From Nocedal's Numerical Optimization book, the Hessian approximation equation is given using Symmetric Rank 1 (SR1) formula as follows:

$$B_{k+1} = B_k + \frac{(y_k - B_ks_k)(y_k - B_ks_k)}{(y_k - B_ks_k)^Ts_k}^T$$

where $y_k = \nabla f(x_{k+1}) - \nabla f(x_k)$ and $sk = x_{k+1} - x_k$

It further states that using Sherman-Morrison formula (Consider $\bar A = A + ab^T$, then $A^{-1} = \bar A^{-1} - \frac {A^{-1}ab^TA^{-1}}{1+b^TA^{-1}a}$ for $\bar A$ matrix)

The equation of inverse of Hessian can be derived using the above formula as follows:

$$H_{k+1} = H_k + \frac{(s_k - H_ky_k)(s_k - H_ky_k)^T}{(s_k - H_ky_k)y_k}$$

The problem is that to me that derivation is not obvious and I've tried solving this algebrically but I'm unable to get the derivation right from $H_{k+1}$ by manipulating $B_{k+1}$.


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