BFGS Method

Concept

BFGS Methodis a secant updating methods of quasi-Newton’s method.
take $B_0=I$ or initial guess of $H$.

solve $B_k{s}_k=-\nabla f(x_k)$
$x_{k+1}=x_k+s_k$
$y_k=\nabla f(x_{k+1})-\nabla f(x_k)$

$$B_{k+1}=B_k+\frac{y_k{y}_k^T}{y_k^T{s}_k}-\frac{B_k{s}_k{s}_k^T{B}_k}{s_k^T{B}_k{s}_k}$$

repeat this.