Borwein algorithm The paper extends the study of the modified Borwein method for the calculation of the Riemann zeta-function. D. The PBB algorithm has the characteristics The algorithm is line-search-free and parameter-free, and essentially provides a convergent variant of the Barzilai-Borwein method for general unconstrained convex optimization. It significantly improves the Mar 1, 2019 · The image reconstruction algorithms can be divided into two categories, that is, non-iterative and iterative algorithms. For a massive multiple-input multiple-output (MI-MO) system, how to cope with the DOI: 10. The best known such algorithms are the Archimedes algorithm, which was derived by Pfaff in 1800, and the Brent-Salamin formula. The theoretical and experimental results demonstrate that our approach has an order-of-magnitude lower complexity with no matrix inversion, a near-optimal Apr 10, 2017 · The Barzilai–Borwein (BB) method, an effective gradient descent method with clever choice of the step length, is adapted from nonlinear optimization to Riemannian manifold optimization. The Barzilai and Borwein subproblem is transformed into a multi-block separable convex quadratic second-order cone programming (CQSOCP) subproblem. Apr 28, 2022 · Preprocessing Barzilai-Borwein algorithm 图8. 1007/s11227-023-05500-z Hamed Jalilian Nov 5, 2015 · The present study is an attempt to extend Barzilai and Borwein’s method for dealing with unconstrained single objective optimization problems to multiobjective ones. 5, p. In the general non-quadratic case, a globalization strategy based on non-monotone line search is suitable to Barzilai-Borwein gradient method. Second we use integral calculus to prove that the Brent-Salamin algorithm approximates $π$. PI. Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different. 1109/NTCI60157. H. In other cases, we choose k= BB k, which results in kx k+1 x kk . Combining These algorithms out perform the standard methods based on Euler-Maclaurin summation, are easier to implement and are easier to analyse. Contribute to BrolanJ/Bailey-Borwein-Plouffe development by creating an account on GitHub. 2018. Jul 28, 2021 · A Barzilai–Borwein gradient algorithm is designed for solving the spatio-temporal internet traffic tensor completion problem. L 1 regularization is used to solve the inverse problem. Sep 1, 2021 · A Barzilai–Borwein gradient algorithm is designed for solving the spatio-temporal internet traffic tensor completion problem. Jan 16, 2024 · 在本文中，我们提出了 AdaBB，一种基于 Barzilai-Borwein 步长的自适应梯度方法。该算法是无线搜索和无参数的，本质上为一般无约束凸优化提供了 Barzilai-Borwein 方法的收敛变体。我们分析了目标函数值的遍历收敛性和求解一般无约束凸优化的迭代 Mar 13, 2017 · The Riemannian Barzilai-Borwein algorithm 3 where h;i x is the Riemannian scalar product in T xMand Df(x) : T xM!T xMis the di er- ential of fat x. Aug 30, 2021 · Jonathan Borwein is perhaps best known for deriving, with his brother Pe-ter, quadratically and higher order convergent algorithms for , including p-th order convergent Feb 21, 2018 · We show that an iteration of the Borwein-Borwein quartic algorithm for $\pi$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $\pi$, in the sense Nov 5, 2023 · David Bailey, Peter Borwein and Simon Plouffe. Many of their results are available in: Dec 2, 2020 · Borwein’s algorithm is a calculation contrived by Jonathan and Peter Borwein to compute the estimation of 1/π. One of their algorithms is the following: Set a Borwein's algorithm was devised by Jonathan and Peter Borwein to calculate the value of . We have tested and compared our algorithms with the projected gradient method of Lin on a variety of randomly generated NMF problems. Moreover, a nonmonotone line search technique Aug 18, 2023 · This paper proposes to use the stabilized Barzilai-Borwein (SBB) method to automatically compute step size for iSARAH algorithm, which leads to a new algorithm called iSARAH-SBB, which can take better advantages of both iSARAH and SBB methods. Let x = (x{sub 1}, x{sub 2} {hor_ellipsis}, x{sub n}) be a vector of real or complex numbers. def bailey_borwein_plouffe(digit_position: int, precision: int = 1000) -> str: """ Implement a popular pi-digit-extraction algorithm known as the. In Mar 4, 2008 · The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. 017 Corpus ID: 44158058; Random Barzilai-Borwein step size for mini-batch algorithms @article{Yang2018RandomBS, title={Random Barzilai-Borwein step size for mini-batch algorithms}, author={Zhuang Yang and Cheng Wang and Zhemin Zhang and Jonathan Li}, journal={Eng. The Brent-Salamin formula is a quadratically converging algorithm. The Inexact SARAH (iSARAH) algorithm as a variant of SARAH algorithm, which does not require Jan 12, 2025 · Here is a Python Program for the Bailey-Borwein-Plouffe spigot algorithm. Sep 1, 2020 · Request PDF | A faster path-based algorithm with Barzilai-Borwein step size for solving stochastic traffic equilibrium models | Step size determination (also known as line search) is an important Feb 28, 2022 · A Barzilai-Borwein type algorithm with an alternating step-length, , is