Suppose we have a parametric convex program with some constraints. \begin{equation} \begin{split} \max_{x} \: & f(x,\mathbf{a})\\ & g_1(x,\mathbf{a})\le 0 \\ & g_2(x,\mathbf{a}) \le 0 \end{split} \end{equation}
where $\mathbf{a}$ is a vector of parameters. I can obtain all KKT points and their corresponding Lagrangian multipliers. I am wondering if it is possible to find the optimal solution based on KKT points and multipliers by conditioning on them? I mean, I want to find conditions and then say if condition 1 is true, the optimal solution will be the first KKT point, and so on.