How to find the nearest/a near positive definite from a given matrix?

Quick sketch of an answer for the Frobenius norm:

1. Replace $X$ with the closest symmetric matrix, $Y=frac12 (X+X^top)$.
2. Take an eigendecomposition $Y=QDQ^top$, and form the diagonal matrix $D_+=max(D,0)$ (elementwise maximum).
3. The closest symmetric positive semidefinite matrix to $X$ is $Z=QD_+Q^top$.
4. The closest positive definite matrix to $X$ does not exist; any matrix of the form $Z+varepsilon I$ is positive definite for $varepsilon>0$. There is no minimum, just an infimum. Positive definite matrices are not a closed set.

To prove (1) and (3), you can use the fact that the decomposition of a matrix into a symmetric and antisymmetric part is orthogonal. In particular, this implies that we can minimize in two succesive steps like we did.

To prove (2), use the Wielandt-Hoffmann theorem.