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Fully leveraging the regulatory role of cascade hydropower in river basins and realizing complementary joint power generation between cascade hydropower and photovoltaic (PV) systems is a crucial approach to promoting the consumption of clean energy. Given the uncertainty of PV outputs, this paper introduces a short-term scheduling model for cascade hydropower–PV systems. The model aims to maximize electricity consumption by considering individual units, hydropower plant constraints, unit constraints, and grid constraints. By allocating loads among hydropower plants and periods, it optimizes hydropower's dual roles, supporting grid power supplies and coordinating with PVs, thus boosting the overall system consumption. In terms of model solution, linearization methods and modeling techniques such as piecewise linear approximation, the introduction of 0–1 integer variables, and the discretization of generation headwater are employed to handle the nonlinear constraints in the original model, transforming it into a mixed-integer linear programming problem. Finally, taking a complementary system constructed by 15 units of 4 hydropower stations and 2 photovoltaic groups in a cascade in a river basin in Southwest China as an example, the results show that through the complementary coordination of cascade hydropower and photovoltaic power, under the same grid constraints, the expected value of the power consumption of the complementary system in the model of this paper increased by 863.2 MW·h, among which the power consumption of photovoltaic group 1 increased by 1035.7 MW·h, and the power consumption of photovoltaic group 2 decreased by 172.5 MW·h. [ABSTRACT FROM AUTHOR]

In the context of global climate change, the multi-city interconnected power system offers the potential for low-carbon and efficient energy utilization, addressing the challenge of ensuring safety, stability, and reduced carbon emissions while meeting diverse demands. This study proposes a security region-based method to evaluate the power supply guarantee capability of such systems, employing a collaborative support framework to characterize the low-carbon feasible space of each city system. A multi-dimensional piecewise linear approximation method and model transformation were applied to construct a scheduling and transformation model for the provincial power grid. The proposed approach enhances power supply security, achieving a 7.21% reduction in system operating costs and a 24.7% decrease in carbon emissions. These findings highlight the effectiveness of the security region approach in balancing safety, efficiency, and environmental objectives, providing a scalable solution for interconnected power grids. [ABSTRACT FROM AUTHOR]

Today, huge amounts of time series data are sensed continuously by AIoT devices, transmitted to edge nodes, and to data centers. It costs a lot of energy to transmit these data, store them, and process them. Data compression technologies are commonly used to reduce the data size and thus save energy. When a certain level of data accuracy is sacrificed, lossy compression technologies can achieve better compression ratios. However, different applications may have different requirements for data accuracy. Instead of keeping multiple compressed versions of a time series w.r.t. different error bounds, HIRE hierarchically maintains a tree, where the root records a constant function to approximate the whole time series, and each other node records a constant function to approximate a part of the residual function of its parent for a particular time period. To retrieve data w.r.t. a specific error bound, it traverses the tree from the root down to certain levels according to the requested error bound and aggregates the constant functions on the visited nodes to generate a new bounded error compressed version dynamically. However, the number of nodes to be visited is unknown before the tree traversal completes, and thus the data size of the new version. In this paper, a time series is progressively decomposed into multiple piecewise linear functions. The first function is an approximation of the original time series w.r.t. the largest error bound. The second function is an approximation of the residual function between the original time series and the first function w.r.t. the second largest error bound, and so forth. The sum of the first, second, ..., and m-th functions is an approximation of the original time series w.r.t. the m-th error bound. For each iteration, Swing-RR is used to generate a Bounded Error Piecewise Linear Approximation (BEPLA). Resolution Reduction (RR) plays an important role. Eight real-world datasets are used to evaluate the proposed method. For each dataset, approximations w.r.t. three typical error bounds, 5%, 1%, and 0.5%, are requested. Three BEPLAs are generated accordingly, which can be summed up to form three approximations w.r.t. the three error bounds. For all datasets, the total data size of the three BEPLAs is almost the same with the size used to store just one version w.r.t. the smallest error bound and significantly smaller than the size used to keep three independent versions. The experiment result shows that the proposed method, referred to as PBEPLA-RR, can achieve very good compression ratios and provide multiple approximations w.r.t. different error bounds. [ABSTRACT FROM AUTHOR]

Ahmad et al. (see [1]) presented a piecewise Lagrange interpolation method for solving tumor-immune interaction models with fractal fractional operators using a power law and exponential kernel. We suggest a convergence anal-ysis for this method and we obtain the order of convergence. Of course, there are some mistakes in this numerical method that were corrected. Furthermore, Numerical illustrations are demonstrated to show the effectiveness of the corrected numerical method. [ABSTRACT FROM AUTHOR]

The use of electronic systems is widespread in various metrology systems related to the monitoring of acoustic noise in urban or industrial environments. A method for analyzing equipments intended for acousticlocation and acoustic measuring devices and systems (DS) based on structural approximation is proposed. The method involves performing a structural approximation of the devices used based on the developed generalized circuit. In this particular instance, the analysis is conducted utilizing the apparatus of continuous piecewise-linear functions and transfer functions of the signal converter. Their utilization facilitates the investigation of devices (systems) of arbitrary structure, characterized by random nonlinear characteristics and a variety of inertial components such as blocks and units. The study of specific devices is based on the finite expressions of the transfer functions of a generalized signal converter connection. Several levels of disclosure of the converter connection are considered. Using the aforementioned methodology, it is feasible to synthesize acousto-electric devices (systems) at both the analytical and structural levels. As nonlinear differential equations of random order that describe the response of devices and systems can be reduced to linear equations of the first order in the general case, the inverse solution (synthesis) is not a complex task, both at the analytical and programmatic levels. [ABSTRACT FROM AUTHOR]

