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

Question

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.

Answer 1

This sounds like parametric programming to me.

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 $a$.

There are MATLAB tools to do this, most prominently the POP and MPT toolboxes, which support linear and quadratic (mixed-integer) problems.