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h search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 search for zeros root finding 2 search for extrema optimization 3 relationship to matrix inversion 4 regular quasi newton methods 5 notable implementations 6 see also 7 references 8 further reading toggle the table of contents quasi newton method 10 languages العربية català deutsch français 日本語 한국어 polski русский українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia optimization algorithm this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources quasi newton method news newspapers books scholar jstor january 2025 learn how and when to remove this message in numerical analysis a quasi newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for newton s method except using approximations of the derivatives of the functions in place of exact derivatives newton s method requires the jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the hessian matrix when used for finding extrema quasi newton methods on the other hand can be used when the jacobian matrices or hessian matrices are unavailable or are impractical to compute at every iteration some iterative methods that reduce to newton s method such as sequential quadratic programming may also be considered quasi newton methods search for zeros root finding edit newton s method to find zeroes of a function g displaystyle g of multiple variables is given by x n 1 x n j g x n 1 g x n displaystyle x_ n 1 x_ n j_ g x_ n 1 g x_ n where j g x n 1 displaystyle j_ g x_ n 1 is the right inverse of the jacobian matrix j g x n displaystyle j_ g x_ n of g displaystyle g evaluated for x n displaystyle x_ n strictly speaking any method that replaces the exact jacobian j g x n displaystyle j_ g x_ n with an approximation is a quasi newton method 1 for instance the chord method where j g x n displaystyle j_ g x_ n is replaced by j g x 0 displaystyle 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 2 using methods developed to find extrema in order to find zeroes is not always a good idea as the majority of the methods used to find extrema require that the matrix that is used is symmetrical while this holds in the context of the search for extrema it rarely holds when searching for zeroes broyden s good and bad methods are two methods commonly used to find extrema that can also be applied to find zeroes other methods that can be used are the column updating method the inverse column updating method the quasi newton least squares method and the quasi newton inverse least squares method more recently quasi newton methods have been applied to find the solution of multiple coupled systems of equations e g fluid structure interaction problems or interaction problems in physics they allow the solution to be found by solving each constituent system separately which is simpler than the global system in a cyclic iterative fashion until the solution of the global system is found 2 3 search for extrema optimization edit the search for a minimum or maximum of a scalar valued function is closely related to the search for the zeroes of the gradient of that function therefore quasi newton methods can be readily applied to find extrema of a function in other words if g displaystyle g is the gradient of f displaystyle f then searching for the zeroes of the vector valued function g displaystyle g corresponds to the search for the extrema of the scalar valued function f displaystyle f the jacobian of g displaystyle g now becomes the hessian of f displaystyle f the main difference is that the hessian matrix is a symmetric matrix unlike the jacobian when searching for zeroes most quasi newton methods used in optimization exploit this symmetry in optimization quasi newton methods a special case of variable metric methods are algorithms for finding local maxima and minima of functions quasi newton methods for optimization are based on newton s method to find the stationary points of a function points where the gradient is 0 newton s method assumes that the function can be locally approximated as a quadratic in the region around the optimum and uses the first and second derivatives to find the stationary point in higher dimensions newton s method uses the gradient and the hessian matrix of second derivatives of the function to be minimized in quasi newton methods the hessian matrix does not need to be computed the hessian is updated by analyzing successive gradient vectors instead quasi newton methods are a generalization of the secant method to find the root of the first derivative for multidimensional problems in multiple dimensions the secant equation is under determined and quasi newton methods differ in how they