Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2009 / B / 6 / a / Solution

The contingent tangent cone consists of feasible first-order velocities:If $x_{\ast }$ is a local minimizer, then $\boxed{\nabla f(x_{\ast }{)}^{T}d\ge 0\text{ for every }d\in T_{\Omega }(x_{\ast })}$. Indeed, $x_n\to x_{\ast }$ along the defining sequence, so eventually $f(x_n)-f(x_{\ast })\ge 0$. Differentiability givesThis proves the inequality. Using feasible limiting velocities avoids assuming that every tangent direction is realized by a straight segment staying in the feasible set.