scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

Quantum Newton’s Method for Solving the System of Nonlinear Equations

SPIN ◽  
2021 ◽  
pp. 2140004
Author(s):  
Cheng Xue ◽  
Yuchun Wu ◽  
Guoping Guo
Keyword(s):  
Quantum Computing ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Data Conversion ◽  
System Of Nonlinear Equations ◽  
Dimensional System ◽  
System Of Linear Equations ◽  
Whole Process

While quantum computing provides an exponential advantage in solving the system of linear equations, there is little work to solve the system of nonlinear equations with quantum computing. We propose quantum Newton’s method (QNM) for solving [Formula: see text]-dimensional system of nonlinear equations based on Newton’s method. In QNM, we solve the system of linear equations in each iteration of Newton’s method with quantum linear system solver. We use a specific quantum data structure and [Formula: see text] tomography with sample error [Formula: see text] to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is [Formula: see text]. Through numerical simulation, we find that when [Formula: see text], QNM is still effective, so the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

Certain efficient iterative methods for bipolar fuzzy system of linear equations

2020 ◽  
Vol 39 (3) ◽  
pp. 3971-3985 ◽  
Author(s):  
Muhammad Saqib ◽  
Muhammad Akram ◽  
Shahida Bashir
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Iterative Methods ◽  
Fuzzy Set ◽  
Fuzzy System ◽  
Linear Equations ◽  
Performance Validity ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Iterative Schemes

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n ×  n BFSLEs into 4n ×  4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs.

Approximate Solution of Generalized Ginzburg-Landau-Higgs System via Homotopy Perturbation Method

2010 ◽  
Vol 65 (4) ◽  
pp. 301-304 ◽  
Author(s):  
Ju-Hong Lu ◽  
Chun-Long Zheng
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Nonlinear Problem ◽  
Homotopy Perturbation Method ◽  
Linear Equations ◽  
Initial Approximation ◽  
System Of Linear Equations ◽  
Arbitrary Degree ◽  
Ginzburg Landau ◽  
Homotopy Perturbation

Using the homotopy perturbation method, a class of nonlinear generalized Ginzburg-Landau-Higgs systems (GGLH) is considered. Firstly, by introducing a homotopic transformation, the nonlinear problem is changed into a system of linear equations. Secondly, by selecting a suitable initial approximation, the approximate solution with arbitrary degree accuracy to the generalized Ginzburg- Landau-Higgs system is derived. Finally, another type of homotopic transformation to the generalized Ginzburg-Landau-Higgs system reported in previous literature is briefly discussed.

Elastic Contact of a Strip Pressed Between Two Cylinders

1968 ◽  
Vol 35 (2) ◽  
pp. 279-284 ◽  
Author(s):  
C. F. Wang
Keyword(s):  
Integral Equation ◽  
Contact Problem ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Thin Layers ◽  
Circular Cylinders ◽  
Elastic Contact ◽  
System Of Linear Equations ◽  
Fourier Cosine ◽  
Cosine Transformation

Methods of solving an integral equation that represents the elastic contact of a strip pressed between two identical cylinders are discussed. Fourier cosine transformation is used to derive the integral equation of the contact problem, and approximation kernels are used to obtain the solutions for the cases of thick and thin layers. The solutions for both cases are given as a truncated series of the weighted Tchebyshev polynomials of the first kind, whose coefficients are determined from a system of linear equations. The problem of two circular cylinders pressing on a strip is given as an example. The nature of two approximate solutions are also briefly discussed.

scholarly journals Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaobin Guo ◽  
Dequan Shang
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Fuzzy Numbers ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Linear Differential

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

scholarly journals Application of Support Vector Regression in Krylov Solvers

2021 ◽  
Vol 5 (5) ◽  
pp. 178-186
Author(s):  
Rehana Thalib ◽  
Maharani Abu Bakar ◽  
Nur Fadhilah Ibrahim
Keyword(s):  
Approximate Solution ◽  
Support Vector Regression ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Initial Guess ◽  
Training Data ◽  
Support Vector ◽  
Lanczos Algorithm ◽  
High Dimensions ◽  
Systems Of Linear Equations

Support vector regression (SVR) is well known as a regression or prediction tool under the Machine Learning (ML) which preserves all the key features through the training data. Different from general prediction, here, we proposed SVR to predict the new approximate solutions after we generated some iterates using an iterative method called Lanczos algorithm, one class of Krylov solvers. As we know that all Krylov solvers, including Lanczos methods, for solving the high dimensions of systems of linear equations (SLEs) problems experiences breakdown which causes the sequence of the iterates is incomplete, or the good approximate solution is never reached. By assuming that some iterates exist after the breakdown, then we could predict what they are. It is realized by learning the previous iterates generated by the Lanczos solvers, which is also called the training data. The SVR is then used to predict the next iterate which is expected the sequence now has similar property as the previous one before breaking down. Furthermore, we implemented the hybrid SVR-Lanczos (or SVR-L) in the restarting frame work, then it is called as hybrid restarting-SVR-L. The idea behind the restarting is that one time running hybrid SVR-L cannot obtain a good approximate solution with small residual norm. By taking one iterate which is resulted by the hybrid SVR-L, putting it as the initial guess, will give us the better solution. To test our idea of prediction of SLEs solutions, we also used the regular regression and compared with the SVR. Numerical results are presented and compared between these two predictors. Lastly, we compared our proposed method with existing interpolation and extrapolation methods to predict the approximate solution of SLEs. The results showed that our restarting SVR-L performed better compared with the regular regression.

scholarly journals Построение трѐхмерного крыловского подпространства для решения линейных систем

2021 ◽  
Vol 70 (2) ◽  
pp. 50-52
Author(s):  
Е.В Богомолова
Keyword(s):  
Approximate Solution ◽  
Krylov Subspace ◽  
Linear Equations ◽  
Three Dimensional ◽  
System Of Linear Equations

The three-dimensional Krylov subspace is considered, which can be used to find an approximate solution of a system of linear equations. The algorithm that implements this method and conditions for its convergence is presented.

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