Initial Iteration Point Research Articles

Application of Orthogonal Polynomial in Orthogonal Projection of Algebraic Surface

Point orthogonal projection onto an algebraic surface is a very important topic in computer-aided geometric design and other fields. However, implementing this method is currently extremely challenging and difficult because it is difficult to achieve to desired degree of robustness. Therefore, we construct an orthogonal polynomial, which is the ninth formula, after the inner product of the eighth formula itself. Additionally, we use the Newton iterative method for the iteration. In order to ensure maximum convergence, two techniques are used before the Newton iteration: (1) Newton’s gradient descent method, which is used to make the initial iteration point fall on the algebraic surface, and (2) computation of the foot-point and moving the iterative point to the close position of the orthogonal projection point of the algebraic surface. Theoretical analysis and experimental results show that the proposed algorithm can accurately, efficiently, and robustly converge to the orthogonal projection point for test points in different spatial positions.

Channel capacity optimization based on Riemannian trust region algorithm in IRS-aided communication system

Intelligent Reflecting Surface (IRS) can offer unprecedented channel capacity gains since it can reconfigure the signal propagation environment. We decide to maximize the channel capacity by jointly optimizing the transmit-power-constrained precoding matrix at the base station and the unit-modulus-constrained phase shift vector at the IRS in IRS-assisted multi-user downlink communication. We first convert the resulting non-convex problem into an equivalent problem, then use the alternate optimization algorithm. While fixing the phase shift vector, we can obtain the optimal precoding matrix directly by adopting standard optimization packages. While fixing the precoding matrix, we propose the Riemannian Trust-Region (RTR) algorithm to solve this optimization problem. And the key of the RTR algorithm is the solution of the trust-region sub-problem. We first adopt the accurate solution based on Newton's (ASNT) method to solve this sub-problem, which can obtain the global solution but cannot guarantee that the solution is optimal since the initial iteration point is difficult to choose. Then, we propose the Improved-Polyline (IPL) method, which can avoid the difficulty of the ASNT method and improve convergence speed and calculation efficiency. The numerical results show that the RTR algorithm has more significant performance gains and faster convergence speed compared with the existing approaches.

A problem-based learning proposal to teach numerical and analytical nonlinear root searching methods

Nonlinear equations are usually solved numerically. However, approximated analytical solutions of nonlinear equations are useful as an initial iteration point for numerical methods. Furthermore, these analytical approximations are sometimes quite close to the actual root. In order to teach at undergraduate level how to perform this kind of approximation, we propose problem-based learning with some nonlinear equations extracted from Applied Sciences. We also present a simple method to analytically solve nonlinear equations which are not solvable by standard computer algebra solvers.

Identification of Tire Model Parameters with Artificial Neural Networks

Accurate modeling of tire characteristics is one of the most challenging tasks. Many mathematical models can be used to fit measured data. Identification of the parameters of these models usually relies on least squares optimization techniques. Different researchers have shown that the proper selection of an initial set of parameters is key to obtain a successful fitting. Besides, the mathematical process to identify the right parameters is, in some cases, quite time-consuming and not adequate for fast computing. This paper investigates the possibility of using Artificial Neural Networks (ANN) to reliably identify tire model parameters. In this case, the Pacejka’s “Magic Formula” has been chosen for the identification due to its complex mathematical form which, in principle, could result in a more difficult learning than other formulations. The proposed methodology is based on the creation of a sufficiently large training dataset, without errors, by randomly choosing the MF parameters within a range compatible with reality. The results obtained in this paper suggest that the use of ANN to directly identify parameters in tire models for real test data is possible without the need of complicated cost functions, iterative fitting or initial iteration point definition. The errors in the identification are normally very low for every parameter and the fitting problem time is reduced to a few milliseconds for any new given data set, which makes this methodology very appropriate to be used in applications where the computing time needs to be reduced to a minimum.

