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Davidon–Fletcher–Powell Formula
The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, Roger Fletcher (mathematician), Roger Fletcher, and Michael J. D. Powell) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first quasi-Newton method to generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function f(x), its gradient (\nabla f), and positive-definite matrix, positive-definite Hessian matrix B, the Taylor series is :f(x_k+s_k) = f(x_k) + \nabla f(x_k)^T s_k + \frac s^T_k s_k + \dots, and the Taylor series of the gradient itself (secant equation) :\nabla f(x_k+s_k) = \nabla f(x_k) + B s_k + \dots is used to update B. The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of B_k: :B_= (I - \gamma_k y_k s_k^T) B_k (I - \gamma_k s_k y_k^T) + \gamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William C
William is a male given name of Germanic origin.Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 276. It became very popular in the English language after the Norman conquest of England in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will, Wills, Willy, Willie, Bill, and Billy. A common Irish form is Liam. Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie or the play ''Douglas''). Female forms are Willa, Willemina, Wilma and Wilhelmina. Etymology William is related to the given name ''Wilhelm'' (cf. Proto-Germanic ᚹᛁᛚᛃᚨᚺᛖᛚᛗᚨᛉ, ''*Wiljahelmaz'' > German ''Wilhelm'' and Old Norse ᚢᛁᛚᛋᛅᚼᛅᛚᛘᛅᛋ, ''Vilhjálmr''). By regular sound changes, the native, inherited English form of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broyden–Fletcher–Goldfarb–Shanno Algorithm
In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations (or approximate gradient evaluations) via a generalized secant method. Since the updates of the BFGS curvature matrix do not require matrix inversion, its computational complexity is only \mathcal(n^), compared to \mathcal(n^) in Newton's method. Also in common use is L-BFGS, which is a limited-memory version of BFGS that is particularly suited to problems with very large numbers of variables (e.g., >1000). The BFGS-B variant handles simple box constraints. The algorithm is named after Charles George Broyden, Roger Fletcher, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Rank-one
The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian) based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem. This update maintains the ''symmetry'' of the matrix but does ''not'' guarantee that the update be ''positive definite''. The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true Hessian than do popular alternatives ( BFGS or DFP), in preliminary numerical experiments. The SR1 method has computational advantages for sparse or partially separable problems. A twice continuously differentiable function x \mapsto f(x) has a gradient (\nabla f) and Hessian matrix B: The function f has an expansion as a Taylor series at x_0, which can be truncated ::f(x_0+\Delta x) \approx f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limited-memory BFGS
Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning. The algorithm's target problem is to minimize f(\mathbf) over unconstrained values of the real-vector \mathbf where f is a differentiable scalar function. Like the original BFGS, L-BFGS uses an estimate of the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense n\times n approximation to the inverse Hessian (''n'' being the number of variables in the problem), L-BFGS stores only a few vectors that represent the approximation implicitly. Due to its resulting linear memory requirement, the L-BFGS method is particularly well suited for optimization problems with many variables. Instead of the inverse Hessian H''k'', L-BFGS maintains a history of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. The "full" Newton's method requires the Jacobian in order to search for zeros, or the Hessian for finding extrema. Search for zeros: root finding Newton's method to find zeroes of a function g of multiple variables is given by x_ = x_n - _g(x_n) g(x_n), where _g(x_n) is the left inverse of the Jacobian matrix J_g(x_n) of g evaluated for x_n. Strictly speaking, any method that replaces the exact Jacobian J_g(x_n) with an approximation is a quasi-Newton method. For instance, the chord method (where J_g(x_n) is replaced by J_g(x_0) for all iterations) is a simple example. The methods given below for optimization refer to an important subclass of quasi-Newton methods, secant methods. Using methods developed to find extrema in order to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method In Optimization
In calculus, Newton's method is an iterative method for finding the roots of a differentiable function , which are solutions to the equation . As such, Newton's method can be applied to the derivative of a twice-differentiable function to find the roots of the derivative (solutions to ), also known as the critical points of . These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function . Newton's method The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case. Given a twice differentiable function f:\mathbb\to \mathbb, we seek to solve the optimization problem : \min_ f(x) . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, ''linear algebra duality'' corresponds in this way to bilinear maps from pairs of vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Fletcher (mathematician)
Roger Fletcher FRS FRSE (29 January 1939 – 15 July 2016) was a British mathematician and professor at University of Dundee. He was a Fellow of the Society for Industrial and Applied Mathematics (SIAM) and was elected as a Fellow of the Royal Society in 2003. In 2006, he won the Lagrange Prize from SIAM. In 2008, he was awarded a Royal Medal of the Royal Society of Edinburgh. See also * BFGS method * Davidon–Fletcher–Powell formula * Nonlinear conjugate gradient method In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function \displaystyle f(x) :: \displaystyle f(x)=\, Ax-b\, ^2, the minimum of f is obtained whe ... Bibliography *''Practical methods of optimization'', Wiley, 1987, ; Wiley, 2000, References {{DEFAULTSORT:Fletcher, Roger 1939 births 2016 deaths British mathematicians Fellows of the Royal Society of Edinburgh Fellows of the Royal Society Aca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
