I recently asked about an optimization problem, which turned out to be a quadratic programming problem. Specifically, I wanted to minimize the objective function $$-\mathbf{w^{\top}XX^{\top}w}$$ with the constraint $||\mathbf{w}|| = 1$. For this problem the optimal solution is proportional to the left singular vector corresponding to the largest singular value.
After analyzing the problem for two weeks, and learning a lot about quadratic programming, I came to the conclusion that I must modify my original problem. Specifically, I would like to maximize
$$f(\mathbf{y}) = \begin{cases}\sum_{i}y_{i}^{2}, \quad &y_{i}^{2} < c\\ 0,\quad &y_{i}^{2} \geq c\end{cases}$$
where $\mathbf{y} = \mathbf{w^{\top}}\mathbf{X}$ and $c$ is some constant value. So, unlike previously, there is a cut-off in the function.
Is it still possible to solve this via SVD (or find other closed-form solution)? Alternatively, could I rewrite the problem so that the piecewise part becomes a constraint to the problem?
The previous question is available here, which might be helpful for context.