This article develops a novel decision-oriented framework that strategically deconstructs systems resilience in a way that focuses on systems' design, capabilities, and management. The framework helps evaluate and compare how system design choices impact system resilience. First, we propose a resilience score based on a piecewise linear approximation to a resilience curve. Using multicriteria decision analysis principles, we score system design alternatives in terms of system-specific capabilities. We estimate the relevance of these capabilities to resilience curve parameters associated with resilience phases. Finally, we interpret the derivatives of resilience with respect to the curve parameter values as the leverage of these parameters. Using multiple levels of weighted sums of the scores, we calculate the first order impact of system design choices first on a proxy for the generic resilience parameters and then on resilience, which allows situational characteristics to be incorporated in their natural terminology while mapping their impact on resilience with a traceable logic. We illustrate the approach by using existing materials to develop an example comparing engineered designs for minimizing post-wildfire flood impacts. [ABSTRACT FROM AUTHOR]

The article considers the problem of estimating the error in calculating the gradient of functions of Sobolev classes for piecewise linear approximation on triangulations. Traditionally, problems of this kind are considered for continuously differentiable functions. In this case, the corresponding estimates reflect both the smoothness class of the functions and the quality of the triangulation simplexes. However, in problems of substantiating the existence of a solution to a variational problem in the nonlinear theory of elasticity, conditions arise for permissible deformations in terms of generalized derivatives. Therefore, for the numerical solution of these problems, conditions are required that provide the necessary approximation of the derivatives of continuous functions of the Sobolev classes. In this article, an integral estimate of the indicated error for continuously differentiable functions is obtained in terms of the norms of the corresponding spaces for functions that reflect, on the one hand, the quality of triangulation of the polygonal region, and on the other hand, the class of functions with generalized derivatives. The latter is expressed in terms of the majorant of the modulus of continuity of the gradient. To obtain the final estimate, the possibility of passing to the limit using the norm of the Sobolev space is proven.

This document is a compilation of articles from the journal "Operations Research" that cover a wide range of topics related to production management, inventory systems, financial networks, audit strategies, online markets, volunteer engagement, and data-driven optimization. The articles provide valuable insights and findings that can enhance efficiency and decision-making in these fields. The research presented in these articles is relevant for practitioners and researchers conducting research in these areas. The articles offer unique perspectives and approaches to solving complex problems in different industries, aiming to provide practical solutions and improve decision-making processes. [Extracted from the article]

Utility Preference Robust Optimization with Moment-Type Information Structure In some decision-making problems, information on the true utility function of the decision maker may be incomplete, which may bring potential modeling risk. In "Utility Preference Robust Optimization with Moment-Type Information Structure," Guo, Xu, and Zhang propose a maximin utility preference robust optimization model where information about the DM's preference is constructed by moment-type conditions. The authors propose a piecewise linear approximation approach to tackle the maximin problem, reformulate the approximate problem as a single mixed integer linear program, and derive error bounds for the approximate ambiguity set, the optimal value, and the optimal solutions. To examine the performance of the model and the computational schemes, they carry out extensive numerical tests and demonstrate the effectiveness of the model and efficiency of the computational methods. Utility preference robust optimization (PRO) models have recently been proposed to deal with decision-making problems where the decision-maker's true utility function is unknown and the optimal decision is based on the worst-case utility function in an ambiguity set of utility functions. In this paper, we consider the case where the ambiguity set is constructed using some moment-type conditions. We propose piecewise linear approximation of the utility functions in the ambiguity set. The approximate maximin problem can be solved directly by derivative-free methods when the utility functions are nonconcave. Alternatively, we can reformulate the approximate problem as a single mixed integer linear program (MILP) and solve the MILP by existing solvers such as Gurobi. To justify the approximation scheme, we derive error bounds for the approximate ambiguity set, the optimal value and optimal solutions of the approximate maximin problem. To address the data perturbation/contamination issues arising from the construction of the ambiguity set, we derive some stability results which quantify the variation of the ambiguity set against perturbations in the elicitation data and its propagation to the optimal value and optimal solutions of the PRO model. Moreover, we extend the PRO models to allow some inconsistencies in the process of eliciting the decision-maker's preferences. Finally, we carry out numerical tests to evaluate the performances of the proposed numerical schemes and show that the computational schemes work fairly efficiently, the PRO model is resilient against small perturbations in data (with respect to both exogenous uncertainty data and preference elicitation data), and there is a potential to improve the efficiency of the preference elicitation by incorporating an optimal selection strategy. Funding: This project is supported by Hong Kong RGC [Grant 14500620] and The Chinese University of Hong Kong start-up grant. The research of S. Guo is supported by the National Key R&D Program of China [Grant 2022YFA1004000] and the Natural Science Foundation of China [Grant 12271077]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.2464. [ABSTRACT FROM AUTHOR]

In the studies of unit commitment or optimal power flow, to formulate a mixed-integer linear programming model that can be efficiently solved with commercial solvers, it is necessary to approximate the quadratic cost curves of thermal units as piecewise linear (PWL) functions. The conventional approach involves evenly spaced piecewise linear (ES-PWL) interpolation, which often results in relatively large approximation errors. In order to reduce the error, this paper proposes a more accurate PWL method for approximating the quadratic cost functions of thermal units. The method employs a linear least-squares fit instead of linear interpolation within each subinterval and introduces a one-terminal-constraint approach to ensure the continuity of the piecewise function. Subsequently, a straightforward equation is derived, applicable to the widely used ES-PWL interpolation, with the potential to enhance the accuracy of the approximation. Mathematical verification attests that the proposed method substantially diminishes the squared 2-norm error, less than 37.5% of the error associated with ES-PWL interpolation. Subsequent numerical investigations are carried out on a 10-unit system, the IEEE RTS-79, and a real industrial system. The findings validate that all the approximation errors of the proposed method are within 37.5% of the errors associated with the ES-PWL interpolation, meaning that a unit commitment solution that closely approximates the outcome of the quadratic function is obtained. The computational time is also acceptable. [ABSTRACT FROM AUTHOR]

