|
Compact Quasi-Newton Representation
The compact representation for quasi-Newton methods is a matrix decomposition, which is typically used in gradient based optimization (mathematics), optimization algorithms or for solving nonlinear systems. The decomposition uses a low-rank representation for the direct and/or inverse Hessian matrix, Hessian or the Jacobian matrix and determinant, Jacobian of a nonlinear system. Because of this, the compact representation is often used for large problems and constrained optimization. Definition The compact representation of a quasi-Newton matrix for the inverse Hessian H_k or direct Hessian B_k of a nonlinear loss function, objective function f(x):\mathbb^n \to \mathbb expresses a sequence of recursive rank-1 or rank-2 matrix updates as one rank-k or rank-2k update of an initial matrix. Because it is derived from quasi-Newton updates, it uses differences of iterates and gradients \nabla f(x_k) = g_k in its definition \_^k . In particular, for r=k or r=2k the rectangular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Methods
In numerical analysis, a quasi-Newton method is an Iterative method, iterative numerical method used either to Root-finding algorithm, find zeroes or to Mathematical optimization, 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 Derivative, derivatives of the functions in place of exact derivatives. Newton's method requires the Jacobian matrix and determinant, Jacobian matrix of all Partial derivative, partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used Newton's method in optimization, 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 method, iterative methods that reduce to Newton's method, such as sequential quadratic programming, may also be considered quasi-Newton methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
