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Let $\mathbf{A}_i \in M_{n,n}(\mathbb{C})$, $\mathbf{u} \in M_{n,1}(\mathbb{C})$, $x_i, c_i \in \mathbb{R}$, and \begin{align} \mathbf{v}(x_i) := \left( \mathbf{I} + \sum \limits_{i=1}^{N} x_i \mathbf{A}_i \right)^{-1} \mathbf{u} \ \ , \end{align}

Compute gradient of $f(x_i)$ with respect to $x_i$, i.e., $\frac{d f(x_i)}{d x_i}$: \begin{align} f(x_i) = \left\| \mathbf{u} - \mathbf{v}(x_i) \right\|_2^2 + \sum_{i=1}^{N} x_i \left(\mathbf{v}(x_i)^* \mathbf{A}_i \mathbf{v}(x_i) - c_i \right ) . \end{align}


EDIT:

The derivative of the first norm component can be obtained as done here: Find $\frac{d}{dx} \left( \lVert u - M^{-1}(x)u\rVert_2^2 \right)$, where $M(x) = \left(B + xA \right)$ and $M^H(x) = M(x)$ . Following the similar procedure, one can derive the complete gradient of $f(x_i)$.

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