Convex optimisation with nonconvex constraint function

Question

May I ask for a convex optimisation problem (convex/concave objective with convex feasible region), if some of the constraint function is not convex, is kkt still sufficient and necessary for optimality (under slater condition)?

Answer 1

So long as the (min) objective function is convex and the feasible region is convex, then the solution is not affected by some (or all) constraints being written in a non-convex form.

The constraints affect the solution's validity only by their description of the feasible region. In particular a non-convex constraint can be added to a problem, and if the new constraint happens to be satisfied on the feasible region already determined by earlier constraints, it does not affect the solution at all.

In other cases one might express a convex feasible region using a non-convex function. For example, we might have constraints:

$$ \begin{align*} x &\ge 0 \\ \sqrt{x} &\le 2 \end{align*} $$

Because these constraints define the convex region $x \in [0,4]$, the choice to present that feasible region using the non-convex square root function does not invalidate the use of criteria suitable for convex optimization problems. One can in principle always rewrite the constraints that determine a convex region using convex functions.