employed. Bailey, The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm,Mathematics of Computation 42 (1988), 283–296. Author links open overlay panel Chongyang He, Yiting Zhang, Dingyu Zhu We propose two stochastic optimization methods based on the competitive Barzilai-Borwein (BB) step size in both the inner and outer loops of the mini-batch semi-stochastic gradient descent In this article, to solve the abovementioned problems, we propose a new fast $\ell _{1/2}$ sparse NMF algorithm, called NewSpr-NMF, which combines two famous fast gradient strategies, i. It utilizes the sparsity and gradient of the solution to transform large-scale problems into small-scale subproblems, and uses the FISTA algorithm to solve subproblems. This algorithm is an iterative method and it evaluates function and gradient values of the nonlinear objective function in each iteration. The Barzilai-Borwein(BB) method is used in the stochastic gradient descent (SGD) and other deep learning optimization algorithms because of its outstanding performance in terms of convergence speed. In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. Start out by setting: a_0 = 6 - 4sqrt{2}: y_0 = sqrt{2} - 1 Then iterate Jun 1, 2018 · Inspired by recent works Sopyła and Drozda, 2015, Tan et al. The only difference between SVRG and SVRG-BB is that in the latter we use BB method to compute the step size k, instead of using a prefixed as in SVRG. Unlike the existing Nov 9, 2023 · In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. However there is a problem that I couldn't understand and couldn't find any proof or explaination. The analysis begins with the assumption that the gradient norms at the first two iterations are fixed. 1). In Nov 1, 2022 · In this paper, we introduce a modified Barzilai-Borwein algorithm for solving the generalized absolute value equation Ax+B|x|=b. Borwein brothers suggested two sequences $\{x_n\}_{ n\ge 0 Jul 11, 2022 · In this paper, an easy to implement and efficient algorithm is proposed for large-scale sparse least squares problems. The purpose of this work is to Jan 1, 2022 · Preprocessing Barzilai-Borwein algorithm 图8. Numerical experiments show that our algorithms are more efective when compared with some existing conjugate gradient methods. The most prominent and oft-used one is explained under the first section. The adaptive Barzilai-Borwein (BB) step sizes [33], as a simple and effective method, Jul 9, 2019 · This paper consists of three independent parts: First we use only elementary algebra to prove that the quartic algorithm of the Borwein brothers has exactly the same output as the Brent-Salamin algorithm, but that the latter needs twice as many iterations. One of the popular mathematical metho ds in the field of image reconstruction is modeling In this paper, we propose a signal detection approach based on an optimized Barzilai- Borwein (BB) algorithm whose initialization is tuned using symmetric successive over-relaxation (SSOR). The BBP formula allows us to find the n-th digit (or a short block of digits near the n-th) in the binary representation of ˇfaster algorithm for computing ˇto arbitrary precision, and used the first few iterations to show that 3:1408 ˇ310 71 <ˇ<3 1 7 ˇ3:1429: Nov 1, 2023 · A mini-batch algorithm for large-scale learning problems with adaptive step size. This paper introduces eight kinds of negative gradient algorithms, compares them according to their characteristics, calculates strictly convex quadratic functions Aug 11, 2018 · general objective unctions. [26] proposed a stabilized BB algorithm (BBSTAB) by using a new method to limit the BB step size to avoid the BB method generating overly long Sep 4, 2015 · Let \(k:=k+1\), and go to Step 1. They published a book: Mar 17, 2020 · Borwein, Borwein and Bailey [15, pg. 2023. However, the numerical performances of the algorithm heavily depend on M, one of the parameters in the nonmonotone line search, even for ill-conditioned problems. Semantic Scholar's Logo. A formal description of our stabilized BB algorithm follows. 8977151 Corpus ID: 211058574; An optimized Low-Complexity Barzilai-Borwein Algorithm for Signal Detection in Massive MIMO Systems @article{Tan2019AnOL, title={An optimized Low-Complexity Barzilai-Borwein Algorithm for Signal Detection in Massive MIMO Systems}, author={Mingshan Tan and Xiaofeng Qiao}, Mar 25, 2023 · BB方法 ，即Barzilai-Borwein (BB) method 是梯度下降方法的一种，他主要是通过近似牛顿方法来实现更快的收敛速度，同时避免计算二阶导数带来的计算复杂度： Jul 1, 2023 · The training effect of the neural network model is directly influenced by optimization algorithms. Finally, numerical results demonstrate the feasibility and efficiency of the proposed algorithm and overmatch some It is shown that an iteration of the Borwein-Borwein quartic algorithm is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for \\(\\pi), in the sense that they produce exactly the same sequence of approximations to \\(\\pi \\) if performed using exact arithmetic. Nov 23, 2024 · The algorithm corresponding to the BBP formula is a spigot algorithm, but it is more than just a spigot algorithm, it is a