Karush-Kuhn-Tucker conditions for non-convex problems

Consider a problem (ICP) with equality and inequality constraints:
\begin{align*}
\text{minimize}\quad &f(x) \\
\text{subject to}\quad &h_i(x) = 0,\quad i=1,\dotsc,m\\
                                    &g_j(x) \le 0,\quad j=1,\dotsc,r
\end{align*} where \(f,\;h_i,\; g_j\) are continuously differentiable functions from \(\mathbb{R}^n\) to \(\mathbb{R}\).

With Lagrange function
\[L(x,\lambda,\mu) = f(x) + \sum_{i=1}^m \lambda_i h_i(x) + \sum_{j=1}^r \mu_j g_j(x)\]
Define the set of active inequality constraints:
\[ A(x) =\{ j \;|\; g_j(x) = 0 \}\] A feasible vector \(x\) is regular if the equality constraint gradients \(\nabla h_i(x),\; i=1,\dotsc,m\), and the active inequality constraint gradients \(\nabla g_j(x),\; j\in A(x)\) are linearly independent.

KKT optimality conditions

KKT necessary conditions for regular \(x^*\)

Let \(x^*\) be a local minimum of the problem
\begin{align*}
\text{minimize}\quad &f(x) \\
\text{subject to}\quad &h_i(x) = 0,\quad  i=1,\dotsc,m\\
                                    &g_j(x) \le 0,\quad j=1,\dotsc,r
\end{align*} where \(f,\;h_i,\; g_j\) are continuously differentiable functions from \(\mathbb{R}^n\) to \(\mathbb{R}\) and \(x^*\) is regular. There exist unique Lagrange multiplier vectors \(\lambda^* = (\lambda^*_1,\dotsc,\lambda^*_m)\), \(\mu^*=(\mu^*_1,\dotsc,\mu^*_r)\) such that
\begin{align*}
\nabla_x L(x^*,\lambda^*,\mu^*) = 0, \\
\mu^*_j\ge 0, \quad &j=1,\dotsc,r, \\
\mu^*_j=0,\quad &\forall j \notin A(x^*)
\end{align*}  Note: The condition can be written as
\[\mu^*_j g_j(x^*) = 0, \quad j=1,\dotsc,r\] and aptly referred to as complementary slackness condition.

If in addition \(f\), \(h\), and \(g\) are twice continuously differentiable, there holds
\[ y^T \nabla^2_{xx} L(x^*,\lambda^*,\mu^*)y \ge 0\] for all \(y\in \mathbb{R}^n\) such that
\begin{align*}
\nabla h_i(x^*)^T y &= 0, \quad \forall i=1,\dotsc,m \\
\nabla g_j(x^*)^T y &= 0, \quad \forall j \in A(x^*)
\end{align*}