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We are given a set of $K_1$ vectors $a_{i} \in \mathbb{R}^{N}$, $1 \leq i \leq K_1$; $K_2$ vectors $b_{j} \in \mathbb{R}^{N}$, $1 \leq j \leq K_2$; and a set of $K_1$ matrices $A_{i} \in \mathbb{R}^{N \times N}$, $1 \leq i \leq K_1$. We are also given two binary vectors, $v \in \mathbb{R}^{K_1}$ and $u \in \mathbb{R}^{K_2}$. Each vector contains at least one $1$.

We seek a matrix $M \in \mathbb{R}^{N \times N}$ that satisfies the following property. First, we use the matrix $M$ to compute another matrix $W \in \mathbb{R}^{K_1 \times K_2}$ with each element $w_{ij} = (a_i - b_j)^{T} (A_i + M)^{-1} (a_i - b_j)$. Then, we take row-wise and column-wise maximum of the matrix $W$, to get two vectors $\tilde{v} \in \mathbb{R}^{K_1}$ and $\tilde{u} \in \mathbb{R}^{K_2}$. The property $M$ should satisfy is that the maximum entries in $\tilde{v}$ should correspond to the $1$-values in $v$; and the maximum entries in $\tilde{u}$ should correspond to the $1$-values in $u$.

What methods can be used to solve such problems?

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