Broyden Family Method

Definition

Broyden family update is obtained from combining the BFGS and the DFP method

Algorithm

$B_n=(1-{\alpha }_{n-1})J_{n-1}^{BFGS}+{\alpha }_{n-1}{J}_{n-1}^{DFP}$ where $\alpha \in [0,1]$

Facts

If the $\alpha$ of Broyden Family is a $\frac{\Delta {\mathbf{f}}_n^{⊺}\Delta {\mathbf{x}}_n}{\Delta {\mathbf{f}}_n^{⊺}\Delta {\mathbf{x}}_n-\Delta {\mathbf{x}}_n^{⊺}{J}_{k-1}\Delta {\mathbf{x}}_n}$, then the update is equal to Symmetric Rank-One Method $B_n=B_{n-1}+\frac{(\Delta {\mathbf{f}}_n-J_{n-1}\Delta {\mathbf{x}}_n)(\Delta {\mathbf{f}}_n-J_{n-1}\Delta {\mathbf{x}}_n{)}^{⊺}}{(\Delta {\mathbf{f}}_n-J_{n-1}\Delta {\mathbf{x}}_n{)}^{⊺}\Delta {\mathbf{x}}_n}$