Consider the following minimization problem,
$$\min_{x\in \mathbb{R}^2} (x_1-x_2^2) \text{ subject to } -x_1^2-x_2^2+2x_1\geq 0.$$
I am asked to find "feasible directions" from the point $(x_1,x_2)=(0,0).$ That is, directions along which the objective decreases.
My notes say the set of feasible directions at a point $x$ is given by:
$$\{ s | <s,\nabla c_i(x)> = 0 \text{ for each equality constraint } c_i, (<s, \nabla c_i(x)>) \geq 0 \text { for each active inequality constraint at } x\} .$$
Are these directions enough to ensure a decrease in the objective? Or do I need to add the condition that $(<s,\nabla f(x)>) < 0$ to ensure that $s$ is descent?
We see that $\{x | f(x) < f(0) \} = \{x | x_1 < x_2^2 \}$, and we see from this (consider $f(0+th)$) that if $h$ is a feasible direction then $h_1 \le 0$.
If we examine feasibility of such $h$, then we need $th$ to be feasible for small $t>0$, or in other words, $-t^2h_1^2-t^2h_2^2+2th_1 \ge 0 $. From this we see that we must have $h_1 \ge 0$ and so $h_1=0$. For $th$ to be feasible, we need $h_2 = 0$ and so there are no (first order) feasible direction.