sorry I don't know how to post Latex here I have to implement in Matlab the following formula and I'd like to know if the expression given here really corresponds to the minimum indicated, and why:
I found several formula errors in the previous equation of the same publication, but here I'm unable to see if this is correct. So, is this correct, and how does one prove it? thanks
PS: the non-bold letters are matrices, I being the identity and the bold letters being column vectors, lambda is a scalar value
Under the assumption the $f,g,\mu$ are finite-dimensional vectors in $\mathbb{R}^n$, $M$ is a square real matrix, $V$ is a tall real matrix and $\lambda$ is a scalar (Lagrange multiplier) and the norm is the Euclidean norm:
I think it is correct (though I wasn't really careful) What you can do is to write out the norms explicitly ($\| x \|^2 = x^Tx$) and then take the derivative with respect to $f$ (vector derivatives are not complicated, check this page) and then set it to zero. From convexity, that would give the unique global minimizer $f^*$.