The main issue in the present article is to investigate how to solve a mixed integer fractional signomial geometric programing problem (MIFSGP). In the first step to achieving this idea, we must convert a fractional signomial programing problem into a non-fractional problem via a simple conversion technique. Then, a convex relaxation with a new modified piecewise linear approximation with integer break points as a pre-solve method is used to reach an integer global optimum solution. A few numerical examples are included to illustrate the advantages of the proposed method [ABSTRACT FROM AUTHOR]

Assessment of the current state of the near-surface part of the geological environment and understanding of its layered structure play an important role in various scientific and applied fields. The presented work is devoted to the application of phasometric modifications of geoelectric control methods to solve the problem of the detailed complex study of the underground layers of the environment in the process of drilling operations with the use of special equipment. These studies are based on the analysis of variations in phase parameters and characteristics of an artificially excited multiphase electric field to assess poorly distinguishable details and changes in the layered structure of the medium. The proposed method has increased accuracy, sensitivity and noise proofness of measurements, which allows for extracting detailed information about the heterogeneity, composition and stratification of underground geological formations not only in the zone where the drill makes contact with the medium, but also in the entire control zone. This paper considers practical mathematical models of phase images for basic scenarios of drill penetration between the layers of the near-surface part of the geological medium with different characteristics, obtained by means of approximation apparatus based on continuous piecewise linear functions, and also suggests the use of modern machine learning methods for intelligent analysis of its structure. Studying the phase shifts in electrical signals during drilling highlights their value for understanding the dynamics of soil response to the process. The observed signal changes during the drilling cycle reveal in detail the heterogeneity in soil structure and its response to changes caused by drilling. The stability of phase shifts at the last stages of the process indicates a quasi-equilibrium state. The results make a significant contribution to geotechnical science by offering an improved approach to monitoring a layered structure without the need for deep drilling. [ABSTRACT FROM AUTHOR]

Introduction The survivable Capacitated Networks Design Problem (SCNDP for short) is one of the most essential issues in telecommunication and transportation networks. A survivable network is a network that is designed to remain operational in the event of a component(s) failure(s) (cable cuts, electronic failures on switching centers, and so forth). In SCNDP, researchers usually consider survivability in two different ways diversification and reservation. Diversification consists of dividing the demand of source-sink node pairs into two or more node-disjoint (or arc-disjoint) paths, and in reservation after the failure of a component(s) of the network, part of the demand for node pairs can still be satisfied by rerouting. To implement reservation, the links on the network ought to have enough capacity to support the new flows after the rerouting has been made. Therefore, in this paper, we first present one model for the problem of survivable capacitated network design based on the reservation strategy. In many real-world applications, the observed values are often uncertain, for example random, fuzzy, etc. Therefore, we consider the demand vector (the right-hand side values) randomly and then by using joint chance-constrained and probability concepts, obtain the corresponding deterministic model. Then, we propose an approximation optimization approach and use piecewise tangent approximation and piecewise linear methods to obtain its lower and upper bounds, respectively. Finally, computational results show the efficiency of the proposed model and approximation methods for relatively large-scale networks. Material and Methods In many real-world applications, the values of parameters are often uncertain. There are several methods available to tackle this uncertainty, including: 1. Fuzzy programming, 2. Stochastic programming based on scenarios, 3. Robust optimization, 4. Probabilistic programming. In this paper, we will use the probabilistic programming (joint chance-constrained) method to obtain the corresponding deterministic model according to the type of the initial model and assume an independent non-deterministic demand vector. The resulting model is the mixed integer non-convex non-linear model which we will use iterative and approximate piecewise tangent and piecewise linear methods to solve it. Results and discussion In this article, we first presented a model for the survivable capacitated network design problem, and then by considering the demand vector as nondeterministic and assuming the independence of random variables, we presented the corresponding deterministic model by using joint-chance constrained and probability concepts. Because the resulting model is the mixed integer non-convex non-linear model, two approximations were used to solve the problem. The numerical results obtained from GAMS software and the CPLEX solver were reported for medium to large-scale networks. Next, we will examine the initial model under the assumption of dependence between random variables. Conclusion The following conclusions were drawn from this research. • This paper develops a cost-capacity efficient model for the SCNDP with different capacity levels and piecewise linear convex objective function based on the reservation strategy. • We suppose the demand vector can be reported as random variables in the case of independence. To deal with uncertainty, we focus on the joint chance-constrained programming method. • This paper develops a mixed integer non-convex non-linear programming model and finds a lower and upper approximation, using the piecewise tangent approximation and piecewise linear approximation, respectively. [ABSTRACT FROM AUTHOR]

The investigation of fatigue behavior typically involves time‐consuming tests, leading some researchers to explore methodologies based on self‐heating tests to reduce the process. For composite materials, the conventional approach involves piecewise linear approximations of the self‐heating curve and the stress transition between the first and second regimes is arbitrarily associated with a fatigue lifetime equal to 106 cycles. This paper proposes a novel methodology to address these simplifications. First, a non‐linear viscoelastic model is used to describe the self‐heating curve. Based on the mechanisms established in the literature, a link is proposed between the dissipation and the fatigue limit. The load leading to a significant contribution of non‐linear mechanisms is associated with an infinite life time. It allows the identification of an S– N curve and prediction of fatigue behavior through a minimal number of tests. The comparison with fatigue results is satisfactory. Highlights: A link is proposed between dissipation and fatigue.Modeling of the dissipation in laminated composites subjected to cyclic loadings.A criterion is proposed to identify the fatigue limit based on the dissipation.A S– N curve is identified and extrapolated up to high cycles fatigue. [ABSTRACT FROM AUTHOR]

We revisit an approximate stochastic dynamic programming method that we proposed earlier for the optimization of multireservoir problems. The method exploited the convexity properties of the value function to sample the reservoir level space based on the local curvature of the value function, which is estimated by the difference between a lower and an upper bounds (error bound). Unlike the previous approach in which the state space is exhaustively partitioned into full dimensional simplices whose vertices formed a discrete grid over which the value function was approximated, here we propose instead a new randomized approach for selecting the grid points from a small number of randomly sampled simplices from which an error bound is estimated. Results of numerical experiments on three literature test problems and simulated midterm reservoir optimization problems illustrate the advantages of the randomized approach which can solve models of higher dimensions than with the exhaustive approach. [ABSTRACT FROM AUTHOR]