digit extraction algorithm. The polylogarithm Li_n(z), also known as the Jonquière's function, is the function Li_n(z)=sum_(k=1)^infty(z^k)/(k^n) (1) defined in the complex plane over the open unit disk. The Hurwitz zeta function may be defined as Dec 4, 2020 · Although these series only converge for , the Riemann and Hurwitz zeta functions are defined through analytic continuation for arbitrary complex (is a pole). The BB gradient method possesses the R-superlinear convergence for two-dimensional convex quadratics. For example, Raydan [17] designed an efficient global Barzilai and Borwein algorithm Nov 3, 2022 · Barzilai and Borwein [8] proved that the above algorithm is R-superlinearly convergent for the quadratic case. The implementation uses three algorithms: the Borwein algorithm for the Riemann zeta function when \(s\) is close to the real line; the Riemann-Siegel formula for the Riemann zeta Jan 1, 2023 · To obtain the numerical solution of the GAVE (1), based on the methodology of Barzilai-Borwein algorithm, in this paper, we propose a modified Barzilai-Borwein (MBB) algorithm for solving the GAVE (1). B. Jan 13, 2011 · We are interested in locations of multiple facilities in the plane with the aim of minimizing the sum of weighted distance between these facilities and regional customers, where the distance between a facility and a regional customer is evaluated by the farthest distance from this facility to the demand region. They conceived a few different algorithms. More generally, global convergence of a nonmonotone line search strategy for Riemannian optimization algorithms is proved under some standard assumptions. At each iteration, the generated search direction demonstrates descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the 1-norm. This class of formulas are now known as BBP-type formulas. Aug 31, 2021 · In this paper, a new alternating nonmonotone projected Barzilai–Borwein (BB) algorithm is developed for solving large scale problems of nonnegative matrix factorization. Contribute to TheAlgorithms/Python development by creating an account on GitHub. The algorithm’s sensitivity is tested by choosing different momentum factors until a suitable momentum factor is found. 1109/ICICM48536. 10403676 Corpus ID: 267339792; An Improved Iterative Reduction FISTA-Barzilai-Borwein Algorithm for Large-Scale LASSO @article{Ma2023AnII, title={An Improved Iterative Reduction FISTA-Barzilai-Borwein Algorithm for Large-Scale LASSO}, author={Runxuan Ma and Lijun Xu and Kai Qu}, journal={2023 International Conference on May 18, 2020 · This paper presents a new projected Barzilai–Borwein method for the complementarity problem over symmetric cone by applying the Barzilai–Borwein-like steplengths to the projected method. Borwein, and S. In order to improve the stability of BB step Nov 1, 2022 · In the first order reliability method (FORM), the Hasofer–Lind and Rackwitz–Flessler (HL–RF) algorithm sometimes encounters numerical instability problems due to the highly nonlinear limit state function (LSF). Algorithm Oct 17, 2006 · The 'Bailey-Borwein-Plouffe' (BBP) algorithm for is based on the BBP formula for , which was discovered in 1995 and published in 1996 [3]: = {sub k=0}{sup } 1/16{sup k} (4/8k+1 - 2/8k+4 - 1/8k+5 This paper focuses on the computational contribution in this topic by developing a variant of the classical Barzilai-Borwein (BB) gradient method to solve the reduced CWPs. It An active set Barzilar–Borwein algorithm is proposed and it is proved that any limit point of the sequence generated by the algorithm is a strong stationary point. Its definition on the whole complex plane then follows uniquely via analytic continuation. Jul 15, 2019 · The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems. Bailey∗ September 17, 2006 1. Raydan [] extended the method to unconstrained optimization by incorporating the nonmonotone line Mar 7, 2008 · Download a PDF of the paper titled Easy Proofs of Some Borwein Algorithms for $\pi$, by Jesus Guillera. , 2016, Bordes et al. 预处理Barzilai-Borwein 算法 从图中可以看出，条件数对一般的负梯度算法影响较大，而BB算法更稳定，在解决大规模问题时更有竞争力。BB-1与BB-2算法相比，后者在矩阵维数较小时没有明显的优势，但随着矩阵维 Sep 14, 2019 · In this paper, we develop an active set identification technique for the $$\\ell _0$$ ℓ0 regularization optimization. Apr 30, 2015 · Borwein's algorithm. 2019. We are interested in locations of multiple facilities in the plane with the aim of minimizing the sum of weighted distance between these facilities and regional customers, where the distance between a facility and a Jun 29, 2023 · Total variation method based on modified Barzilai–Borwein algorithm to noise reduction in MRI The Journal of Supercomputing ( IF 3. 