Consider $x^*$ is a local minimum of the non-convex optimization problem \begin{align} \min_x \quad & f_0(x) \\ \text{subject to} \quad & f_i(x) \leq 0,\quad i=1,\dots,m \\ & h_i(x) = 0, \quad i = 1,\dots,p \end{align}
If regularity conditions hold, then there exist lagrange multipliers $ \lambda^* $ and $\nu^*$ such that $(x^*, \lambda^*, \nu^*) $ all satisfy the KKT conditions, since they are neccesary conditions.
The dual function is $$g(\lambda,\nu) = \inf_x \left(f_0(x) + \sum_{i=1}^m \lambda_i f_i(x) + \sum_{i=1}^{p} \nu_i h_i(x)\right) $$
My question is: If strong duality does not hold , are the lagrange multipliers $ \lambda^* $ and $\nu^*$ from above always dual optimal variables for the dual problem with the dual function $g(\lambda,\nu)$, i.e. they maximize the following problem: \begin{align} \text{maximize} \quad & g(\lambda,\nu) \\ \text{subject to} \quad & \lambda \succeq 0 \end{align} ?
If not, can someone give an easy counterexample?
