I want to find optimal $\beta \in \mathbb{R}^r$ that solves the following quadratic program: $$ \min_{\beta \in \mathbb{R}^r}\;\; \left(G\beta\right)^{\prime}W\left(G\beta\right)\;\;\; \text{s.t.}\;\; ||\beta||_1 \leq c_1, $$ where $G$ is a $q \times r$ matrix, and $W$ is a $q\times q$ positive definite and symmetric matrix.
As you can see, I am adding a "Lasso" constraint to a quadratic problem that has, as one of its solutions, the vector $\beta =\left(0,0,\cdots,0\right)$. I want to avoid this trivial solution. This is not the only solution.
I thought about adding a second constrait of the form $||\beta||_1 \geq c_2$. The problem with this approach is that the constraint is not convex.
My question is if there is a more convenient approach (that preserves convexity) to incorporate some restriction in the previous problem such that the trivial solution is not feasible.