The purpose of the article is to substantiate the directions of health tourism development in Czechia. The purpose of the study is the subjects of health tourism: well-known rehabilitation centers in Czechia, medical institutions, health resorts. The methodological basis was formed by the following methods: abstract-logical; statistical; expert evaluation; cluster analysis; piecewise linear approximation. Among the most popular regions of Czechia for health tourism, the main places are Prague, Moravian-Silesia, Olomouc, Plzeň, South Moravian and South Bohemian. The increase in the number of patients among residents of the country and foreigners who use health tourism services is studied. The five leaders among the countries that used the services are identified. The main problems and advantages of health tourism in Czechia are identified. The Czech tourism development strategy for 2021–2030 is analyzed and directions for its implementation are proposed. Originality / value: the originality of the study lies in forecasting the directions of health tourism development in Czechia using the method of piecewise linear approximation in the following areas: adults with health insurance; adults at their own expense, children and teenagers, which will increase the volume of income for health tourism institutions in Czechia. According to the forecasting results, the demand for health services will increase by 3% annually. The results of this study can be useful for substantiating the directions of further health tourism development in the country, which will ensure the competitiveness and investment attractiveness of the country.

WPREs (wind power ramp events) are one of the most critical factors affecting the security and protection of the electrical system. Accurate ramp event detection may help power systems better manage extreme events and reduce financial damage. In this study, We present an improved piecewise linear approximation for recognizing wind ramps in Kanyakumari district. In practise, wind power ramps can be decreased by properly managing and dispatching flexible reserve and associated services. This necessitates the use of proper ramp detection techniques as well as precise ramp forecasts. The method’s plan to break down wind power signal into increasing with increasing ramps, making ramp identification easier and ensuring that all conceivable ramps of varying lengths are identified. Using observed wind power data, the ramp detection method is used to assess the performance of an energy wind farm. The results reveal that identifying wind power ramps using the segmentation method is equivalent to optical ramp identification.

Stationary operation of a thermoelectric system is often not economical due to the impossibility of soft adjustment of the microclimate. The aspect makes relevant the research of the transient modes of Peltier elements and thermoelectric systems built on their basis. The study of the dynamic properties of thermoelectric devices and systems with high accuracy requires the use of complex high-order models, which makes it difficult to obtain analytical solutions. Using the spectral approach and piecewise linear approximation, analytical formulae of the transient modes of Peltier thermoelectric elements for various deviations of control currents are obtained. The calculated relations of piecewise linear approximating functions, spectral densities and time forms of input and output parameters of the model under study are presented. Time diagrams of the temperature response of a particular Peltier element in linear and nonlinear modes are constructed, calculated for the sinusoidal deviation of the control current from the stationary value. [ABSTRACT FROM AUTHOR]

The purpose of the article is to substantiate the directions of health tourism development in Czechia. The purpose of the study is the subjects of health tourism: well-known rehabilitation centers in Czechia, medical institutions, health resorts. The methodological basis was formed by the following methods: abstract-logical; statistical; expert evaluation; cluster analysis; piecewise linear approximation. Among the most popular regions of Czechia for health tourism, the main places are Prague, Moravian-Silesia, Olomouc, Plzeň, South Moravian and South Bohemian. The increase in the number of patients among residents of the country and foreigners who use health tourism services is studied. The five leaders among the countries that used the services are identified. The main problems and advantages of health tourism in Czechia are identified. The Czech tourism development strategy for 2021-2030 is analyzed and directions for its implementation are proposed. Originality / value: the originality of the study lies in forecasting the directions of health tourism development in Czechia using the method of piecewise linear approximation in the following areas: adults with health insurance; adults at their own expense, children and teenagers, which will increase the volume of income for health tourism institutions in Czechia. According to the forecasting results, the demand for health services will increase by 3% annually. The results of this study can be useful for substantiating the directions of further health tourism development in the country, which will ensure the competitiveness and investment attractiveness of the country. [ABSTRACT FROM AUTHOR]

The functioning of a thermoelectric system in a stationary mode is often inefficient because it does not allow flexible control of the temperature regime. The design of thermoelectric devices and systems in transient modes requires identifying their dynamic models, first of all, the control object – the Peltier thermoelectric module. As a rule, well-known identification techniques involve calculating the parameters of a fractional-rational transfer function (or, equivalently, an autoregressive model) of the object of control. At the same time, the increase in the accuracy of identification requirements is associated with significant computational costs. The proposed identification technique requires the determination of the time dependencies of the control current and temperature deviation, the calculation of their spectra based on piecewise linear approximation, and the calculation of the transfer coefficient as the ratio of the spectra of the output and input signals. The calculated relations of piecewise linear approximating functions, spectral densities, and time forms of input and output parameters of the model under study are presented. The amplitude and phase frequency response of the Peltier module are calculated. The low RMS error in the identification of the amplitude-frequency characteristic and phase-frequency characteristic showed the effectiveness of the proposed identification technique. [ABSTRACT FROM AUTHOR]

Multi-energy systems (MESs) combining different energy carriers like electricity and heat allow for more efficient and sustainable energy solutions. However, optimizing the design and operation of MESs is challenging due to non-linearities in the mathematical models used, especially the performance curves of technologies like combined heat and power units. Unlike similar work from the literature, this paper proposes an improved piecewise linearization method to efficiently handle the non-linearities, models an MES as a multi-objective mixed-integer linear program (MILP), and solves the optimization problem over a year with hourly resolution to enable detailed operation and faithful system design. The method uses fewer linear pieces to approximate non-linear functions compared to a standard technique, resulting in lower complexity while preserving accuracy. The MES design and operation problem maximizes cost reduction and the rate of renewable energy sources. A case study on an MES with electricity and heat over one year with hourly resolution demonstrates the effectiveness of the new method. It allows for solving a long-term MES optimization problem in reasonable computation times. [ABSTRACT FROM AUTHOR]

