Davidon–Fletcher–Powell formula
The Davidon–Fletcher–Powell formula finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first quasi-Newton method to generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix.
Given a function, its gradient, and positive-definite Hessian matrix, the Taylor series is
and the Taylor series of the gradient itself
is used to update.
The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of :
where
and is a symmetric and positive-definite matrix.
The corresponding update to the inverse Hessian approximation is given by
is assumed to be positive-definite, and the vectors and must satisfy the curvature condition
The DFP formula is quite effective, but it was soon superseded by the Broyden–Fletcher–Goldfarb–Shanno formula, which is its dual.
