Projection onto convex set defined by $\|\mathbf{t} -\mathbf{W}^T\mathbf{y}\|^2 \leq k$

Question

I want to use the method of Projections Onto Convex Sets, and for the problem at hand I need to find a closed form solution for $\mathbf{P}_C$, the projection onto set $C$, defined as:

$$C = \{ \mathbf{y}\ |\ \|\mathbf{t} -\mathbf{W}^T\mathbf{y}\|^2 \leq k \}$$

I am new to convex optimization, and I was wondering if such a closed form can be found.

Answer 1

You should solve the following SOCP, I don't think that a closed form solution exists: $$\text{minimize}_y~~ \|y-y_0\|^2_2\\ \text{subject to}~~ \|t-W^Ty\|\leq k$$