L1.2 - Introduction to unconstrained optimization: first- and second-order conditions (vector case) 

True, the upper index 2 is missing in the box. Note that it should not be interpretted as SQUARING. It is just one possible notation for the matrix of second (mixed) derivatives.

Exactly that - you select epsilon and then stay in a neighborhood within distance that is less than epsilon from the critical point.

You are perfectly right. It should be O(alpha^3).

@@aa4cc Thanks for your reply. Have a nice day!

Indeed, the Hessian matrix being positive semidefinite is a necessary condition of optimality while the the stricter requirement of Hessian being positive definite is a sufficient condition of optimality.

Do you mean second-order necessary condition? Or second-order sufficient one? In the former case, the point is then neither minimum nor maximum. Saddle point. To explain the latter case, consider two functions f(x)=x^3 and g(x)=x^4, both at x=0. Both satisfy the first-order necessary condition and second-order necessary condition at x=0 (first derivative equal zero and second derivative nonnegative). So far so good. But both fail to satisfy the second-order sufficient condition (second derivative strictly positive), and yet one of them is minimized at x=0 while the other is not (just picture the graphs). To detect this, we would have to study higher-order derivatives. For f(x), the third derivative is nozero. Recall that in Taylor's expansion, you have odd powers of independent variables with the third derivative. But that means that the corresponding contribution can have both signs and the point can by neither minimum nor maximum. For g(x), it is the fourth derivative that is nonzero and the corresponding term in Taylor only contains even powers of independent variables, hence for positive fourth derivative the function is minimized. Hope this helps.

@@aa4cc Thank you. Now I see things better!