Quasi-Newton method

The search for a minimum or maximum of a scalar-valued function is nothing else than 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 ${\displaystyle g}$ is the gradient of ${\displaystyle f}$, then searching for the zeroes of the vector-valued function ${\displaystyle g}$ corresponds to the search for the extrema of the scalar-valued function ${\displaystyle f}$; the Jacobian of ${\displaystyle g}$ now becomes the Hessian of ${\displaystyle f}$. 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 property.


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 are based on Newton's method to find the stationary point of a function, 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) ${\displaystyle B}$ 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 ${\displaystyle B^{-1}}$ directly.


As in Newton's method, one uses a second-order approximation to find the minimum of a function ${\displaystyle f(x)}$. The Taylor series of ${\displaystyle f(x)}$ around an iterate is


where (${\displaystyle \nabla f}$) is the gradient, and ${\displaystyle B}$ an approximation to the Hessian matrix.^[4] The gradient of this approximation (with respect to ${\displaystyle \Delta x}$) is


and setting this gradient to zero (which is the goal of optimization) provides the Newton step:


The Hessian approximation ${\displaystyle B}$ is chosen to satisfy


which is called the secant equation (the Taylor series of the gradient itself). In more than one dimension ${\displaystyle B}$ is underdetermined. In one dimension, solving for ${\displaystyle B}$ 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 (${\displaystyle B^{T}=B}$); furthermore, the variants listed below can be motivated by finding an update ${\displaystyle B_{k+1}}$ that is as close as possible to ${\displaystyle B_{k}}$ in some norm; that is, ${\displaystyle B_{k+1}=\operatorname {argmin} _{B}\|B-B_{k}\|_{V}}$, where ${\displaystyle V}$ is some positive-definite matrix that defines the norm. An approximate initial value ${\displaystyle B_{0}=\beta I}$ is often sufficient to achieve rapid convergence, although there is no general strategy to choose ${\displaystyle \beta }$.^[5] Note that ${\displaystyle B_{0}}$ should be positive-definite. The unknown ${\displaystyle x_{k}}$ is updated applying the Newton's step calculated using the current approximate Hessian matrix ${\displaystyle B_{k}}$:


is used to update the approximate Hessian ${\displaystyle B_{k+1}}$, or directly its inverse ${\displaystyle H_{k+1}=B_{k+1}^{-1}}$ using the Sherman–Morrison formula.


The most popular update formulas are:


Other methods are Pearson's method, McCormick's method, the Powell symmetric Broyden (PSB) method and Greenstadt's method.^[2]