Piecewise Linear Functions Research Articles

Efficient continuous piecewise linear regression for linearising univariate non-linear functions

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

Multitargets Orientation Technique Based on Reflection Characteristic Analysis Using an Inverse Diffraction Parabolic Equation

In this article, the inverse diffraction parabolic equation (IDPE) model based on the finite difference method is proposed, which is first applied in the multiple nonradiation targets orientation technology. In principle, the electromagnetic signal propagating in the transmission path will produce a reflected signal back to the source end while encountering the discontinuous objects. The distribution of the reflection or refraction intensity is directly associated with the distances and heights of the objects, so the location can be determined by means of analyzing the distribution. Here, according to the profile data of field intensity at the source end, the distribution of backward propagating electromagnetic waves are calculated rapidly by the IDPE. Then, the local extreme searching method is applied to search the coordinate of the convergence point of field intensity and the positions of multiple objects are finally determined. The piecewise linear function is used to model the irregular terrain. The influence of discontinuous terrain slopes on the false alarm probability of objects localization is also analyzed. The results show that the localization accuracy of the IDPE algorithm is affected by multiple factors, such as the radio frequency and sampling interval of field intensity. It is proved that the IDPE is a novel and efficient algorithm for multiple nonradiation targets orientation technology in long-range complicated terrain environment.

Contraction of variational principle and optical soliton solutions for two models of nonlinear Schrödinger equation with polynomial law nonlinearity

<abstract><p>Our study analyzes the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by powerful and comprehensible techniques, such as the variational principle method and the amplitude ansatz method. We will derive the functional integral and the Lagrangian of these equations, which illustrate the system's dynamic. The solutions of these models will be extracted by selecting the trial ansatz functions based on the Jost linear functions, which are continuous at all intervals. We start with the Jost function that has been approximated by a piecewise linear function with a single nontrivial variational parameter in three cases from a region of a rectangular box, then use this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by piecewise linear ansatz function in two cases of the two-box potential, then approximate it by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function, and finally, will be approximated by the tanh function. Also, we utilize the amplitude ansatz method to extract the new solitary wave solutions of the proposed equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. Furthermore, conditions for the stability of the solutions will be submitted. These answers are crucial in applied science and engineering and will be introduced through various graphs such as 2D, 3D, and contour plots.</p></abstract>

Approximate Equivalence of Higher-Order Feedback and Its Application in Chaotic Systems

Based on the feature of piecewise linear (PWL) functions, a nonlinear feedback of a higher-order system can be transformed to an approximately equivalent PWL function so as to ease system implementation in engineering. As an example, a cubic feedback term can be approximately equivalently transformed to be a PWL function. Since the PWL function can be expressed by many simple functions such as signum function and absolute-valued function, the cubic term can be approximately equivalently replaced with these functions. Consequently, the method of approximate equivalence is employed in the JCS-08-13-2022 (JCS) chaotic system for simple circuit design and implementation. In this approach, the widely used multipliers are avoided and the circuits become more economical and also more robust. In this paper, the cubic Chua’s resistor is equivalently approximately replaced by a PWL function. To show the effectiveness of the approximate equivalence, numerical simulations are demonstrated and verified by circuit implementation.

Using infeasible path cuts to solve Electric Vehicle Routing Problems with realistic charging functions exactly within a branch-and-cut framework

The paper investigates the Electric Vehicle Routing Problem with a non-linear concave and strictly monotonic increasing charging function. In the literature, the non-linear charging function is typically approximated by a piecewise linear charging function which does not overestimate the real charging function in any point. As the piecewise linear charging function underestimates the real state-of-charge in some points, such an approximation excludes feasible solutions from the solution space. To overcome this drawback we introduce a new method to determine a piecewise linear charging function overestimating the real charging function in a way that the area between both functions is minimized as well as an adaptation of a known linearization to include the piecewise linear charging function in a branch-and-cut approach. Thereby, we include infeasible solutions in the solution space. To declare them infeasible again we check every integer solution obtained in the branch-and-cut procedure and add an infeasible path cut if the solution is infeasible for the real charging function such that the procedure terminates with an optimal solution for the real charging function. Our approach is evaluated in a computational study in which instances with up to 100 customers were solved to optimality. Moreover, we evaluate the trade-off between a more complex model formulation due to more binary variables if the number of supporting points for the piecewise linear approximation is increased and the higher approximation error if fewer supporting points are used.

