The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, Roger Fletcher, and Michael J. D. Powell) 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.
and the Taylor series of the gradient itself (secant equation)
is used to update .
The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of :
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
- Newton's method
- Newton's method in optimization
- Quasi-Newton method
- Broyden–Fletcher–Goldfarb–Shanno (BFGS) method
- L-BFGS method
- SR1 formula
- Nelder–Mead method
- Davidon, W. C. (1991), "Variable metric method for minimization", SIAM Journal on Optimization, 1: 1–17, CiteSeerX 10.1.1.693.272, doi:10.1137/0801001
- Fletcher, Roger (1987), Practical methods of optimization (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-91547-8.
- Nocedal, Jorge; Wright, Stephen J. (1999), Numerical Optimization, Springer-Verlag, ISBN 0-387-98793-2