I am trying to understand the geometric interpretation of KKT conditions for nonlinear programs.
Consider the following nonlinear program: $$\min\ f(x)$$ $$s.t.\ g_i(x)\leq 0,\ i=1,...,m$$ $$h_j(x)=0,\ j=1,...,n$$
$\textbf{Definition}$. For a feasible point $x$, the linearizing cone is defined as $$L(x)=\{d| \langle \nabla g_i(x),d \rangle \leq 0,(i,g_i(x)=0),\ \langle \nabla h_j(x),d \rangle = 0, \forall j \}$$.
$\textbf{Question}$. What is the significance of this cone? why is it called linearizing? and why does it contain all the tangent cone at $x$?