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
$$\begin{align*} \nabla h_i(x^*)^T y &= 0, \quad \forall i=1,\dotsc,m \\ \nabla g_j(x^*)^T y &= 0, \quad \forall j \in A(x^*) \end{align*}$$

Braess' paradox

An interesting anomaly in non-cooperative behavior.  (An example where Nash equilibrium is not optimal) See link