WPREs (wind power ramp events) are one of the most critical factors affecting the security and protection of the electrical system. Accurate ramp event detection may help power systems better manage extreme events and reduce financial damage. In this study, We present an improved piecewise linear approximation for recognizing wind ramps in Kanyakumari district. In practise, wind power ramps can be decreased by properly managing and dispatching flexible reserve and associated services. This necessitates the use of proper ramp detection techniques as well as precise ramp forecasts. The method's plan to break down wind power signal into increasing with increasing ramps, making ramp identification easier and ensuring that all conceivable ramps of varying lengths are identified. Using observed wind power data, the ramp detection method is used to assess the performance of an energy wind farm. The results reveal that identifying wind power ramps using the segmentation method is equivalent to optical ramp identification. [ABSTRACT FROM AUTHOR]

Because of the COVID-19 situation, selection for a teaching assistant position to get a TA scholarship in a university in Thailand needs to be performed online by the formed committee. Due to the online process and the limited number of scholarships offered by the university, beyond the face-to-face interview, multiple-criteria decision analysis can help to select a proper student. In this study, we use the extended VIKOR method with fuzzy numbers to help committees to select the students from the applicants. The criteria and the weights of the criteria are provided with the help of committees. Both trapezoidal and triangular linguistic variables are used to find the solution and to observe the range of the possible result. The different weights supporting the strategy of maximum group utility are varied to detect the potential alternatives. The ranking results are also compared with the one obtained from the TODIM approach to illustrate the appropriate alternative. [ABSTRACT FROM AUTHOR]

We propose a direct digital frequency synthesizer (DDFS) by using an error-controlled piecewise linear (PWL) approximation method. For a given function and a preset max absolute error (MAE), this method iterates continuously from right to left within the input interval, dividing the entire interval into multiple segments. Within each segment, the least squares method is used to approximate the objective function, ensuring that each segment meets the error requirements. Based on this method, We first implemented a set of DDFS under different MAE to study the relationship between SFDR and MAE, and then evaluated its hardware overhead. In order to increase the frequency of the output signal, we implement a multi-core DDFS using time interleaving scheme. The experimental results show that our DDFS has significant advantages in SFDR, using fewer hardware resources to achieve high SFDR. Specifically, the SFDR of proposed DDFS can reach 114 dB using 399 LUTs, 66 flip flops and 3 DSPs. More importantly, we demonstrate through experiments that proposed DDFS breaks the SFDR theoretical upper bound of DDFS based on piecewise linear approximation methods. [ABSTRACT FROM AUTHOR]

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions. [ABSTRACT FROM AUTHOR]

The exponential function a x is widespread in many fields of science. Its calculation is a complicated issue for Central Processing Units (CPUs) and Graphics Processing Units (GPUs), as well as for specialised Digital Signal Processing (DSP) processors, such as Intelligent Processor Units (IPUs), for the needs of neural networks. This article presents some simple and accurate exponential function calculation algorithms in half, single, and double precision that can be prototyped in Field-Programmable Gate Arrays (FPGAs). It should be noted that, for the approximation, the use of effective polynomials of the first degree was proposed in most cases. The characteristic feature of such algorithms is that they only contain fast 'bithack' operations ('bit manipulation technique') and Floating-Point (FP) addition, multiplication, and (if necessary) Fused Multiply-Add (FMA) operations. We published an article on algorithms for this class of function recently, but the focus was on the use of approximations of second-degree polynomials and higher, requiring two multiplications and two additions or more, which poses some complications in FPGA implementation. This article considers algorithms based on piecewise linear approximation, with one multiplication and one addition. Such algorithms of low complexity provide decent accuracy and speed, sufficient for practical applications such as accelerators for neural networks, power electronics, machine learning, computer vision, and intelligent robotic systems. These are FP-oriented algorithms; therefore, we briefly describe the characteristic parameters of such numbers. [ABSTRACT FROM AUTHOR]

A new O(n) time complexity numerical method for computing the solutions of Basset integro-differential equations is presented. The method is based on the use of Laplace and bilinear transformations, leading after a precise Chebyshev rational interpolation, to the construction of first order infinite impulse response (IIR) filters. The resulting recurrence relations allow for a very accurate O(n) resolution of Basset equations. Different numerical examples are presented in order to demonstrate the applicability of the method to the study of the transient creeping flow motion of a rigid spherical particle. The results are compared on the one hand to those provided by the Talbot inversion method and on the other hand to those provided by an iterative method specially elaborated here to solve the Basset problem.

Economic model predictive control (EMPC) is an effective control strategy for wastewater treatment plants (WWTPs), which are crucial for preventing water pollution and improving the water quality. However, industrial processes typically involve large-scale nonlinear and strongly coupled systems, and it is computationally expensive to solve nonlinear optimization problems based on nonlinear prediction models such as EMPC. In this study, to facilitate the application of EMPC to large-scale nonlinear systems such as WWTPs, the lattice trajectory piecewise linear (PWL) model is used to approximate the nonlinear system with a predefined error bound. The resulting optimization problem can be expressed as a continuous PWL programming problem if the cost function of the EMPC is linear. Therefore, an iterative descent algorithm is proposed to transform the PWL programming problem into a series of linear programming problems. The stability of the nonlinear system operating under EMPC is analyzed. The EMPC scheme based on the lattice trajectory PWL model is applied to a WWTP benchmark problem, in which the state is 78-dimensional and the EMPC cost is linear. The proposed strategy outperforms the EMPC schemes based on the nonlinear prediction model and trajectory PWL prediction model. Overall, the EMPC scheme based on the lattice trajectory PWL prediction model can improve the computational efficiency of optimization problems while ensuring control performance. • Lattice trajectory piecewise linear approximation of the wastewater process. • A novel descent algorithm is proposed for piecewise linear programming problem. • Stability of the nonlinear wastewater process is proved. • The efficacy of the proposed strategy is shown through simulation. [ABSTRACT FROM AUTHOR]

Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an ε -approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error ε , univariate ε -approximations require fewer simplices than bivariate ε -approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation. [ABSTRACT FROM AUTHOR]

Achieving cost-effective hydrogen production is key to facilitating the use of hydrogen as an alternative to fossil fuels. This study adopts a mixed integer linear programming approach to simultaneously derive the optimal sizing and operation of hydrogen generation sites. Waste heat recovery solutions to serve a district heating network were considered, resulting in an 18.9% reduction in hydrogen production costs compared to conventional designs, with up to 5.3% of the reduction attributable to waste heat utilization. A global sensitivity analysis indicated that renewable availability, imported electricity costs, and electrolyzer conversion efficiency are the most influential parameters in reducing total costs. Additionally, the study explores the impact of modeling fidelity on optimization results, revealing that nominal efficiency assumptions commonly used in the literature can lead to sizing errors and cost inefficiencies. Implementing piecewise linear approximations for electrolyzer performance significantly enhances the accuracy of these predictions. Overall, the findings underscore the crucial interconnection between the optimal sizing and operation of hydrogen generation sites, emphasizing that these should not be treated as separate steps in the design process. This work provides a robust foundation for advancing the optimal design of hydrogen generation sites and offers practical recommendations to reduce hydrogen costs. • Concurrent optimization of sizing and operation of a hydrogen generation site. • Waste heat recovery is considered to reduce total cost of the site. • Levelized cost of hydrogen reduced by 18.9% compared to conventional design solution. • Global Sensitivity Analysis identifies most influential parameters. • Neglecting partial load operation impacts the optimal sizing. [ABSTRACT FROM AUTHOR]

In order to improve the performance of a micro-electro-mechanical system (MEMS) accelerometer, three algorithms for compensating its temperature drift are proposed in this paper, including deep long short-term memory recurrent neural network (DLSTM-RNN, short DLSTM), DLSTM based on sparrow search algorithm (SSA), and DLSTM based on improved SSA (ISSA). Moreover, the piecewise linear approximation (PLA) method is employed in this paper as a comparison to evaluate the impact of the proposed algorithm. First, a temperature experiment is performed to obtain the MEMS accelerometer's temperature drift output (TDO). Then, we propose a real-time compensation model and a linear approximation model for neural network methods compensation and PLA method compensation, respectively. The real-time compensation model is a recursive method based on the TDO at the last moment. The linear approximation model considers the MEMS accelerometer's temperature and TDO as input and output, respectively. Next, the TDO is analyzed and optimized by the real-time compensation model and the three algorithms mentioned before. Moreover, the TDO is also compensated by the linear approximation model and PLA method as a comparison. The compensation results show that the three neural network methods and the PLA method effectively compensate for the temperature drift of the MEMS accelerometer, and the DLSTM + ISSA method achieves the best compensation effect. After compensation by DLSTM + ISSA, the three Allen variance coefficients of the MEMS accelerometer that bias instability, rate random walk, and rate ramp are improved from 5.43 × 10 − 4 mg , 4.33 × 10 − 5 mg / s 1 2 , 1.18 × 10 − 6 mg / s to 2.77 × 10 − 5 mg , 1.14 × 10 − 6 mg / s 1 2 , 2.63 × 10 − 8 mg / s , respectively, with an increase of 96.68% on average. [ABSTRACT FROM AUTHOR]

Nonlinear functions are widely used as activation functions in artificial neural networks, which have a great impact on the fitting ability of artificial neural networks. Due to the complexity of the activation function, the computation of the activation function and its derivative requires a lot of computing resources and time during training. In order to improve the computational efficiency of the derivatives of the activation function in the back-propagation of artificial neural networks, this paper proposes a method based on piecewise linear approximation method to calculate the derivative of the activation function. This method is hardware-friendly and universal, it can efficiently compute various nonlinear activation functions in the field of neural network hardware accelerators. In this paper, we use least squares to improve a piecewise linear approximation calculation method that can control the absolute error and get less number of segments or smaller average error, which means fewer hardware resources are required. We use this method to perform a segmented linear approximation to the original or derivative function of the activation function. Both types of activation functions are substituted into a multilayer perceptron for binary classification experiments to verify the effectiveness of the proposed method. Experimental results show that the same or even slightly higher classification accuracy can be achieved by using this method, and the computation time of the back-propagation is reduced by 4–6% compared to the direct calculation of the derivative directly from the function expression using the operator encapsulated in PyTorch. This shows that the proposed method provides an efficient solution of nonlinear activation functions for hardware acceleration of neural networks. [ABSTRACT FROM AUTHOR]

This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of the fin and the surface on which the fin is placed. This work presents an equivalent minimum principle, represented by a convex and coercive functional, ensuring the solution's existence and uniqueness. In order to illustrate the use of the proposed functional for reaching approximations, the heat-transfer process in a trapezoidal fin considering a piecewise linear approximation is simulated. The Appendix presents a case in which the exact solution in a closed form has been achieved. [ABSTRACT FROM AUTHOR]

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the high-order numerical scheme of right Caputo time fractional differential equations with uniform accuracy. Firstly, we construct the high-order finite difference method for the right Caputo fractional ordinary differential equations (FODEs) based on piecewise quadratic interpolation. The local truncation error of right Caputo FODEs is given, and the stability analysis of the right Caputo FODEs is proved in detail. Secondly, the time fractional partial differential equations (FPDEs) with right Caputo fractional derivative is studied by coupling the time-dependent high-order finite difference method and the spatial central second-order difference scheme. Finally, three numerical examples are used to verify that the convergence order of high-order numerical scheme is in time with uniform accuracy. [ABSTRACT FROM AUTHOR]

