# How to optimize an approximated matrix multiplication?

Suppose the objective I try to maximize is $$\max_{X} \|(I - \alpha X)^{-1}XA\|_F$$ where $$X$$ is the matrix needs to be pinned down, $$\alpha$$ is a scalar, and $$\|\cdot\|_F$$ is the Frobenius norm. Note that $$(I-\alpha X)^{-1}$$ is invertible only when $$\alpha\neq1$$, so assume $$\alpha<1$$. All matrices are $$n$$ by $$n$$ square matrices. An extra condition of matrix $$X$$ is that the vectors $$X(:,k)$$ for $$k=1,\cdots,n$$ is selected from identity by the optimization. So $$X(:,k)$$ is also a standard basis vector and $$X$$ may have repeated columns, i.e., $$X$$ is a permutation matrix.

One possible way to obtain the maxima of the objective is to apply the matrix multiplication via random sampling, where the objective is rewritten as

$$\max_{p_k}\left\| \frac{1}{n}\sum_{k=1}^n\frac{1}{p_k}B(:,k)X(k,:)A\right\|_F$$

where $$B$$ is the inverse and be approximated by Neumann series such that $$B = (I - \alpha X)^{-1} = \sum_{i=0}^n (\alpha X)^k$$

Note that we do not know the form or the specification of $$X$$ in advance but choose each column of $$X$$ by random sampling with probability $$p_k$$.

Now I have 3 questions:

(1) How to compute $$B$$ when it is random sampling approximation?

(2) Is this method too complex to implement on numerical calculation?

(3) Is there any better way to solve this puzzle?