A Fully Distributed ADP Algorithm for Real-Time Economic Dispatch of Microgrid

This paper proposes a fully distributed approximate dynamic programming (FD-ADP) algorithm framework for real-time economic dispatch of a microgrid (MG). Firstly, a modified distributed consensus-based algorithm (DCBA) framework with power restriction factors is proposed to distributedly obtain the real-time ED decision of MG containing distributed energy resources with linear cost functions, which improves the convergence stability compared with the traditional DCBA framework. Then, combined with the approximate dynamic programming (ADP) theory, a FD-ADP algorithm is proposed to deal with data uncertainties. To this end, the piece-wise linear (PWL) function is applied to approximate the value function, and a distributed slope update approach is proposed to embed empirical knowledge into the value function while ensuring the distributed property in off-line training. With the well-trained PWL function assisting in decision, the distributed real-time ED strategy of MG with approximate global optimality is obtained by the proposed FD-ADP algorithm during the entire schedule horizon (e.g., 24 hours). Finally, case studies on the modified IEEE 6-bus MG system and IEEE 123-bus MG system demonstrate the superiority of solution convergence, validity of distributed training, and optimality of distributed schedule of the proposed FD-ADP algorithm.

An optimization method for wireless sensor networks coverage based on genetic algorithm and reinforced whale algorithm.

In response to the problem of coverage redundancy and coverage holes caused by the random deployment of nodes in wireless sensor networks (WSN), a WSN coverage optimization method called GARWOA is proposed, which combines the genetic algorithm (GA) and reinforced whale optimization algorithm (RWOA) to balance global search and local development performance. First, the population is initialized using sine map and piecewise linear chaotic map (SPM) to distribute it more evenly in the search space. Secondly, a non-linear improvement is made to the linear control factor 'a' in the whale optimization algorithm (WOA) to enhance the efficiency of algorithm exploration and development. Finally, a Levy flight mechanism is introduced to improve the algorithm's tendency to fall into local optima and premature convergence phenomena. Simulation experiments indicate that among the 10 standard test functions, GARWOA outperforms other algorithms with better optimization ability. In three coverage experiments, the coverage ratio of GARWOA is 95.73, 98.15, and 99.34%, which is 3.27, 2.32 and 0.87% higher than mutant grey wolf optimizer (MuGWO), respectively.

A dual method for polar cuts in disjoint bilinear programming

As one branch of deterministic approaches to disjoint bilinear programming, cutting plane methods are renowned for its ability to systematically reduce the search space by adding cutting planes that are able to cut off regions deemed infeasible or suboptimal. Polar cuts have been widely utilized as a dominating type of cut in terms of deepness. During the establishment of a polar cut, the modified Newton's method is employed to derive the cutting points along the positive or negative extensions of edges emanating from a local solution. Nonetheless, its performance can be further improved along the positive extensions. Drawing inspiration from integer programming, we develop a new approach based on the LP duality theory for this purpose. It re-formulates the original program with a piece-wise linear concave objective function as a single LP. Moreover, we propose a new technique to derive the edges as accommodation to degeneracy. Numerical results show that, by utilizing our newly developed dual method, computing time can be gradually saved as the percentage of generated cutting points along the positive extensions of edges rises.

Smoothing piecewise linear activation functions based on mollified square root functions

Smoothing piecewise linear activation functions based on mollified square root functions