A new O (n) time complexity numerical method for computing the solutions of Basset integro-differential equations is presented. The method is based on the use of Laplace and bilinear transformations, leading after a precise Chebyshev rational interpolation, to the construction of first order infinite impulse response (IIR) filters. The resulting recurrence relations allow for a very accurate O (n) resolution of Basset equations. Different numerical examples are presented in order to demonstrate the applicability of the method to the study of the transient creeping flow motion of a rigid spherical particle. The results are compared on the one hand to those provided by the Talbot inversion method and on the other hand to those provided by an iterative method specially elaborated here to solve the Basset problem. [ABSTRACT FROM AUTHOR]

We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky (Neural Netw 94:103–114, 2017), formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. Further, we show that our formulation represents feasible points via a Gray code. We close with computational results on problems with quadratic objectives and/or constraints, showing that our proposed method (i) across the board outperforms existing MIP relaxations from the literature, and (ii) on hard instances produces better bounds than exact solvers within a fixed time budget. [ABSTRACT FROM AUTHOR]

This paper presents a computation-efficient stochastic dynamic programming algorithm for solving energy storage price arbitrage considering variable charge and discharge efficiencies. We formulate the price arbitrage problem using stochastic dynamic programming and model real-time prices as a Markov process. Then we propose an analytical solution algorithm using a piecewise linear approximation of the value-to-go function. Our solution algorithm achieves extreme computation performance and solves the proposed arbitrage problem for one operating day in less than one second on a personal computer. We demonstrate our approach using historical price data from four price zones in New York Independent System Operator, with case studies comparing the performance of different stochastic models and storage settings. Our results show that the proposed method captures 50% to 90% of arbitrage profit compared to perfect price forecasts. In particular, our method captures more than 80% of arbitrage profit in three out of the four price zones when considering batteries with more than two-hour duration and realistic degradation cost. [ABSTRACT FROM AUTHOR]

Service Level Agreement (SLA) is an important basis to ensure the availability of cloud service. Cloud service providers seek to not only fulfill the SLA, but also simultaneously minimize the risk uncertainty. Under such a mechanism, service providers look forward to determine the optimal number of backup virtual machines (VMs) to minimize operational costs. In this paper, we propose a data-driven risk measurement method under multi-stage SLA and empirically portray the likelihood and risk values of compensation events at different stages under several service periods. Furthermore, a multi-objective optimization function is established to balance the provisioning cost and risk volatility for cloud service providers with different risk preferences. This paper provides a practical method to cloud service providers by addressing the risk uncertainty in multi-stage SLA from cloud computing resource management perspective. [ABSTRACT FROM AUTHOR]

For a piecewise linear system of conic inequalities, we establish the stability of its global error bound when the piecewise linear vector-valued objective function undergoes small linear perturbations. In particular, our results improve the corresponding ones about linear conic inequalities in the literature. [ABSTRACT FROM AUTHOR]

Featured Application: Data analysis and processing system, Neural network system, Accelerator and Coprocessor. In this paper, we propose an optimized method for nonlinear function approximation based on multiplierless piecewise linear approximation computation (ML-PLAC), which we call OML-PLAC. OML-PLAC finds the minimum number of segments with the predefined fractional bit width of input/output, maximum number of shift-and-add operations, user-defined widths of intermediate data, and maximum absolute error (MAE). In addition, OML-PLAC minimizes the actual MAE as much as possible by iterating. As a result, under the condition of satisfying the maximum number of segments, the MAE can be minimized. Tree-cascaded 2-input and 3-input multiplexers are used to replace multi-input multiplexers in hardware architecture as well, reducing the depth of the critical path. The optimized method is applied to logarithmic, antilogarithmic, hyperbolic tangent, sigmoid and softsign functions. The results of the implementation prove that OML-PLAC has better performance than the current state-of-the-art method. [ABSTRACT FROM AUTHOR]

For including piecewise linear (PWL) functions in MILP problems, the logarithmic convex combination (Log) formulation has been shown to yield very fast solving times. However, identifying approximations that can be used with Log is a big challenge since the approximation has to be compatible with a J1 triangulation. In this article, an algorithm is proposed that identifies approximations using J1 compatible triangulations. It seeks to satisfy the specified error tolerance with the minimum number of linear pieces, so that the MILP formulation is small. To evaluate the performance of the J1 approach it is applied to two sets of benchmark functions from literature and results are compared to state-of-the-art approaches. Overall the J1 approach is shown to efficiently approximate functions in up to 3 dimensions. Especially for tight error tolerances, these J1 approximations require fewer auxiliary variables in MILP compared to alternative approaches. • Identify piecewise-linear approximations of multi-variable functions. • Approximations are compatible with logarithmic MILP formulation. • Evaluation based on two sets of benchmark functions from literature. [ABSTRACT FROM AUTHOR]

• The model is applicable to the peak shaving operation of the retrofitted cascade hydropower station. • Novel linearization methods to enhance the efficiency of model solving. • A 4.6% reduction in the peak-to-valley difference of residual load after retrofitting. • Retrofitting the leading power station enables optimal peak shaving. The integration of pumped storage units with conventional cascade hydropower to form a cascade hybrid pumped storage hydropower station (CHPHPS) is considered one of the effective approaches to expedite the development of pumped storage. However, more complex hydraulic and electrical coupling relationships pose challenges to the operation of these stations. In this study, the typical peak shaving mode of CHPSHS is initially analyzed, and a corresponding peak shaving model is proposed. The objective function of the model is to minimize the peak-valley difference in the remaining load on the grid. In terms of modelling, the refined modelling method using the unit as the scheduling unit is used to successfully reflect the differentiated operating characteristics of different types of units. In terms of model solving, state variables are introduced to facilitate the decoupling and switching of operational states. Furthermore, linearization techniques such as piecewise linear approximation, integration of 0–1 integer variables, and discretization of generation head are utilized to handle nonlinear constraints. The model is converted into a mixed-integer linear programming (MILP) problem and solved using the CPLEX tool with Java programming. A case study conducted in southwest China shows that CHPSHS can enhance the peak shaving ability and reduce the peak-valley difference in the remaining load on the grid, providing valuable reference for retrofitting conventional cascade hydropower stations. [ABSTRACT FROM AUTHOR]

