In lecture 2 page 22 formula (2) I don't understand why the condition is stated as positive SEMI definite.
isn't a minimum guaranteed only when second order derivative is positive (not semi-definite positive). For ex: x^3 at x=0 is not a minimum but "hessian" (f''(x)) is semi-definite positive since it is >=0.
The condition holds when the problem is convex. Quoting:
"For a convex optimization problem...".
Indeed, if you are analysing a "general" function, the condition holds if the hessian is PSD for each possible value in the domain of the function.
So, x^3 is not convex (since its second derivative is not >= 0 for each x), and thus you can't state that when its first derivative is 0, that is a minimum.
second order condition
Hi,
In lecture 2 page 22 formula (2) I don't understand why the condition is stated as positive SEMI definite.
isn't a minimum guaranteed only when second order derivative is positive (not semi-definite positive). For ex: x^3 at x=0 is not a minimum but "hessian" (f''(x)) is semi-definite positive since it is >=0.
Thanks for your help
The condition holds when the problem is convex. Quoting:
"For a convex optimization problem...".
Indeed, if you are analysing a "general" function, the condition holds if the hessian is PSD for each possible value in the domain of the function.
So, x^3 is not convex (since its second derivative is not >= 0 for each x), and thus you can't state that when its first derivative is 0, that is a minimum.
1
Add comment