289] confirm this (28 hours for Algorithm BB4 versus 40 hours for Algorithm BB1). We will see that by using only a formula of Gauss's and elementary Mar 7, 2023 · Although these series only converge for \(\Re(s) > 1\), the Riemann and Hurwitz zeta functions are defined through analytic continuation for arbitrary complex \(s \ne 1\) (\(s = 1\) is a pole). They devised several other algorithms. Unlike the existing algorithms available in the literature, a nonmonotone line search strategy is proposed to find suitable step lengths, and an adaptive BB spectral Nov 11, 2021 · A Fast Barzilai-Borwein Gradient Projection for Sparse Reconstruction Algorithm Based on 3D Modeling: Application to ERT Imaging Abstract: Image reconstruction for electrical resistance tomography (ERT) is an ill-posed inverse problem. , 2009, Byrd et al. . We analyze the ergodic convergence of the objective function value and the convergence of the iterates for solving general unconstrained convex optimization. Nov 23, 2018 · Our SVRG-BB algorithm is described in Algorithm 1. Most search results for "spigot algorithm" are about the algorithm for π from Rabinowitz and Wagon, which is not a digit extraction algorithm. The 'Bailey-Borwein-Plouffe' (BBP) algorithm for {pi} is based on the BBP formula for {pi}, which was discovered in 1995 and published in 1996 [3]: {pi} = {summation}{sub k=0}{sup {infinity}} 1/16{sup k} (4/8k+1 - 2/8k+4 - 1/8k+5 - 1/8k+6). The point of view of the Borweins is explained in [], and their proofs require a good knowledge of the elliptic modular Aug 9, 2018 · We show that an iteration of the Borwein-Borwein quartic algorithm for p is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for p, in the sense that they produce exactly the same sequence of approximations to p if performed using exact arithmetic. Both algorithms are superior to the simple Taylor’s series or direct summation. Numerical experiments on Abilene and GÉANT datasets report that the proposed tensor Jan 8, 2024 · utilizing a modied Barzilai–Borwein method (MBBM) algorithm. 3) Here BB BB k = BB 1 k or k = 2 k, depending on the speci c BB method in (1. It reduces the original large-scale problem into a sequence of small-scale subproblems by utilizing the “sparsity” of the gradient. Nov 11, 2016 · Main features of the Barzilai-Borwein (BB) method • The BB method was published in a 8-page paper1 in 1988 • It is a gradient method with modified step sizes, which are motivated by Newton’s method but not involves any Hessian • At nearly no extra cost, the method often significantly improves the performance of a standard gradient method Sep 1, 2021 · In this paper, a new alternating nonmonotone projected Barzilai–Borwein (BB) algorithm is developed for solving large scale problems of nonnegative matrix factorization. 1 (i) The \(\lambda _k^{BB}\) in Step 7 is the BB steplength and is introduced by Barzilai and Borwein in []. x is said to The iterative reduction FISTA algorithm (IR-FISTA-B) is an extremely excellent algorithm for solving large-scale LASSO problems. engappai. In this paper, we improve the algorithm by using the Barzilai-Borwein step In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. Key words: Barzilai-Borwein gradient method, quasi-Newton equation, BF-GS method, adaptive method Mar 11, 2023 · 优化算法直接影响神经网络模型的训练效果。Barzilai-Borwein(BB)方法因其在收敛速度方面的突出表现而被用于随机梯度下降(SGD)和其他深度学习优化算法中。为了提高深度学习中BB步长的稳定性和网络模型的输出质量，本文提出了两种基于BB方法的 DOI: 10. By applying the well-known location-allocation heuristic, the Jan 1, 2025 · This section introduces a new three-term conjugate Barzilai-Borwein algorithm to search the MPP. Alert. thereby reducing the efficiency of the algorithms. Numerical experiments on Abilene and GÉANT datasets report that the proposed tensor Feb 21, 2018 · In a recent work [6], Borwein and Borwein derived a class of algorithms based on the theory of elliptic integrals that yield very rapidly convergent approximations to elementary constants. 预处理Barzilai-Borwein 算法 从图中可以看出，条件数对一般的负梯度算法影响较大，而BB算法更稳定，在解决大规模问题时更有竞争力。BB-1与BB-2算法相比，后者在矩 Liu et al. Note that the similar notation Li(z) is used for the logarithmic integral. As compared with Newton, Quasi-Newton and steepest descent multi-objective optimization methods, Barzilai and Borwein multiobjective optimization (BBMO) method requires simple and quick . Aug 16, 2017 · In the mid 1980s, the Borwein brothers observed that the modular equations given by Ramanujan could be connected to the number \(\pi \) via the elliptic alpha function, to construct extraordinarily rapid algorithms to calculate \(\pi \) [6, Chap. DOI: 10. , 2016, instead of using a diminishing step size or a tuning step size by hand in the mini-batch algorithms, we equip the state of the art mini-batch algorithm, mS2GD , with the ability to automatically compute step size by using the improved CodePal. Sep 1, 2021 · In this paper, a new alternating nonmonotone projected Barzilai–Borwein (BB) algorithm is developed for solving large scale problems of nonnegative matrix factorization. Based on the identification technique, we propose an active set Barzilar–Borwein algorithm and prove that any limit point The iterative reduction FISTA algorithm (IR-FISTA-B) is an extremely excellent algorithm for solving large-scale LASSO problems. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $\ell_1$-norm. in general, there is a paucity of guidance for making good choices. Search Oct 18, 2007 · The BBP Algorithm for Pi David H. This approach (known as the Projected Alternating Barzilai-Borwein or PABB algorithm) is based on recent work by Dai and Fletcher [8] who have investigated a variant of the PABB method. Bailey, P. Oct 18, 2022 · formance of mini-batch algorithms can vary significantly based on the choice of the step size sequence, and, in general, there is a paucity of guidance for making good choices. Jan 1, 2023 · Semantic Scholar extracted view of "A modified Barzilai-Borwein algorithm for the generalized absolute value equation" by Shuang Yang et al. Borwein, Pi and the AGM - A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987. Jonathan Borwein is perhaps best known for deriving, with his brother Pe-ter, quadratically and higher order convergent algorithms for ˇ, including p-th order convergent algorithms for any prime p, and similar algorithms for certain other funda-mental constants and functions [12, 13, 15]. Please cite this paper if you use this code in Feb 15, 2023 · [4], the Barzilai–Borwein (BB) method often requires less computational work and speeds up the convergence greatly. In this paper, a new alternating nonmonotone projected Aug 25, 2017 · generated by the SPBB algorithm is a stationary point associated with the smoothing function used in the algorithm. Linear back-projection (LBP) and Landweber iterative algorithm is the most commonly used algorithms of each category respectively, which are selected to make comparison with the presented sparse image reconstruction algorithm. It is related to rate of convergence of the algorithm. Sep 5, 2018 · 该项工作将Barzilai-Borwein算法引入大规模MIMO检测，并提出了改进版本最陡下降Barzilai-Borwein（SDBB）算法。 该算法可以很好地应用于5G的大规模MIMO系统，在负载因数较大时，仍可以提供出色的检测性能及较低的复杂度。图示为论文标题及核心仿真 Jun 29, 2023 · Yang S, Wu S-L (2022) A modified Barzilai–Borwein algorithm for the generalized absolute value equation. Skip to search form Skip to main content Skip to account menu. Aug 25, 2017 · analyzes, and tests a Barzilai–Borwein gradient algorithm. Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. The first part primarily seeks a reference point close to the limit state surface in a reliable and fast manner. Author links open overlay panel Xuexue Zhang a, Sanyang Liu a, Nannan Zhao b. It Nov 5, 2023 · David Bailey, Peter Borwein and Simon Plouffe. [25] provided a new adaptive step range for BB step calculation to accelerate the convergence speed of BB, and proposed a new adaptive Barzilai and Borwein algorithm (NABB). Jan 29, 2024 · The project Barzilai-Borwein (PBB) algorithm is regarded as a popular and effective method for solving which was originated by Barzilai and Borwein . We discuss the global convergence of the proposed algorithm under Sep 14, 2019 · Based on the identification technique, we propose an active set Barzilar–Borwein algorithm and prove that any limit point of the sequence generated by the algorithm is a strong stationary point Dec 1, 1997 · The Mathematical Intelligencer - D. Search 223,100,932 papers from all fields of science. A feature of the BB method is that it may generate too long steps, Nov 3, 2023 · The iterative reduction FISTA algorithm is improved by using the Barzilai-Borwein step size as the step factor in the inner loop algorithm, and appropriate parameter update criteria to accelerate the convergence rate of the algorithm. We consider some of Jonathan and Peter Borweins’ contributions to the high-precision computation of These algorithms out perform the standard methods based on Euler-Maclaurin summation, are easier to implement and are easier to analyse. We show that there is a Apr 1, 2016 · It is shown that an iteration of the Borwein-Borwein quartic algorithm is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for \(\pi), in the sense that they produce exactly the same sequence of approximations to \(\pi \) if performed using exact arithmetic. This paper presents a new analysis of the BB method for two-dimensional strictly convex quadratic functions. The Ultimate Coding Companion Our mission is to revolutionize coding by making it easy, joyful, and accessible to anyone. Unlike the existing algorithms available in the literature, a nonmonotone line search Jan 20, 2023 · It essentially generalizes an algorithm given by Peter Borwein for computing the Riemann zeta function. The Hurwitz zeta function already includes a great many useful special cases and extensions. This and other algorithms can be found in the book Pi and the AGM – A Amazingly, this formula is a digit-extraction algorithm for in base 16. IEEE (opens in a new tab) Save. Examples and discussions. Higher Order Transformations. Conghui Tan, Shiqian Ma, Yu-Hong Dai, Yuqiu Qian. A Barzilai and Borwein scaling conjugate gradient method Jun 15, 2016 · We present a smoothing projected Barzilai–Borwein (SPBB) algorithm for solving a class of minimization problems on a closed convex set, where the objective function is nonsmooth nonconvex, perhaps even non-Lipschitz. 