Strict local minimiser

Question

Let $\Omega$ be a convex subset of $R^n$ adm f is a real valued, twice differentiable function. Let $x^*$ to be a point in $\Omega$ and suppose that there exists $c \in R \, c >0$ s.t. for all feasible directions d $d\neq0$ at $x^*$ we have that $d^Tf'(x^*) \geq 0$ and $d^T f''(x^*)d \geq c||d||^2$. How to show that $x^*$ is a strict local minimizer of f.

Answer 1

Use the Taylor expansion $$f(x^* + t \, d) = f(x^*) + f'(x^*) \, d + \frac{t^2}2 \, d^\top f''(x^*) \, d + o(\|t\,d\|^2). $$ If $d$, $\|d\| \le 1$, is a feasible direction, you obtain for some $T > 0$, that for all $t \in [0,T]$ you have $$ f(x^* + t \, d) \ge f(x^*) + \frac c4 \, \|t \, d\|^2. $$ This shows that $x^*$ is a strict local minimizer.