I am struggling in formulating the dual of a very simple problem:
\begin{align} \max_{x_{i},y_{j}} \sum_{i=1}^{I}c_{i}x_{i} - \sum_{j=1}^{J}(\delta_{j}y_{j}+\gamma_{j}y_{j}^{2}) \hspace{0.2in} \\ s.t \hspace{0.2in} \sum_{i=1}^{I}a_{ij}x_{i}\leq y_{j}\hspace{0.1in}\forall\hspace{0.1in}j,\hspace{0.1in}y_{j}\geq 0,\hspace{0.1in}x_{i}\geq 0 \end{align}
Furthermore, all the parameters $c_{i},\delta_{j},\gamma_{j},a_{ij}$ are strictly positive, such that the problem is concave.
When the problem is linear (all $\delta_{j}$ are equal to 0), the dual is easy to compute. However, once I add the quadratic term, I am not really sure how to proceed.
Any suggestion is very much appreciated!