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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] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...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] |
Confidence Interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. The confidence level represents the long-run proportion of corresponding CIs that contain the true value of the parameter. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value. Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level. All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval. Definition Let be a random sample from a probability distribution with statistical parameter , which is a quantity to be estimate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Credible Interval
In Bayesian statistics, a credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals and confidence regions in frequentist statistics, although they differ on a philosophical basis: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value. Also, Bayesian credible intervals use (and indeed, require) knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not. For example, in an experiment that determines the distribution of possible values of the parameter \mu, if the subjective probability that \mu lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood Estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when all observed outcomes are assumed to have Normal distributions with the same variance. From the perspective of Bayesian inference, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sherman–Morrison Formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications. Statement Suppose A\in\mathbb^ is an invertible square matrix and u,v\in\mathbb^n are column vectors. Then A + uv^\textsf is invertible iff 1 + v^\textsf A^u \neq 0. In this case, :\left(A + uv^\textsf\right)^ = A^ - . Here, uv^\textsf is the outer product of two vectors u and v. The general form shown here is the one published by Bartlett. Proof (\Leftarrow) To prove that the backward direction 1 + v^\textsfA^u \neq 0 \Rightarrow A + uv^\textsf is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y (in this case the right-ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Definiteness
In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite function * Positive-definite function on a group * Positive-definite functional * Positive-definite kernel * Positive-definite matrix * Positive-definite quadratic form In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ... References *. *. {{Set index article, mathematics Quadratic forms ... [...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] |
Wolfe Conditions
In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969. In these methods the idea is to find ::\min_x f(\mathbf) for some smooth f\colon\mathbb R^n\to\mathbb R. Each step often involves approximately solving the subproblem ::\min_ f(\mathbf_k + \alpha \mathbf_k) where \mathbf_k is the current best guess, \mathbf_k \in \mathbb R^n is a search direction, and \alpha \in \mathbb R is the step length. The inexact line searches provide an efficient way of computing an acceptable step length \alpha that reduces the objective function 'sufficiently', rather than minimizing the objective function over \alpha\in\mathbb R^+ exactly. A line search algorithm can use Wolfe conditions as a requirement for any guessed \alpha, before finding a new search direction \mathbf_k. Armijo rule and curvature A step length \alpha_k is said to satisfy the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the quadratic function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function whose graph is shaped like a cup \cup, while a concave function's graph is shaped like a cap \cap. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
