What are the properties of the solution of a nonlinear programming with linear objective function

Question

In a constrained nonlinear programming with a linear objective function, are there any results/properties of the solution?

The constraints are nonlinear nonconvex.

Answer 1

Any optimization problem whatsoever can be expressed as a constrained nonlinear programming problem with a linear objective function, because the problem \begin{align} \operatorname{minimize} & \quad f(x) \\ \text{subject to} & \quad x \in S \end{align} (with optimization variable $x$) is equivalent to the problem \begin{align} \operatorname{minimize} & \quad t \\ \text{subject to} & \quad t \geq f(x) \\ &\quad x \in S \end{align} (with optimization variables $x$ and $t$).

So no, there is not much that you can say about the solution to such a general problem.