Define the Euclidean projection for a convex set $C$ as follows
$$\pi_C(y) := \min_{x \in C} \| y – x \|_2^2$$
How would we find the projection map when $C$ is the cone of positive semidefinite matrices, i.e., $\displaystyle C := \{ M : M \succeq 0 \}$?
I'm not really sure how to proceed since I can't really get a hold of how to think about how 'close' a non positive semidefinite matrix is to a positive semidefinite one.
Best Answer
It is pretty much the same, you can define for instance the distance
$$\pi_C(Y):=\min_{X\in C}||Y-X||_F^2$$
where $C:=\{X\in\mathbb{S}^n:X\succeq0\}$, and $||\cdot||_F$ is the Frobenius norm defined as $||X||_F^2:=\mathrm{trace}(X^TX)$.
This can be cast as a semidefinite program of the form:
$$\begin{array}{rcl} \underset{X}{\min} && \mathrm{trace}(A^TA-A^TX-XA+M)\\ \mathrm{s.t.}&& X\succeq0,\\ &&\begin{bmatrix} M & X\\X & I \end{bmatrix}\succeq0. \end{array}$$