Writing objective that depend on minimization according to DCP ruleset


I want to minimize the following concave function:

G(\boldsymbol{u},\boldsymbol{v})=\min_{y\in{Y}} \sum_{i=1}^M -2u_{i}\sqrt{\boldsymbol{a}'\cdot \boldsymbol{y}+\alpha_{i}}+v_i(\boldsymbol{b_{i}}'\cdot \boldsymbol{y}+\beta_{i})

with the vectors
and the scalars \alpha_{i},\beta_{i} all being positive.

Is there a way to write this objective according to the DCP ruleset?
I’ve tried to solve the minimization problem on y to get an explicit expression but could not.