Data field application in removing large P-phase arrival picking errors and relocating a mine microseismic event

Seismic source location is a key parameter in the microseismic (MS) monitoring technology, and its accuracy is correlated with the calculation of some other event source parameters (e.g., event magnitude and focal mechanism). The conventional location methods usually take advantages of objective functions based on all arrival-time dataset and solve them through optimization algorithms, while a local optimization may be obtained due to the influence of initial iteration point. Even if a global grid search algorithm is applied, it is likely to obtain a poor location result when there are large P-phase arrival picking errors. Therefore, this study proposed a method to remove large picking errors from P-phase arrival time dataset and relocate the seismic source based on bootstrap sampling selected sub datasets and data field theory. Its basic principles are shown as follows: by using the time difference (TD2) method through the simplex algorithm, seismic sources are located based on sub datasets of P-phase arrival time from multiple bootstrap sampling, and the location mean value of points with the 50 largest potential values in the data field is taken as the approximate location result of a MS event. Furthermore, according to the approximate location, the difference between theoretical travel time of P waves at each sensor and travel time based on the observed arrival time is calculated, and then a threshold value is set to remove large picking errors. Furthermore, by repeating the above steps of bootstrap sampling and data field based location, an accurate relocation is obtained. The synthetic tests and application were based on sensor locations of the Institute of Mine Seismology (IMS) acquisition system settled in the Yongshaba mine (China). Two typical synthetic events were separately set inside and outside the sensor array, and eight blasting events with known locations were treated as test data. These events were located and studied by utilizing the TD2 method, data field based TD2 method (DF1–TD2) and data field based TD2 method with large picking errors removed (DF2–TD2). Results demonstrate that the TD2 method shows a poor location stability when large picking errors are present, while location errors obtained by the DF1–TD2 and DF2–TD2 methods are obviously smaller than that of the TD2 method. Moreover, location results obtained by the DF2–TD2 method with large picking errors removed are superior to those without removing large picking errors. Furthermore, the DF2–TD2 method was applied to locate 300 MS events, the location elevations are well consistent with the stope distribution, showing that the method has a good application prospect. Finally, the data field theory was combined with other location objective functions and P-phase arrival time picked by the Akaike information criterion (AIC) method, and we obtained good location results. In addition, the data field based location results obtained through a grid search algorithm and the simplex algorithm are similar. Therefore, the combinations of data field with different objective functions, automatic P-phase arrival picking methods and iteration algorithms show broad prospects.

Point Orthogonal Projection onto a Spatial Algebraic Curve

Point orthogonal projection onto a spatial algebraic curve plays an important role in computer graphics, computer-aided geometric design, etc. We propose an algorithm for point orthogonal projection onto a spatial algebraic curve based on Newton’s steepest gradient descent method and geometric correction method. The purpose of Algorithm 1 in the first step of Algorithm 4 is to let the initial iteration point fall on the spatial algebraic curve completely and successfully. On the basis of ensuring that the iteration point fallen on the spatial algebraic curve, the purpose of the intermediate for loop body including Step 2 and Step 3 is to let the iteration point gradually approach the orthogonal projection point (the closest point) such that the distance between them is very small. Algorithm 3 in the fourth step plays an important double acceleration and orthogonalization role. Numerical example shows that our algorithm is very robust and efficient which it achieves the expected and ideal result.

Vehicle pose estimation algorithm for parking automated guided vehicle

Parking automated guided vehicle is more and more widely used for efficient automatic parking and one of the tough challenges for parking automated guided vehicle is the problem of vehicle pose estimation. The traditional algorithms rely on the profile information of vehicle body and sensors are required to be mounted at the top of the vehicle. However, the sensors are always mounted at a lower place because the height of a parking automated guided vehicle is always beyond 0.2mm, where we can only get the vehicle wheel information and limited vehicle body information. In this article, a novel method is given based on the symmetry of wheel point clouds collected by 3-D lidar. Firstly, we combine cell-based method with support vector machine classifier to segment ground point clouds. Secondly, wheel point clouds are segmented from obstacle point clouds and their symmetry are corrected by iterative closest point algorithm. Then, we estimate the vehicle pose by the symmetry plane of wheel point clouds. Finally, we compare our method with registration method that combines sample consensus initial alignment algorithm and iterative closest point algorithm. The experiments have been carried out.

