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#### numbersense

##### New member

- Mar 26, 2013

- 5

The gradient is $\nabla q(\textbf{x}) = G \textbf{x} + \textbf{d}$ and the Hessian is $\nabla^2 q(\textbf{x}) = G$.

If $q(\textbf{x})$ is a strictly convex function then show that $G$ is positive definite.

I am not sure whether I should start with the convex function definition or start by considering the gradient or the Hessian.

I tried expanding the inequality in the convex function definition but didn't get anywhere.

There is a proposition that says $f$ is strictly convext on $\mathbb{R}^n$ $\implies$ any stationary point is the unique global minimizer. (I can't even prove that a stationary point exists) Another theorem says that positive definiteness is a sufficient condition for being a unique global minimizer and positive semi definiteness is a necessary condition for being a local minimizer. I can't see how to use these statements to prove what the question is asking.