In the space $\mathcal{M}_n(\mathbb{R})$ embedded with the Frobenius norm, I would like to compute the orthogonal projection of a symmetric matrix $$M = \pmatrix{a & b \\ b & 1}$$ onto the closed convex set of matrices that are symmetric positive semidefinite and write $$\pmatrix{\alpha & \beta \\ \beta & 1}$$ So, there is actually a way of solving this problem using a Lagrangian formulation, but I would like to know why another idea I had doesn't work. I thought:
In the space of symmetric matrices, matrices that write $\pmatrix{\alpha & \beta \\ \beta & 1}$ form an affine subspace. The orthogonal projection $p_1(M)$ of $M = \pmatrix{a & b \\ b & c}$ onto this space is just $M = \pmatrix{a & b \\ b & 1}$.
In the space of symmetric matrices, you can compute the orthogonal projection onto the space of positive semidefinite matrices $p_2$ by diagonalizing and setting negative eigenvalues to 0.
So I thought if I define $p=p_1 \circ p_2$, and compute $p \circ \dots \circ p (M)$, this should converge towards the desired result, but it's actually numerically not working. I just made a few drawings and couldnt figure a counter example, with an affine and a convex set. Do you know why this doesn't work?
