For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices:

$$\Pi(A)=\mathrm{argmin}_{M\succeq0}\|A-M\|_{\infty}.$$

Here $\|A\|_{\infty}=\mathrm{max}_{ij}|A_{ij}|$ is the entry-wise max-norm. This problem has exact solution for the Frobenius and spectral matrix norm (see for example here). Is there a closed-form solution for the max-norm? Is there an efficient algorithm to calculate the projection and what is the computational cost (for $n\times n$ complex matrix)?