constrain the solution typically by adding a simple low rank update to the current estimate of the hessian the first quasi newton algorithm was proposed by william c davidon a physicist working at argonne national laboratory he developed the first quasi newton algorithm in 1959 the dfp updating formula which was later popularized by fletcher and powell in 1963 but is rarely used today the most common quasi newton algorithms are currently the sr1 formula for symmetric rank one the bhhh method the widespread bfgs method suggested independently by broyden fletcher goldfarb and shanno in 1970 and its low memory extension l bfgs the broyden s class is a linear combination of the dfp and bfgs methods the sr1 formula does not guarantee the update matrix to maintain positive definiteness and can be used for indefinite problems the broyden s method does not require the update matrix to be symmetric and is used to find the root of a general system of equations rather than the gradient by updating the jacobian rather than the hessian one of the chief advantages of quasi newton methods over newton s method is that the hessian matrix or in the case of quasi newton methods its approximation b displaystyle b does not need to be inverted newton s method and its derivatives such as interior point methods require the hessian to be inverted which is typically implemented by solving a system of linear equations and is often quite costly in contrast quasi newton methods usually generate an estimate of b 1 displaystyle b 1 directly as in newton s method one uses a second order approximation to find the minimum of a function f x displaystyle f x the taylor series of f x displaystyle f x around an iterate is f x k δ x f x k f x k t δ x 1 2 δ x t b δ x displaystyle f x_ k delta x approx f x_ k nabla f x_ k mathrm t delta x frac 1 2 delta x mathrm t b delta x where f displaystyle nabla f is the gradient and b displaystyle b an approximation to the hessian matrix 4 the gradient of this approximation with respect to δ x displaystyle delta x is f x k δ x f x k b δ x displaystyle nabla f x_ k delta x approx nabla f x_ k b delta x and setting this gradient to zero which is the goal of optimization provides the newton step δ x b 1 f x k displaystyle delta x b 1 nabla f x_ k the hessian approximation b displaystyle b is chosen to satisfy f x k δ x f x k b δ x displaystyle nabla f x_ k delta x nabla f x_ k b delta x which is called the secant equation the taylor series of the gradient itself in more than one dimension b displaystyle b is underdetermined using n displaystyle n different secants would be enough to determine b displaystyle b but is equivalent to computing a finite difference hessian in one dimension solving for b displaystyle b and applying the newton s step with the updated value is equivalent to the secant method the various quasi newton methods differ in their choice of the solution to the secant equation in one dimension all the variants are equivalent most methods but with exceptions such as broyden s method seek a symmetric solution b t b displaystyle b t b furthermore the variants listed below can be motivated by finding an update b k 1 displaystyle b_ k 1 that is as close as possible to b k displaystyle b_ k in some norm that is b k 1 argmin b b b k v displaystyle b_ k 1 operatorname argmin _ b b b_ k _ v where v displaystyle v is some positive definite matrix that defines the norm an approximate initial value b 0 β i displaystyle b_ 0 beta i is often sufficient to achieve rapid convergence although there is no general strategy to choose β displaystyle beta 5 note that b 0 displaystyle b_ 0 should be positive definite the unknown x k displaystyle x_ k is updated applying the newton s step calculated using the current approximate hessian matrix b k displaystyle b_ k δ x k α k b k 1 f x k displaystyle delta x_ k alpha _ k b_ k 1 nabla f x_ k with α displaystyle alpha chosen to satisfy the wolfe conditions x k 1 x k δ x k displaystyle x_ k 1 x_ k delta x_ k the gradient computed at the new point f x k 1 displaystyle nabla f x_ k 1 and y k f x k 1 f x k displaystyle y_ k nabla f x_ k 1 nabla f x_ k is used to update the approximate hessian b k 1 displaystyle b_ k 1 or directly its inverse h k 1 b k 1 1 displaystyle h_ k 1 b_ k 1 1 using the sherman morrison formula a key property of the bfgs and dfp updates is that if b k displaystyle b_ k is positive definite and α k displaystyle alpha _ k is chosen to satisfy the wolfe conditions then b k 1 displaystyle b_ k 1 is also positive definite the most popular update formulas are method b k 1 displaystyle displaystyle b_ k 1 h k 1 b k 1 1 displaystyle h_ k 1 b_ k 1 1 bfgs b k y k y k t y k t δ x k b k δ x k b k δ x k t δ x k t b k δ x k