Given the following minimization problem $$min f(x,y)=-x-y$$ in the region described by $ x-4y+3 \geq 0$, $-x+y^2 \geq 0$, $x \geq 0$,
I notice that the origin is an irregular point. In fact, gradients of constraints, evaluated in the origin, are $$\nabla g_1(x,y) = (1,-4) $$ $$\nabla g_2(x,y) = (-1,2y) = (-1,0)$$ $$\nabla g_3(x,y) = (1,0)$$.
If I consider the Jacobian matrix given by $ ( \nabla g_2 \nabla g_3 ) $ (g2 and g3 are the active constraints in the origin), I notice that I don't have full rank, so the gradients are linearly dependent (LICQ violated! Linear independence constraint qualification: the gradients of the active inequality constraints and the gradients of the equality constraints are linearly independent). The exercise solution just states that in this case, we go on because we can't apply KKT conditions, so we aren't able to exclude that (0,0) is a point of minimum.
However, I tried to follow the proceeding that many use to explain the geometrical meaning of KKT conditions and I think I can say something more.
$g_1$ and $g_2$ are the constraints gradients (let's ignore their amplitude). If (0,0) is a min point, then there's no descent direction that is also feasible. In (0,0) the only feasible directions (I need to stay in the region) are $d_1$ and $d_2$. If I want to "forbid" descent feasible direction, I need to force the gradient to belong to x axis. In that way (and this is the only case!) both of feasible directions aren't descent directions. In the exercise situation, $\nabla f$ is $(-1,-1)$ so we just conclude that 0 is not a point of minimum, because one of the feasible directions defines with $\nabla f$ an obtuse angle (i.e. descent direction).
I tried to generalize the reasoning further, I came to the conclusion (probably false!) that when I deal with irregular points, I should consider the cone generated by the gradients. If the cone is not well defined (i.e. the gradients are aligned but opposite) and I have two cones/half-planes generated by them, then I should consider the intersection between half-planes that contain at least a feasible direction. You won't believe me, but I found a variety of cases in which the rule seems to work. Is it a correct idea?
Thanks a lot