Let $B=A+\rho u u^\top$, where $A\in\mathbb{R}^{d\times d}$ is a positive semidefinite (PSD) matrix with $\text{rank}(A)=k (k<d)$, $\rho>0$ is a scalar, $u\in\mathbb{R}^d, u^\top u=1$. And here I assume that $\text{rank}(B)=k+1$.
I was wondering if there is an efficient way to solve
$\arg\min_{B'}||B-B'||_\mathtt{F}, \text{ s.t. rank}(B')=k$,
where $||\cdot||_\mathtt{F}$ is the Frobenius norm.
Thanks!