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far-away assumption", and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the L 1 -minimization-based methods and do not recommend such methods due to their similar behavior to L 2 -based methods and more complicated implementations. [ABSTRACT FROM AUTHOR]

Approximation of piecewise linear and generalized functions is an important and difficult problem. These functions are widely used in mathematical modeling of various processes and systems, such as: automatic control theory, electrical engineering, radio engineering, information theory and transmission of signals and images, equations of mathematical physics, oscillation theory, differential equations and many others. The widespread use of such functions is explained by their positive properties. For example, piecewise linear functions are characterized by a simple structure over segments. However, these features also have disadvantages. For example, in the case of using piecewise linear functions, solutions have to be built in segments. In this case, the problem of matching the obtained solutions at the boundaries of the segments arises, which leads to the complication of the research results. The use of generic functions has similar disadvantages. To eliminate shortcomings in practice, one resorts to the approximation of these functions. There are a large number of well-known methods for approximating piecewise linear and generalized functions. Recently, new methods for their approximation have been developed. In this study, an attempt was made to generalize and discuss the existing methods for approximating the considered functions. Particular emphasis is placed on the description of new approximation methods and their applications in various fields of science and technology. The publication-based review discusses the strengths and weaknesses of each method, compares them, and considers suitable application examples. The review will undoubtedly be interesting not only for mathematicians, but also for specialists and scientists working in various applied fields of research. [ABSTRACT FROM AUTHOR]

A new discontinuous Galerkin time-domain (DGTD) method based on generalized sheet transition conditions (GSTCs) for analyzing the transient response of electromagnetic metasurfaces is presented. The discontinuities in electromagnetic fields around the vicinity of the metasurface are modeled by placing virtual edges on both sides of the metasurface. The GSTCs are incorporated in the updating equations of the standard DGTD algorithm by integrating the time-domain GSTCs formula with Galerkin’ s-weighted residual method. The DGTD-GSTC formulation can also simulate a bf piecewise linear approximation of a curved metasurfaces due to the unstructured mesh. This is the first time that DGTD is applied to the simulation of GSTCs. The implementation of this new method is verified by numerical cases. [ABSTRACT FROM AUTHOR]

An adaptive approach to the numerical differentiation of difficult-to-compute functions is considered. Complex dependencies, which are the result of multiple superpositions of functions or the product of various algorithmic processes, are knowingly difficult to study directly. To establish the nature of the behavior of such dependencies, one has to resort to numerical analysis. One of the important characteristics of functions is a derivative, which indicates the direction and rate of change of a dependence. However, with difficult-to-compute functions, the available a priori information is not always sufficient to achieve the appropriate accuracy of the solution by known means. The loss of accuracy occurs due to the accumulation of round-off errors that grow in proportion to the number of calculated values of a function. In this case, it is necessary to pass on to the posterior approach in order to determine the behavior of the function and move away from the scheme of equidistant nodes, relying on an adaptive way of studying the local situation in the domain of the function. This paper implements an adaptive method for finding derivatives of a function with a minimum of restrictive requirements for the class of functions and the form of their assignment. Due to this, the costs of calculating the function have been significantly reduced with the result that their number has been brought to almost the optimal level. At the same time, the amount of RAM used has sharply decreased. There is no need for a preliminary analysis of the problem of establishing the class of the function under study, in the involvement of special functions or transformation of initial conditions for using standard tables of weight coefficients, etc. For research, it is enough to assign a continuous and bounded function on a fixed segment and a minimum step, which is indirectly responsible for ensuring the required accuracy of differentiation. The effectiveness of the proposed method is demonstrated on a number of test examples. The developed method can be used in more complex problems, for example, in solving some types of differential and integral equations, as well as for a wide range of optimization problems in a wide variety of areas of applied analysis and synthesis.

We study extremum norm problems for the terminal state of a linear dynamical system using methods of parameterization of admissible controls. Piecewise continuous controls are approximated in the class of piecewise linear functions on a uniform grid of nodes of the time interval by linear combinations of special support functions. In this case, the restriction of a control of the original problem to the interval induces the same restrictions for the variables of the finite-dimensional problems. The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional minimization problem for a parabola over a segment. For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed approach provides further insights into global resolution of non-convex optimal control problems and is exemplified by some illustrative problems.

Introduction. One of the directions associated with identification, analysis of the shape of objects, their size, orientation, marking and other geometric characteristics is contour analysis. Various methods for contour approximation are described in the literature. The proposed method is based on a well-known method. Its essence lies in the sequential search for possible directions and end points of approximating straight line segments belonging to the contour. The number of approximation nodes should be as small as possible. The calculation is carried out only for the next point of the contour, without returning to check the criterion of approximation to all previous points. The computational complexity of the algorithm is proportional to the number of points in the contour. The purpose of the paper to propose a method of piecewise linear approximation of the contours of objects in images, which will allow to use the parallel computations at all stages of computer processing using vector operations. Results. The paper proposes an improved method for piecewise linear approximation of a closed contour of an object in an image by a polygon, the vertices of which are directly the points of this contour. Approximation criterion: the distance from each point of the approximated section of the contour to the approximating segment should not exceed the approximation error. The method is focused on parallel computing using vector operations. A method for parallel computation of integral vectors of extreme values of a sequence of numbers for the implementation of parallel computations using vector operations at all stages of approximation is also proposed. Conclusions. Methods are proposed that are implemented using vector operations and provide an opportunity to speed up the solution of contour analysis problems, as well as other similar problems in real time. The gain in computing speed is proportional to the amount of data that a vector processor can simultaneously process. The presence of developed subsystems of vector instructions in Intel and ARM processors makes it possible to use the proposed computation methods in practice.