differentiation of a norm of matrix function

Question

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in \mathscr{R}^n,\mathbf{v_2}\in [\mathscr{R}^n]^-$ are vectors,$\mathbf{I}$ is identity matrix, $\mathbf{W}$ is a doubly stochastic matrix. And $b = -\frac{\sum{{v_1}_i}}{\sum{{v_2}_i}}$ , is the limit point of $x(\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1}$ as $x\to 0$. I had seen by simulation that $y$ is strictly increasing in the given range of $x$ as x goes from $0$ to max.

Answer 1

Define a few variables for convenience $$\eqalign{ v &= v_1 \cr D &= {\rm Diag}(v_2) \cr E &= I-W-DWx \cr dE &= -DW\,dx \cr\cr h &= E^{-1}vx-b \cr dh &= E^{-1}v\,dx - E^{-1}\,dE\,E^{-1}vx \cr &= E^{-1}v\,dx - E^{-1}(-DW)E^{-1}vx\,dx \cr &= (E^{-1} + xE^{-1}DWE^{-1})\,v\,\,dx \cr\cr }$$ Now write the objective function and find its differential $$\eqalign{ y &= h^Th \cr dy &= 2h^Tdh \cr &= 2h^T(E^{-1} + xE^{-1}DWE^{-1})\,v\,dx \cr\cr }$$ From that last expression, the derivative wrt $x$ must be $$\eqalign{ \frac{dy}{dx} &= 2h^T(E^{-1} + xE^{-1}DWE^{-1})\,v \cr }$$