displaystyle b_ k frac y_ k y_ k mathrm t y_ k mathrm t delta x_ k frac b_ k delta x_ k b_ k delta x_ k mathrm t delta x_ k mathrm t b_ k delta x_ k i δ x k y k t y k t δ x k h k i y k δ x k t y k t δ x k δ x k δ x k t y k t δ x k displaystyle left i frac delta x_ k y_ k mathrm t y_ k mathrm t delta x_ k right h_ k left i frac y_ k delta x_ k mathrm t y_ k mathrm t delta x_ k right frac delta x_ k delta x_ k mathrm t y_ k mathrm t delta x_ k broyden b k y k b k δ x k δ x k t δ x k δ x k t displaystyle b_ k frac y_ k b_ k delta x_ k delta x_ k mathrm t delta x_ k delta x_ k mathrm t h k δ x k h k y k δ x k t h k δ x k t h k y k displaystyle h_ k frac delta x_ k h_ k y_ k delta x_ k mathrm t h_ k delta x_ k mathrm t h_ k y_ k broyden family 1 φ k b k 1 bfgs φ k b k 1 dfp φ 0 1 displaystyle 1 varphi _ k b_ k 1 text bfgs varphi _ k b_ k 1 text dfp quad varphi in 0 1 dfp i y k δ x k t y k t δ x k b k i δ x k y k t y k t δ x k y k y k t y k t δ x k displaystyle left i frac y_ k delta x_ k mathrm t y_ k mathrm t delta x_ k right b_ k left i frac delta x_ k y_ k mathrm t y_ k mathrm t delta x_ k right frac y_ k y_ k mathrm t y_ k mathrm t delta x_ k h k δ x k δ x k t δ x k t y k h k y k y k t h k y k t h k y k displaystyle h_ k frac delta x_ k delta x_ k mathrm t delta x_ k mathrm t y_ k frac h_ k y_ k y_ k mathrm t h_ k y_ k mathrm t h_ k y_ k sr1 b k y k b k δ x k y k b k δ x k t y k b k δ x k t δ x k displaystyle b_ k frac y_ k b_ k delta x_ k y_ k b_ k delta x_ k mathrm t y_ k b_ k delta x_ k mathrm t delta x_ k h k δ x k h k y k δ x k h k y k t δ x k h k y k t y k displaystyle h_ k frac delta x_ k h_ k y_ k delta x_ k h_ k y_ k mathrm t delta x_ k h_ k y_ k mathrm t y_ k other methods are pearson s method mccormick s method the powell symmetric broyden psb method and greenstadt s method 2 these recursive low rank matrix updates can also represented as an initial matrix plus a low rank correction this is the compact quasi newton representation which is particularly effective for constrained and or large problems relationship to matrix inversion edit when f displaystyle f is a convex quadratic function with positive definite hessian b displaystyle b one would expect the matrices h k displaystyle h_ k generated by a quasi newton method to converge to the inverse hessian h b 1 displaystyle h b 1 this is indeed the case for the class of quasi newton methods based on least change updates 6 regular quasi newton methods edit an attempt to provide an overview of the various approaches to quasi newton methods was made in 1985 in the article regular quasi newton methods 7 here a comprehensive class of these methods was developed a representation of all rank 1 formulas of the so called symmetric novelized huang class which includes well known methods such as the davidon fletcher powell dfp broyden fletcher goldfarb shanno bfgs and self scaling variable metric ssvm methods suggestions for further optimization of the solution behavior of quasi newton methods are also given the following class of regular i e preferred for use due to special properties quasi newton update formulas was constructed h i 1 b h i p i q i θ i r i ρ i displaystyle h_ i 1 b h_ i p_ i q_ i theta _ i r_ i rho _ i with b h p q θ r ρ r h ρ σ r τ θ σ 2 p p t r θ 1 τ h q q t h r θ σ p q t h h q p t displaystyle b h p q theta r rho rh rho sigma r tau theta over sigma 2 pp t r theta 1 over tau hqq t h r theta over sigma pq t h hqp t h r n x n displaystyle h in mathbb r nxn positive definite p q r n ϵ p t h 1 p displaystyle p q in mathbb r n epsilon p t h 1 p σ p t q τ q t h q θ r ρ r r 0 ρ σ 0 displaystyle sigma p t q tau q t hq theta r rho in mathbb r r 0 rho sigma 0 θ ϵ τ σ 2 σ 2 r τ ρ σ θ r τ ρ σ 0 displaystyle theta epsilon tau sigma 2 sigma 2 r tau rho sigma theta r tau rho sigma geq 0 for approximate sufficiently accurate beam minimization positive definite h 0 r n x n displaystyle h_ 0 in mathbb r nxn and arbitrary x 0 r n displaystyle x_ 0 in mathbb r n the following applies to these regular methods which are derived from the above formula 1 the methods are quasi newton methods 2 the matrices h i displaystyle h_ i are positive definite for all iterations thus f x i 1 f x i displaystyle f x_ i 1 f x_ i applies for all iterations 3 for all iterations i 0 displaystyle i geq 0 we obtain solutions to the minimization problem for exact beam minimization and quadratic objective functions each of these methods also terminates at the minimum point after at most n iteration steps in particular the regular quasi newton methods have the good properties of both the extended greenstadt class and the symmetric extended huang class with regard to convergence and stability it can be assumed that all particularly powerf...