Irregular Subarray Tiling via Heuristic Iterative Convex Relaxation Programming

The design of irregular subarrayed planar phased arrays composed of modular tiles is a research area of increasing interest. In this article, a heuristic iterative convex relaxation programming (H-ICRP) framework is proposed. By formulating the tiling of irregular subarray as a binary programming problem, this framework is suitable for arbitrary array aperture and subarray structure. To achieve exact aperture cover, the nonconvex binary constraint is relaxed to a convex approximation and solved through multiple iterations. Furthermore, for better performance of the scanning radiation pattern, the multitask Bayesian compressive sensing (MT-BCS) solver is adopted to find a heuristic initial iteration point instead of randomizing, such that the robustness and effectiveness can be improved remarkably. Good performance and potentialities of the proposed method are verified by comparing with existing approaches in various numerical examples.

An Improved Curvature Circle Algorithm for Orthogonal Projection onto a Planar Algebraic Curve

Point orthogonal projection onto planar algebraic curve plays an important role in computer graphics, computer aided design, computer aided geometric design and other fields. For the case where the test point p is very far from the planar algebraic curve, we propose an improved curvature circle algorithm to find the footpoint. Concretely, the first step is to repeatedly iterate algorithm (the Newton’s steepest gradient descent method) until the iterated point could fall on the planar algebraic curve. Then seek footpoint by using the algorithm (computing footpoint q ) where the core technology is the curvature circle method. And the next step is to orthogonally project the footpoint q onto the planar algebraic curve by using the algorithm (the hybrid tangent vertical foot algorithm). Repeatedly run the algorithm (computing footpoint q ) and the algorithm (the hybrid tangent vertical foot algorithm) until the distance between the current footpoint and the previous footpoint is near 0. Furthermore, we propose Second Remedial Algorithm based on Comprehensive Algorithm B. In particular, its robustness is greatly improved than that of Comprehensive Algorithm B and it achieves our expected result. Numerical examples demonstrate that Second Remedial Algorithm could converge accurately and efficiently no matter how far the test point is from the plane algebraic curve and where the initial iteration point is.

Image quality optimization towards lidar registration based on iterative termination

Image quality optimization is a key technique in image processing, whose goal is to improve image quality by image enhancement or image format transform. This paper aims at optimizing image acquisition using Lidar registration, which can cope with disadvantages of conventional algorithms such as low-resolution. Specifically, we propose an iterative termination optimization strategy based on image quality perception features and local mean estimation. First, fuzzy images with different types and degrees of distortion are incorporated to form a representative natural image set, and feature maps of fuzzy images are extracted by the natural scene statistical method in the spatial domain. Noticeably, the proposed algorithm which performs iterative deblurring operation records the optimal iteration point based on recording the quality value FSIM of the restored image, and calibrates the corresponding feature vector in the sample library with the optimal iteration point (step number). Afterwards, we leverage LME method to implement an estimate of the number of iteration steps. Based on these two steps, the estimation of the initial iterative monitoring point is completed, so that the subsequent adaptive iterative termination work is more purposeful to monitor the defuzzification metric. The optimization operation can be completed faster effectively.

Statistical Process Monitoring with Biogeography-Based Optimization Independent Component Analysis

Independent Component Analysis (ICA), a type of typical data-driven fault detection techniques, has been widely applied for monitoring industrial processes. FastICA is a classical algorithm of ICA, which extracts independent components by using the Newton iteration method. However, the choice of the initial iterative point of Newton iteration method is difficult; sometimes, selection of different initial iterative points tends to show completely different effects for fault detection. So far, there is still no good strategy to get an ideal initial iterative point for ICA. To solve this problem, a modified ICA algorithm based on biogeography-based optimization (BBO) called BBO-ICA is proposed for the purpose of multivariate statistical process monitoring. The Newton iteration method is replaced with BBO here for extracting independent components. BBO is a novel and effective optimization method to search extremes or maximums. Comparing with the traditional intelligent optimization algorithm of particle swarm optimization (PSO) and so on, BBO behaves with stronger capability and accuracy of searching for solution space. Moreover, numerical simulations are finished with the platform of DAMADICS. Results demonstrate the practicability and effectiveness of BBO-ICA. The proposed BBO-ICA shows better performance of process monitoring than FastICA and PSO-ICA for DAMADICS.

A semismooth Newton method for a kind of HJB equation

In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported.

