In Nocedal/Wright Numerical Optimization book
at pages 138-139 the approximate Hessian $B_k$ update for the quasi-Newton method: (DFP method)
$$B_{k+1} = \left(I-\frac{y_ks_k^T}{y_k^Ts_k}\right)B_k\left(I-\frac{s_ky_k^T}{y_k^Ts_k}\right)+ \frac{y_ky_k^T}{y_k^Ts_k}\tag{1}$$
is explained as the solution to the problem:
$$\min_B\|B-B_k\|_{F,W} \\ \text{subject to}~B=B^T,~Bs_k=y_k \tag{2}$$
for which $\|A\|_{F,W}$ is the weighted Frobenius norm :
$$\|A\|_{F,W} = \|W^{1/2}AW^{1/2}\|_F$$
for $W$ being any symmetric matrix satisfying the relation $Wy_k=s_k$
How can I prove that $B_{k+1}$ given by equation (1) is the solution to the problem (2)?
Best Answer
This answer goes basically along the lines of my answer about BFGS update.