Suppose we have $\theta=(\theta_1,\ldots,\theta_n)$, with $v_i:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ being a continuous, differentiable, concave function. Now I want to solve the following maximization problem: $$ \max_{\theta} \sum_{i=1}^{n} v_i(\theta_i,a_i) $$ subject to $$ \theta_i\geq 0,\quad i=1,\ldots,n $$ and $$ \sum_{i=1}^{n} \theta_i \leq c,\quad c\in\mathbb{R}. $$ So far so good. We assume that $a_i$ is a negligible parameter and write down the Lagrangian and use the KKT conditions to derive the optimal solution.
But what if I want to consider $a_i$ as a variable of the objective function and also impose a constraint, that is $$ 0\leq a_i \leq 1,\quad i=1,\ldots,n. $$
Can I still use the KKT conditions in this latter case? Would I have to define two Lagrange multipliers, one for each constraint? I am a little bit confused on how to do it.
I'd appreciate any help. Thank you.