derivate from the confined analytic function

In the article the author finds out the best straight-line method to recover the derivate in zero-point from the confined analytic function determined in the unit circle according to values of the function and values of its derivate in given points creating the regular polygon with zero as the center. The paper consists of three parts. The first part one contains notions and results necessary for the solution of the objective. The second part determines the measure of inaccuracy of the best approximation method and draws out the corresponding extremum function. It is established that the extremum function is the only one to an approximation of the constant multiplier modulo equal to one. The third part calculates coefficients of the best straight-line method of recovery. It is established that the best straight- linear method of recovery is the only one.

Method of determining time for preventive diagnosis in city bus operation systems

The article presents a mathematical model of the process of diagnosing and repairing preventive public transport buses. The mathematical model was developed using the theory of semi-Markov processes. The presented model considers two types of corrective repairs - carried out by the Technical Emergency Service and at Service Stations, and preventive repair. The presented model of the exploitation process system has been described to a small extent in the literature so far. The considered system is analyzed on the basis of two criteria: the availability of public transport buses and the profit per unit time. For the adopted assumptions, the conditions for the existence of an extremum of the criterion function were formulated. In the article, the presented model is illustrated based on sample results.

Throughput maximization scheme of the IRS-aided wireless powered NOMA systems

The computation complexity of the existing resource allocation schemes to maximize sum throughput is very high for intelligent reflecting surface (IRS)-assisted wireless powered non-orthogonal multiple access (NOMA) system. Therefore, a low complexity resource allocation method to maximize sum throughput is considered by joint optimizing phase shifts and time allocation. Firstly, to reduce the variables that need to be optimized, we derive the problem with only phase shifts as variables, and further deduce optimization problem of phase shifts only for downlink wireless energy transmission (WET) process. Then, the problem of phase shifts for WET process is solved directly by transforming multi-variable optimization problem into that with single variable, and the phase shifts for the uplink wireless information transmission process are got as well with given phase shifts for WET process. Finally, with given phase shifts, the optimal time allocation is obtained by using the function extremum method. Theoretical analysis demonstrates that the proposed scheme has low computational complexity, and the simulation results show that sum throughput of the proposed scheme is no less than that of the existing schemes in the same scenario.

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

AbstractEnergetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.

Low complexity resource allocation scheme for IRS‐assisted downlink non‐orthogonal multiple access systems

AbstractA low complexity resource allocation method is proposed for downlink non‐orthogonal multiple access system assisted by intelligent reflecting surface (IRS). Firstly, an optimisation problem is formulated to minimise power consumption, with power allocation and IRS phase shifts as variables. Then, the joint optimisation problem of power allocation and IRS phase shifts is transformed into the optimisation of IRS phase shifts, which is further decomposed into multiple single variable sub‐problems, the solutions to which can be obtained by using the function extremum method. Next, based on these sub‐problems, one‐level iterative algorithm is developed to optimise IRS phase shifts. Finally, the minimum power required by each user is calculated based on the IRS phase shifts obtained through iteration. Simulation results show that the proposed scheme is better than the existing schemes in the same scenario under the same rate requirement in terms of computational complexity and power consumption.