# How to find the optimal solution of a convex program given all KKT points?

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.

If there is any example, I would be thankful if you can share it.

• I think for convex program, any KKT point is optimal.
– xd y
Aug 13, 2021 at 3:23

In short, you can calculate a set of regions, typically called "critical regions", within which the same set of constraints is active. This in return enables you to calculate $$x$$, $$\lambda$$ and $$\mu$$ as an explicit function of your parameter $$a$$.