THE MATHEMATICAL MODEL OF TWO-DIMENSIONAL ELECTROMAGNETHIC FIELD

The purpose of this work is to develop a mathematical model of two-dimensional electromagnetic field, i.e. to construct a system of axioms (assumptions) that we unconditionally accept, although they do not correspond to the real electromagnetic field, but approximate it quite well. A mathematical model of two-dimensional electromagnetic field is constructed based on four assumptions: 1. Boundedness: we assume that on an infinite plane there is a bounded convex domain, outside which the field is zero. 2. Existence of nodes: we assume that in the domain there is a finite set of points, called nodes. All calculations of potential values are carried out only at internal nodes․ 3. Existence of neighbors: we assume that all pairs of nodes are divided into two types: neighboring and non-neighboring, and only neighboring nodes interact. 4. Discretization: we assume that the magnetic permeability of the medium is discrete, i.e. is a piecewise constant function. It is shown that within our mathematical model, from Assumptions 1 – 4, it follows that the electromagnetic potential is necessarily a piecewise linear function. It is shown that within our mathematical model, minimization of the energy functional leads to a system of nonlinear equations, the number of equations is equal to the number of internal nodes. It is proved that within our mathematical model, from Assumptions 1 – 4, it follows that for the numerical solution of the resulting system of equations, the optimal mesh is the Delaunay triangulation. The results are illustrated by an example of numerical solution of a field problem for electromagnetic systems with magnetorheological fluid using the FEMM tool. It is shown that the FEMM application package corresponds to our model.

Разработка технологических схем и обоснование параметров крепления восточного борта карьера «Железный» тросово-канатными анкерами на основе формирования и анализа больших данных

Productivity of a mining enterprise is limited by the possibility of its transportation subsystem, which efficiency improvement is an urgent scientific and practical task. Studying the processes of formation and transformation of material flows allows us to clarify the design methodology of continuous haulage systems. The normative methodology of belt conveyors selection is based on treating the material flows as normally distributed random variables, with the point value, i.e. the irregularity coefficient, being used as the main indicator. There are doubts in methodological justification of such an approach. The normal law of distribution in the classical variant adequately describes a random variable changing within infinite limits, and real mine material flows via the conveyor lines are random variables with unilateral constraints. The majority of tasks on finding the design ranges of material flows can be solved using experimentally established distribution functions (or probability density functions) of random quantities of material flows. Based on the publications in recent years, as well as on our own experimental studies, it is proposed to describe material flows by piecewise linear probability density functions, in particular the triangular ones, and to summarize material flows on the basis of general theoretical provisions of classical probability theory. The paper solves a particular problem of finding an analytical solution of the sum of two random material flows defined by the triangular distribution laws and compares the results of numerical integration of the probability densities of the material flows.

Уточнение методов расчета грузопотоков конвейерных транспортных систем угольных шахт

Productivity of a mining enterprise is limited by the possibility of its transportation subsystem, which efficiency improvement is an urgent scientific and practical task. Studying the processes of formation and transformation of material flows allows us to clarify the design methodology of continuous haulage systems. The normative methodology of belt conveyors selection is based on treating the material flows as normally distributed random variables, with the point value, i.e. the irregularity coefficient, being used as the main indicator. There are doubts in methodological justification of such an approach. The normal law of distribution in the classical variant adequately describes a random variable changing within infinite limits, and real mine material flows via the conveyor lines are random variables with unilateral constraints. The majority of tasks on finding the design ranges of material flows can be solved using experimentally established distribution functions (or probability density functions) of random quantities of material flows. Based on the publications in recent years, as well as on our own experimental studies, it is proposed to describe material flows by piecewise linear probability density functions, in particular the triangular ones, and to summarize material flows on the basis of general theoretical provisions of classical probability theory. The paper solves a particular problem of finding an analytical solution of the sum of two random material flows defined by the triangular distribution laws and compares the results of numerical integration of the probability densities of the material flows.

Bifurcation, chaos and multi-stability regions in an asset pricing model with three subsystems

An asset pricing model with two types of chartists and fundamentalists and trend followers is considered, it is driven by a two-dimensional piecewise linear discontinuous map with three subsystems. There are great differences in the dynamic behaviour between expected offset (the expectations of trend followers offset the difference between the expectations of Type 1 traders in bull and bear markets) and expected non-offset. It is proven that there is no chaos in the dynamic of system with expected offset. However, chaos may exist in the dynamic of system with expected non-offset. We present a systematic approach to the problem of analysing the bifurcation phenomena associated with the appearance/disappearance of cycles, which may be related to several bifurcations. The multi-stability regions in parameter plane and related basins of multi-attractors in phase space are investigated. This paper aims to uncover the endogenous law of the unpredictability and excessive volatility in financial markets.

