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Nelder–Mead Method
The Nelder–Mead method (also downhill simplex method, amoeba method, or polytope method) is a numerical method used to find the minimum or maximum of an objective function in a multidimensional space. It is a direct search method (based on function comparison) and is often applied to nonlinear optimization problems for which derivatives may not be known. However, the Nelder–Mead technique is a heuristic search method that can converge to non-stationary points * * (algorithm summary online). on problems that can be solved by alternative methods. * Yu, Wen Ci. 1979. "Positive basis and a class of direct search techniques". ''Scientia Sinica'' 'Zhongguo Kexue'' 53—68. * Yu, Wen Ci. 1979. "The convergent property of the simplex evolutionary technique". ''Scientia Sinica'' 'Zhongguo Kexue'' 69–77. * * The Nelder–Mead technique was proposed by John Nelder and Roger Mead in 1965, as a development of the method of Spendley et al. Overview The method uses the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946 his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's ''t''-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Evolution
In evolutionary computation, differential evolution (DE) is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as they make few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found. DE is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means DE does not require the optimization problem to be differentiable, as is required by classic optimization methods such as gradient descent and quasi-newton methods. DE can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining ... [...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] |
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 when the gradient is 0: :: \nabla_x f=2 A^T(Ax-b)=0. Whereas linear conjugate gradient seeks a solution to the linear equation \displaystyle A^T Ax=A^T b, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient \nabla_x f alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there. Given a function \displaystyle f(x) of N variables to minimize, its gradient \nabla_x f indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: :: \Delta x_0=-\nabla_x f (x_0) with an adjustable s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LINCOA
Michael James David Powell (29 July 193619 April 2015) was a British mathematician, who worked in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. Education and early life Born in London, Powell was educated at Frensham Heights School and Eastbourne College. He earned his Bachelor of Arts degree followed by a Doctor of Science (DSc) degree in 1979 at the University of Cambridge. see alsAn Interview with M. J. D. Powellby Philip J. Davis, 6 April 2005 Career and research Powell was known for his extensive work in numerical analysis, especially nonlinear optimisation and approximation. He was a founding member of the Institute of Mathematics and its Applications and a founding Managing Editor of the ''Journal for Numerical Analysis''. His mathematical contributions include quasi-Newton methods, particularly the Davidon-Fletcher-Powell formula and the Powell's Symmetric Broyden formula, augmented Lagrangian function (also c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NEWUOA
Michael James David Powell (29 July 193619 April 2015) was a British mathematician, who worked in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. Education and early life Born in London, Powell was educated at Frensham Heights School and Eastbourne College. He earned his Bachelor of Arts degree followed by a Doctor of Science (DSc) degree in 1979 at the University of Cambridge. see alsAn Interview with M. J. D. Powellby Philip J. Davis, 6 April 2005 Career and research Powell was known for his extensive work in numerical analysis, especially nonlinear optimisation and approximation. He was a founding member of the Institute of Mathematics and its Applications and a founding Managing Editor of the ''Journal for Numerical Analysis''. His mathematical contributions include quasi-Newton methods, particularly the Davidon-Fletcher-Powell formula and the Powell's Symmetric Broyden formula, augmented Lagrangian function (also c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
COBYLA
Michael James David Powell (29 July 193619 April 2015) was a British mathematician, who worked in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. Education and early life Born in London, Powell was educated at Frensham Heights School and Eastbourne College. He earned his Bachelor of Arts degree followed by a Doctor of Science (DSc) degree in 1979 at the University of Cambridge. see alsAn Interview with M. J. D. Powellby Philip J. Davis, 6 April 2005 Career and research Powell was known for his extensive work in numerical analysis, especially nonlinear optimisation and approximation. He was a founding member of the Institute of Mathematics and its Applications and a founding Managing Editor of the ''Journal for Numerical Analysis''. His mathematical contributions include quasi-Newton methods, particularly the Davidon-Fletcher-Powell formula and the Powell's Symmetric Broyden formula, augmented Lagrangian function (also c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative-free Optimization
Derivative-free optimization is a discipline in mathematical optimization that does not use derivative information in the classical sense to find optimal solutions: Sometimes information about the derivative of the objective function ''f'' is unavailable, unreliable or impractical to obtain. For example, ''f'' might be non-smooth, or time-consuming to evaluate, or in some way noisy, so that methods that rely on derivatives or approximate them via finite differences are of little use. The problem to find optimal points in such situations is referred to as derivative-free optimization, algorithms that do not use derivatives or finite differences are called derivative-free algorithms. Introduction The problem to be solved is to numerically optimize an objective function f\colon A\to\mathbb for some set A (usually A\subset\mathbb^n), i.e. find x_0\in A such that without loss of generality f(x_0)\leq f(x) for all x\in A. When applicable, a common approach is to iteratively improve a pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Termination
Termination may refer to: Science *Termination (geomorphology), the period of time of relatively rapid change from cold, glacial conditions to warm interglacial condition *Termination factor, in genetics, part of the process of transcribing RNA *Termination type, in lithic reduction, a characteristic indicating the manner in which the distal end of a lithic flake detaches from a core *Chain termination, in chemistry, a chemical reaction which halts polymerization *Termination shock, in solar studies, a feature of the heliosphere * Terminating computation, in computer science **Termination analysis, a form of program analysis in computer science ** Termination proof, a mathematical proof concerning the termination of a program ** Termination (term rewriting), in particular for term rewriting systems Technology *Electrical termination, ending a wire or cable properly to prevent interference *Termination of wires to a **Crimp connection ** Electrical connector ** Solder joint *Abor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenbrock Function Nelder-Mead
Rosenbrock is a surname. Notable people with the surname include: * Eddie Rosenbrock (1908–1978), Australian rules footballer *Howard Harry Rosenbrock Howard Harry Rosenbrock (16 December 1920 – 21 October 2010) was a leading figure in control theory and control engineering. He was born in Ilford, England in 1920, graduated in 1941 from University College London with a 1st class honors degre ... (1920–2010), English control theorist and engineer * Peter Rosenbrock (1939–2005), Australian rules footballer {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