1016/j. Download PDF Abstract: In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. Mar 22, 2023 · All Algorithms implemented in Python. Oper Res Lett. Theta Functions and the Arithmetic-Geometric Mean Iteration. Appl. In principle, a merges quartic ally to 1/π. In this paper, an improved HL–RF algorithm introducing the hybrid conjugate gradient method with adaptive Barzilai–Borwein Jan 11, 2014 · Barzilai and Borwein showed that the corresponding iterative algorithm is R-superlinearly convergent for the quadratic case. The Barzilai and Borwein parameters contain second order information without estimating Hessian of convex objective function f (x). A nonmonotone line search technique is incorporated to find a Jan 10, 2016 · In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. Borwein, Peter B. 202] say “[Algorithm BB4] is arguably the most efficient algorithm currently known for the extended precision calculation of \(\pi \),” and 贝利－波尔温－普劳夫公式(BBP 公式) 提供了一个计算圆周率π的第n位二进制数的spigot算法（spigot algorithm）。这个求和公式是在1995年由西蒙·普劳夫提出的，并以公布这个公式的论 Aug 30, 2021 · 1. The primary, concrete result of this paper is an algorithm allows the exploration Jul 19, 2012 · This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. The ndings reveal that the reconstructed images have less noise and sharper edges. Borwein's algorithm. Algorithm 1 SVRG with BB step size (SVRG-BB) Parameters: update frequency m, initial point ~x 0, initial step size 0 (only used in the first Python implementation of SVRG-BB and SGD-BB algorithms of the following paper: "Barzilai-Borwein Step Size for Stochastic Gradient Descent". Author links open overlay panel Zhuang Yang a, Cheng Wang a, Yu Zang a, Jonathan Li a b. The iterative reduction FISTA algorithm (IR-FISTA-B) is an extremely excellent algorithm for solving large-scale LASSO problems. In this section, five examples extracted from literatures are used to evaluate the performance of the proposed method compared with some existing methods. 4). An effective method of Barzilai-Borwein gradient Jan 1, 2025 · We present a smoothing projected Barzilai–Borwein (SPBB) algorithm for solving a class of minimization problems on a closed convex set, where the objective function is nonsmooth nonconvex Mar 15, 2009 · These methods extend the work of Borwein [1], are easily implemented, While Algorithm 4 covers the Lerch zeta function and many others including the polylogarithm, we have concentrated on algorithms for the Hurwitz zeta function. At each iteration, the SPBB algorithm applies the projected gradient strategy that alternately uses the two Barzilai–Borwein stepsizes to the Apr 3, 2009 · As the authors will see in the examples below, an integer relation algorithm can be used to recognize a computed constant in terms of a formula involving known constants, or to discover an underlying relation between quantities that can be computed to high precision. We present an alternative perspective on the proof of the central limit theorem for Expand. A new descent direction is employed and a non-monotone line search is used in the method in order to guarantee the global convergence. MATH Google Scholar . Numerical experiments on 2-p problems, image restoration problems, and stochastic linear complementarity problems show that the SPBB algorithm is promising. Jun 28, 2021 · In this paper, a new alternating nonmonotone projected Barzilai–Borwein (BB) algorithm is developed for solving large scale problems of nonnegative matrix factorization. This summation was discovered in 1995 by This paper considers four methods of selecting the initial step size, and puts them into the algorithm for numerical experiments, and shows that the reciprocal of the minimum eigenvalue of the Hessian matrix is the best. A step of a gradient-like method in Rninvolves summing points and gradient vectors. The first two examples are highly nonlinear problems with normally Jan 14, 2025 · pi may be computed using a number of iterative algorithms. In this paper, we propose to use the Barzilai–Borwein (BB) update step to automatically compute step sizes for the state of the art mini- Jun 1, 2018 · Inspired by recent works Sopyła and Drozda, 2015, Tan et al. Remark 2. Moreover, we compare our algorithms with popular algorithms in various neural networks. The implementation uses three algorithms: the Borwein algorithm for the Riemann zeta function when is close to the real line; the Riemann-Siegel formula for the Riemann zeta function when is large imaginary, Python Program for computing the digits of pi. The Complexity of Algebraic Functions. To build the algorithms, the series consists in choosing one singular p and then to introduce modular forms a(q), b(q) and c(q). The principle result is an algo-rithm that efficiently obtains values of the Hurwitz zeta in t he critical strip s =σ+iτwith 0 ≤σ≤1. The numerical experiments on solving the test problems with dimensions from 50 to 20,000 in the CUTEr library are reported to show efficiency of the algorithm. In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing π 𝜋 \pi (see [] and []). P. Other developments of Barzilai–Borwein gradient algorithm can be found in [5, 13, 15–17, 31, 49]. Jan 1, 2025 · This section introduces a new three-term