A Unified Algorithm for Penalized Convolution Smoothed Quantile Regression

Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this article, we propose a unified algorithm for fitting penalized convolution smoothed quantile regression with various commonly used convex penalties, accompanied by an R-language package conquer available from the Comprehensive R Archive Network. We perform extensive numerical studies to demonstrate the superior performance of the proposed algorithm over existing methods in both statistical and computational aspects. We further exemplify the proposed algorithm by fitting a fused lasso additive QR model on the world happiness data.

FSR-SPD: an efficient chaotic multi-image encryption system based on flip-shift-rotate synchronous-permutation-diffusion operation

Images are a crucial component in contemporary data transmission. Numerous images are transmitted daily through the open-source network. This paper presents a multi-image encryption scheme that utilises flip-shift-rotate synchronous-permutation-diffusion (FSR-SPD) processes to ensure the security of multiple images in a single encryption operation. The proposed encryption technique distinguishes itself from current multi-image encryption methods by utilising SPD operation and rapid FSR-based pixel-shuffling and diffusion operation. The SPD is a cryptographic technique that involves the simultaneous application of permutation and diffusion methods. The FSR-based process involves the manipulation of pixels through three different operations, namely flipping, shifting, and rotating. In the process of encryption, the image components of red, green, and blue colours are merged into a single composite image. The large image is partitioned into non-overlapping blocks of uniform size. The SPD technique is employed to tackle each specific block. The encryption method is efficient and expeditious as it exhibits high performance with both FSR and SPD procedures. The method employs a single, fixed-type, one-dimensional, piecewise linear chaotic map (PWLCM) for both the permutation and diffusion phases, resulting in high efficiency in both software and hardware. The proposed method is assessed using key space, histogram variance, neighbouring pixel correlation, information entropy, and computational complexity. The proposed method has a much bigger key space than the comparative method. Compared to comparison approaches, the suggested solution reduces encrypted picture histogram variance by 6.22% and neighbouring pixel correlations by 77.78%. Compared to the comparison technique, the proposed scheme has a slightly higher information entropy of 0.0025%. Other multiple-color image encryption methods are more computationally intensive than the suggested method. Computer simulations, security analysis, and comparison analysis evaluated the proposed methodology. The results show it outperforms multiple images encrypting methods.

The price-elastic knapsack problem

This paper introduces the price-elastic knapsack problem (PEKP), an extension of the classic knapsack problem where instead of fixed item characteristics, the weight of each item and the profit from including an item in the knapsack are a function of a parameter, namely the price. PEKP is first formulated as a generic nonlinear optimization problem and three special cases are investigated. First, we show a polynomial-time solvable case. Next, we formulate the case where the item weights are affine-linear functions as a quadratic program. The computational results show that solving the quadratic program to optimality is computationally challenging, and thus, an approach that decomposes the problem into three mixed integer programs is proposed. Similarly, the case where the weights of the items are piecewise-linear functions is investigated and a quadratic formulation is proposed. A solution approach based on decomposing the problem into three mixed integer programs that are solved independently is also proposed. Using randomly generated instances of varying sizes, the computational results show that the proposed decomposition leads to significant computational advantages compared to solving the quadratic program in the cases of affine-linear and piecewise-linear functions.

An approximated dynamic programming model for the supply vessel fleet sizing problem

The offshore oil and gas exploration and production activities heavily rely on a fleet of supply vessels, which play a crucial role in regularly providing necessary provisions from an onshore supply depot to the offshore installations. For Petrobras, the decision-making process concerning the number of supply vessels to be hired for the upcoming two years holds significant economic implications, given that the daily rates for a single supply vessel can soar to tens of thousands of USDs.In this study, we address the challenges of this heterogeneous fleet sizing problem by presenting an innovative approximated dynamic programming (ADP) model. The model takes into account the uncertainties associated with sea conditions and future demand, making it well-suited for real-world applications. To tackle this problem, we explored two types of approximate value functions: a piecewise linear concave value function and a neural network value function. Both methods underwent rigorous testing and demonstrated their capability to effectively address real-world problems. Consequently, our proposed method exhibits promising potential for practical implementation, allowing companies like Petrobras to accurately assess and determine the ideal fleet composition. By introducing this ADP model, we aim to provide valuable insights and solutions that enhance decision-making processes related to supply vessel fleet sizing in offshore operations. Through considering various uncertainties and complexities, our approach offers a robust and reliable framework to support critical economic decisions in this dynamic industry.

