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Trust Region
In mathematical optimization, a trust region is the subset of the region of the objective function that is approximated using a model function (often a quadratic). If an adequate model of the objective function is found within the trust region, then the region is expanded; conversely, if the approximation is poor, then the region is contracted. The fit is evaluated by comparing the ratio of expected improvement from the model approximation with the actual improvement observed in the objective function. Simple thresholding of the ratio is used as the criterion for expansion and contraction—a model function is "trusted" only in the region where it provides a reasonable approximation. Trust-region methods are in some sense dual to line-search methods: trust-region methods first choose a step size (the size of the trust region) and then a step direction, while line-search methods first choose a step direction and then a step size. The general idea behind trust region methods is kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Objective Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic". For example, a univariate (single-variable) quadratic function has the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the -axis. If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables and has the form : f(x,y) = a x^2 + bx y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line-search
In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach is trust region. The line search approach first finds a descent direction along which the objective function f will be reduced and then computes a step size that determines how far \mathbf should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either exactly or inexactly. Example use Here is an example gradient method that uses a line search in step 4. # Set iteration counter \displaystyle k=0, and make an initial guess \mathbf_0 for the minimum # Repeat: # Compute a descent direction \mathbf_k # Choose \displaystyle \alpha_k to 'loosely' minimize h(\alpha_k)=f(\mathbf_k+\alpha_k\mathbf_k) over \alpha_k\in\mathbb R_+ # &nb ... [...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 The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, 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 ... * Nonlinear conjugate gradient method 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 Acad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Goldfeld
Stephen Michael Goldfeld (August 9, 1940 – August 25, 1995) was a Princeton University economics professor and provost who served on the Council of Economic Advisers during the Carter administration. Goldfeld received a bachelor's degree from Harvard University in 1960 at the age of twenty and a doctorate in economics from the Massachusetts Institute of Technology in 1963 at the age of twenty three, when he joined the Princeton faculty. As an academic he specialized in financial institutions and in econometrics. He was an associate editor of the ''American Economic Review The ''American Economic Review'' is a monthly peer-reviewed academic journal published by the American Economic Association. First published in 1911, it is considered one of the most prestigious and highly distinguished journals in the field of ec ...'' and other major economic journals. He died in 1995 at the age of 55 of lung cancer. Noted publications * * * References Econometricians 1995 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard E
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", " Rich", "Rick", " Rico", " Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hale Trotter
Hale Freeman Trotter (30 May 1931 – 17 January 2022)biographical information from ''American Men and Women of Science'', Thomson Gale 2004 was a Canadian-American mathematician, known for the Lie–Trotter product formula, the Steinhaus–Johnson–Trotter algorithm, and the Lang–Trotter conjecture. He was born in Kingston, Ontario. He died in Princeton, New Jersey on January 17, 2022. Biography The son of historian Reginald George Trotter, Hale Trotter studied at Queen's University in Kingston with bachelor's degree in 1952 and master's degree in 1953. He received in 1956 his PhD from Princeton University under William Feller with thesis ''Convergence of semigroups of operators''. Trotter was from 1956 to 1958 at Princeton University the ''Fine Instructor'' for mathematics and from 1958 to 1960 an assistant professor at Queen's University. He was from 1962 to 1963 a visiting associate professor, from 1963 to 1969 an associate professor, and from 1969 until his retirement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Econometrica
''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is Guido Imbens. History ''Econometrica'' was established in 1933. Its first editor was Ragnar Frisch, recipient of the first Nobel Memorial Prize in Economic Sciences in 1969, who served as an editor from 1933 to 1954. Although ''Econometrica'' is currently published entirely in English, the first few issues also contained scientific articles written in French. Indexing and abstracting ''Econometrica'' is abstracted and indexed in: * Scopus * EconLit * Social Science Citation Index According to the '' Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levenberg–Marquardt Algorithm
In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA. LMA can also be viewed as Gauss–Newton using a trust region approach. The algorithm was first published in 1944 by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, who worked as a statistician at DuPont, and independently by Girard, Wynne and Morrison. The LMA is used in many software applications for solving gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Surface
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in ''D'' + 1 variables; for example, in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press, from The Geometry Center at University of Minnesota : \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: : x Q x^\mathrm + P x^\mathrm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry ( Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