conjugate Barzilai-Borwein algorithm to search the. Burdakov et al. In recent years, a large number of studies [12,13,14,15,16] have shown that the PBB algorithm is a very effective algorithm in solving optimal problems. Borwein et al. In this paper, we improve the algorithm by using the Barzilai Mar 21, 2023 · on the Barzilai-Borwein algorithm to reduce Gaussian noise in MRI images. Raydan presented a globalization strategy based on nonmonotone line search for the general non-quadratic case. Borwein; Published 1995; Computer Science, Mathematics; A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. This formula as it stands permits {pi} to be computed fairly rapidly to any given precision (although it Each of our algorithms alternately solves a nonnegative linear least squares subproblem in matrix form using a proje cted Barzilai–Borwein method with a nonmonotone line search or no line search. This paper proposes an adaptive nonmonotone Mar 14, 2013 · Due to its simplicity and efficiency, the Barzilai and Borwein (BB) gradient method has received various attentions in different fields. Plouffe, On the rapid computation of various polylogarithmic constants, (to appear in Mathematics of Computation). , 10 and 60 seconds for the MNL and CNL SUE models on the Winnipeg network; 600 and 1200 seconds for the MNL and CNL SUE models on the Chicago Sketch network). We will see that by using only a formula of Gauss's and elementary algebra we are able to Jan 1, 2023 · In this paper, we introduce a modified Barzilai-Borwein algorithm for solving the generalized absolute value equation A x + B | x | = b. Modular Equations and Algebraic Approximations to pi. They published a book: Jonathon M. In this paper, we improve the algorithm by using the Barzilai-Borwein step Download Citation | On Nov 3, 2023, Runxuan Ma and others published An Improved Iterative Reduction FISTA-Barzilai-Borwein Algorithm for Large-Scale LASSO | Find, read and cite all the research Feb 1, 2022 · In the algorithm, a regularization penalty term based on the Barzilai and Borwein parameters is added to the Frank–Wolfe linearization objective function of the direction generating subproblem. In this paper, we develop an active set identification technique for the ℓ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} May 1, 2021 · All algorithms are terminated when RGAP=1E-10 is achieved or the maximum CPU time is taken (i. , 2016, instead of using a diminishing step size or a tuning step size by hand in the mini-batch algorithms, we equip the state of the art mini-batch algorithm, mS2GD , with the ability to automatically compute step size by using the improved Nov 5, 2023 · The iterative reduction FISTA algorithm (IR-FISTA-B) is an extremely excellent algorithm for solving large-scale LASSO problems. The BBP formula allows us to find the n-th digit (or a short block of digits near the n-th) in the binary representation of ˇfaster (in In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. Once the reference point is determined, the second part starts searching for the target MPP. Owing to this, it converges much faster than the Cauchy method. The proposed method is tested on three noisy and blurred images (IXI data set section 3. , Barzilai–Borwein gradient and Nesterov's optimal gradient, to alternately optimize two subproblems in NMF framework under the block coordinate descent Feb 21, 2018 · We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $π$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). The advantage of the proposed algorithm is that the iterative sequence is without solving any subproblems, moreover, the gradient information of Mar 17, 2020 · Borwein, Borwein and Bailey [15, pg. Introduction The “Bailey-Borwein-Plouffe” (BBP) algorithm for π is based on the BBP formula for π, which was The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pi discovered by Simon Plouffe in 1995, pi=sum_ (n=0)^infty (4/ (8n+1)-2/ (8n+4)-1/ (8n+5)-1/ (8n+6)) (1/ (16))^n. One of the popular mathematical methods in the field of image reconstruction is modeling a noisy image as a convolution between the image and point spread function (PSF). A well-designed FISTA algorithm is presented to solve the subproblems. (1989) discuss pth-order iterative algorithms. In the paper, we refer to r(R)f(x) as the Riemannian gradient of fat xand frequently use the notation g(x) := r (R)f(x) and g k:= r f(x k). Jul 13, 1998 · Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Such a technique has a strong ability to identify the zero components in a neighbourhood of a strict L-stationary point. The Barzilai-Borwein (BB) step size is a special step size determination scheme Jun 29, 2023 · In this work, we have proposed the TV regularization method based on the Barzilai–Borwein algorithm to reduce Gaussian noise in MRI images. 03. The search direction is the negative gradient direction. e. Keywords Smoothing projected Barzilai–Borwein algorithm · Constrained The algorithm’s robustness has been demonstrated through experiments on the initial learning rate and random seeds. 