Rescheduling optimization of steelmaking-continuous casting process based on the Lagrangian heuristic algorithm

This study investigates a challenging problem of rescheduling a hybrid flow shop in the steelmaking-continuous casting (SCC) process, which is a major bottleneck in the production of iron and steel. In consideration of uncertain disturbance during SCC process, we develop a time-indexed formulation to model the SCC rescheduling problem. The performances of the rescheduling problem consider not only the efficiency measure, which includes the total weighted completion time and the total waiting time, but also the stability measure, which refers to the difference in the number of operations processed on different machines for the different stage in the original schedule and revised schedule. With these objectives, this study develops a Lagrangian heuristic algorithm to solve the SCC rescheduling problem. The algorithm could provide a realizable termination criterion without having information about the problem, such as the distance between the initial iterative point and the optimal point. This study relaxes machine capacity constraints to decompose the relaxed problem into charge-level subproblems that can be solved using a polynomial dynamic programming algorithm. A heuristic based on the solution of the relaxed problem is presented for obtaining a feasible reschedule. An improved efficient subgradient algorithm is introduced for solving Lagrangian dual problems. Numerical results for different events and problem scales show that the proposed approach can generate high-quality reschedules within acceptable computational times.

A new superlinearly convergent algorithm of combining QP subproblem with system of linear equations for nonlinear optimization

In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly convergent algorithm is proposed. The initial iteration point can be chosen arbitrarily for the algorithm. At each iteration, the new algorithm solves one quadratic programming subproblem which is always feasible, and one or two systems of linear equations with a common coefficient matrix. Moreover, the coefficient matrix is uniformly nonsingular. After finite iterations, the iteration points can always enter the feasible set of the problem, and the search direction is obtained by solving one quadratic programming subproblem and only one system of linear equations. The new algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some numerical results are reported to show that the algorithm is promising.

An Inexact Accelerated Proximal Gradient Method and a Dual Newton-CG Method for the Maximal Entropy Problem

This paper describes an algorithm to solve large-scale maximal entropy problems. The algorithm employs an inexact accelerated proximal gradient method to generate an initial iteration point which is important; then it applies the Newton-CG method to the dual problem. Numerical experiments illustrate that the algorithm can supply an acceptable and even highly accurate solution, while algorithms without generating a good initial point may probably fail.

Novel Node Localization Algorithm Based on Nonlinear Weighting Least Square for Wireless Sensor Networks

Positioning of the location information of wireless sensor network nodes is one of the key issues in wireless sensor network applications. The traditional node positioning method based on least-square algorithm heavily depends on the ranging accuracy, and therefore cannot guarantee high precision. This paper presents a new method for wireless sensor network node positioning based on nonlinear weighting least-square algorithm. Regarding ranging equation error-weighted sum as a whole, this method starts with the initial iteration point of stepwise refinement to explore the optimal solution and further reduces the positioning computational complexity by the simplification of the Taylor equation. Experimental results demonstrate that promising results have been achieved by using this method.

A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization

In this paper, the nonlinear optimization problems with inequality constraints are discussed. Combining the ideas of the strongly sub-feasible directions method and the ɛ-generalized projection technique, a new algorithm starting with an arbitrary initial iteration point for the discussed problems is presented. At each iteration, the search direction is generated by a new ɛ-generalized projection explicit formula, and the step length is yielded by a new Armijo line search. Under some necessary assumptions, not only the algorithm possesses global and strong convergence, but also the iterative points always get into the feasible set after finite iterations. Finally, some preliminary numerical results are reported.

Semismooth Newton and Newton iterative methods for HJB equation

In this paper, some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well-known Hamilton–Jacobi–Bellman equation. By using the slant differentiability introduced by Chen, Nashed and Qi in 2000, a semismooth Newton method is proposed. The method is proved to have monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate. Moreover, an inexact version of the proposed method is introduced, which reduces the cost of computations and still preserves nice convergence properties. Some numerical results are also reported.

A feasible QP-free algorithm combining the interior-point method with active set for constrained optimization

In this paper, by means of “working set” technique for determining the active set and the idea of primal–dual interior-point method, a new feasible QP-free algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only three reduced systems of linear equations with common coefficient matrix. Moreover, the initial iteration point can be at constraint boundary and the coefficient matrix is uniformly nonsingular without the strict complementarity. We also prove that the proposed algorithm obtains global and superlinear convergence. At last, preliminary numerical results are reported.