PCA with $\mathcal{l}$1-regularization / LASSO with linear constraints

Question

I want to solve the following optimization problems.
For some $\lambda>0$, matrix, $X$ and vectors, $\mathbf{v_1}, \mathbf{v_2}, ..., \mathbf{v_k}$ $$\textrm{max}_\mathbf{v}\lambda\|\mathbf{v}\|_1+\mathbf{v}^TX^TX\mathbf{v}$$ s.t. $$\mathbf{v}\perp\mathbf{v_1}, \mathbf{v_2}, ..., \mathbf{v_k}$$ When $k=0$, this question simplifies to $$\textrm{max}_\mathbf{v}\lambda\|\mathbf{v}\|_1+\mathbf{v}^TX^TX\mathbf{v}$$ without the contraint. This is essentially the LASSO regression with target being all zeros
Could anyone point me any references, existent work or directions on solving this problem?