3) Pub Date : 2023-06-29, DOI: 10. 7. Keywords: arithmetic-geometric mean, Borwein-Borwein algorithm, Borwein- Apr 7, 2008 · In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. Due to its simplicity and efficiency, the BB method has been extended or generalized in many occasions or applications. Another quadratically converging algorithm Feb 1, 2022 · In the algorithm, the Barzilai and Borwein regularization penalty term is added to the Frank–Wolfe linearization objective of the direction generating subproblem. Algorithms for the Elementary Jun 8, 2016 · According to the Barzilai-Borwein (BB) algorithm [37], the iteration step factor can be decided by the information from the current iteration and the former one which represents the rate of Jul 29, 2016 · Under some reasonable conditions, global convergence of the proposed algorithm is established in the trust region framework. We prove the convergence of this algorithm and analyze its convergence rate with the tool of the Kurdyka-Łojasiewicz property. Bailey 8 Sept 2006 Background: Computing binary digits of log 2 What is now known as the “Bailey-Borwein-Plouffe” (BBP) algorithm for π was discovered in 1995 and published in 1996 [3]. 170]. 202] say “[Algorithm BB4] is arguably the most efficient algorithm currently known for the extended precision calculation of \(\pi \),” and the times given in Bailey’s paper [4, pg. Combined with the nonmonotone line search, the two-point stepsize gradient method has successfully been applied for large-scale unconstrained optimization. Jan 23, 2023 · Stabilized Barzilai-Borwein method 5 BB k > =kg kk. The proposed algorithm consists of two parts. With the desired termination criterion, the The iterative reduction FISTA algorithm is improved by using the Barzilai-Borwein step size as the step factor in the inner loop algorithm, and appropriate parameter update criteria to accelerate the convergence rate of the algorithm. Oct 1, 2018 · A new iterative algorithm called Barzilai-Borwein (BB) is introduced that outperforms SD with inexpensive operations and its improved form entitled SDBB is proposed to accelerate the convergence even in bad conditions to address the detection difficulties brought by increasing antennas. Google Scholar [35] Gifani P, Behnam H, and Sani ZA Noise reduction of echocardiographic images based on temporal information IEEE Trans Ultrason Ferroelectr Freq Control 2014 61 4 620-630. Artif. We discuss the global convergence of the proposed algorithm under the proper conditions. Thus, denoting stab k = kg kk; we propose to choose k = minf BB; stabg: (2. They claim that this implementation performs better Sep 1, 2015 · In this paper with Barzilai-Borwein (BB) minimization algorithm, which needs only gradient for minimum searching, instead of Newton-Raphson method, is worked. The projected Jun 28, 2021 · In terms of numerical efficiency, noise robustness and quality of matrix factorization, the BB algorithm is promising and applicable to face image reconstruction, and deep mining of transcriptomic profiles of the sub-genomes in hybrid fish lineage, compared with the state-of-the-art algorithms. The following implementation of Borwein’s algorithm with quartic assembly in Java in fact ascertains Pi, however, converges excessively fast. Nov 7, 2018 · Mini-batch algorithms with Barzilai–Borwein update step. Following the discovery of this and related formulas, similar formulas in other bases were investigated. In this paper, we propose to use the Barzilai–Borwein (BB) update step to automatically compute step sizes for the state of Mar 1, 2008 · It is shown that an iteration of the Borwein-Borwein quartic algorithm is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for \(\pi), in the sense that they produce exactly the same sequence of approximations to \(\pi \) if performed using exact arithmetic. Jacobi's Triple-Product and Some Number Theoretic Applications. Expand. Because the AGM converges quadratically, it can be combined with fast Jan 1, 2023 · An accelerated algorithm for distributed optimization with Barzilai-Borwein step sizes. However their proofs are difficult for the nonspecialist in the theory of elliptic modular functions. Unlike the existing algorithms available in the literature, a nonmonotone line search strategy is proposed to find suitable step lengths, and an adaptive BB spectral parameter is employed to generate Jul 28, 2021 · A Barzilai–Borwein gradient algorithm is designed for solving the internet traffic tensor completion model based on the triple decomposition of tensors. This algorithm allows calculation of any nth digit of pi without having to calculate any preceeding digits. NIPS 2016. The existence of a relation between a(q) , b(q) , c(q) and a(q P ) , b (q P ) , c(q P ) supplies modular equation Jul 2, 2021 · We show that an iteration of the Borwein-Borwein quartic algorithm for p is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for p, in the sense that they Nov 7, 2012 · The BBP Algorithm for Pi David H.
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