Kernel principal component analysis fault diagnosis method based on improving Golden Jackal optimization algorithm

Aiming at the shortcomings of the Golden Jackal optimization algorithm, such as low convergence accuracy and easy falling into the optimal local solution, an improved Golden Jackal optimization algorithm was proposed. First, sine and piecewise linear (SPM) chaotic mapping was introduced to increase the population number to achieve the purpose of initial population diversity. The self-adaptive weight and sine–cosine algorithm improved the position update formula of the Golden Jackal optimization algorithm, so the global search ability of the golden jackal algorithm is improved, and avoid the algorithm that fell into local optimality. Second, simulation experiments with eight standard test functions are performed to prove that the algorithm has excellent optimization ability. The improved Golden Jackal optimization algorithm was applied to optimize the kernel parameters of hybrid kernel principal component analysis. A fault diagnosis model is proposed to improve the golden jackal algorithm to optimize the kernel principal component analysis. Finally, the proposed method is used to fault diagnosis in the hot strip mill process. According to the study of simulation results, the faulty data can be identified effectively by this method, the accuracy is up to 100%, and the fault false alarm rate is greatly reduced.

A new analytical method for estimating hydraulic parameters of a nonlinear consolidated aquitard with variable drawdown in adjacent aquifers

Estimation of hydraulic parameters and groundwater depletion of aquitards is necessary to evaluate the groundwater resources and land subsidence. In this study, an analytical solution for water release from a nonlinear consolidated aquitard was derived by approximating the drawdown history as a continuous piecewise linear drawdown function (PLD) with several stages, while the water level in adjacent aquifers varies arbitrarily over time. According to dimensionless and parameter sensitivity analyses, groundwater depletion of an aquitard increases with an increase in compression index, aquitard thickness, and water drawdown rate in adjacent aquifers, decreases with an increase in initial void ratio and effective stress, and is not affected by the consolidation coefficient. In addition, we proposed a type-curve fitting method to calculate the hydraulic parameters of the aquitard under the condition of PLD in adjacent aquifers. The proposed method was applied to estimate the hydraulic parameters of aquitards at two field sites in the Yangtze River Delta in China, i.e., Changzhou and Shanghai. Results show that the compression index of the aquitards in Changzhou and Shanghai are 0.074 and 0.24, respectively, while the consolidation coefficients are 7.62 × 10-7 m2/s and 2.16 × 10-6 m2/s, respectively. The hydraulic conductivity and specific storage estimated by the proposed method are in good agreement with previous studies and the results of the geological method of Konikow and Neuzil (2007). The proposed analytical method for estimating hydraulic parameters and groundwater depletion of aquitards will contribute to the sustainable management of groundwater resources in multi-layer aquifer systems.

Trajectories associated with parabolic and hyperbolic periodic points of piecewise linear area-preserving map

This paper investigates the characteristics of trajectories and invariant sets associated with parabolic and hyperbolic periodic points of a piecewise linear area-preserving map. When the fixed points in both partitions are parabolic, the trajectories may diverge or rotate repeatedly around the origin. By analyzing the trajectory motion in different regions of the phase plane, the characteristics of the trajectory repeated rotation around the origin are shown, and the trajectories may be either divergent or periodic. The necessary conditions for generating stable periodic orbits are presented. It is proved that there are forward invariant sets and that the trajectories from the invariant sets diverge in different quadrants and along a family of invariant lines parallel to the diagonal bisector of the quadrant. We reveal why there are island chains with invariant line segments in a chaotic sea. The method for determining invariant line segments is presented. For the map with two hyperbolic fixed points in both partitions, it is proved that there are forward invariant sets in the phase plane and all trajectories from the invariant sets diverge along a family of hyperbolic invariant sets.