To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the intuition behind the claim that, if the Hessian $H$ of a multidimensional differentiable function $f(x_1,…,x_n)$ is positive semi-definite, it must be […]

Let $U$ be an open convex subset of $\mathbb R^n$ and $f:U\to\mathbb R$ a convex function on it. It is a well-known fact that if the second partial derivatives exist everywhere on $U$ and are all continuous (i.e., if $f\in\mathcal C^2$), then the Hessian of $f$ is symmetric, that is, $\partial^2 f/(\partial x_i\partial x_j)=\partial^2 f/(\partial […]

I am having a hard time understanding how to determine the second derivative of a matrix. I have researched Hessian matrices and do not see how i would apply it to vector funciton. problem statement: compute the derivative of the following $$ f(x) =\begin{bmatrix} x_1+x_1x_2^2\\ -x_2+x_2^2+x_1^2\\ \end{bmatrix}$$ I have found $$ DF =\begin{bmatrix} 1+x_2^2&2x_1\\ 2x_1&-1+x_2\\ […]

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le \lambda f(x) + (1- \lambda)f(y)$ for any $x,y \in \mathbb{R}^2$. Hessian matrix is nonnegative-definite if $f_{xx}” x^2 + f_{x,y}(x+y) + f_{yy}”y^2 \ge 0$ I know […]

Are there any functions, $f:U\subset \mathbb{R}^n \to \mathbb{R}$, with Hessian matrix which is asymmetric on a large set (say with positive measure)? I’m familiar with examples of functions with mixed partials not equal at a point, and I also know that if $f$ is lucky enough to have a weak second derivative $D^2f$, then $D^2 […]

Intereting Posts

Is there a function that is contiunous at all the rationals, and discontinuous at all the irrationals?
Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$
Why does every countable limit ordinal have cofinality $\omega$?
Hartshorne exercise III.6.2 (b) – $\mathfrak{Qco}(X)$ need not have enough projectives
About the interior of the union of two sets
Integers can be expressed as $a^3+b^3+c^3-3abc$
Would the answer of the square root of a square root be positive or negative?
Proof that retract of contractible space is contractible
Basic question about mod
Is this a correct/good way to think interpret differentials for the beginning calculus student?
How come $32.5 = 31.5$?
Incorrect logic in popular proof of the irrationality of $\sqrt2$?
Does $\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0 $ for all $n$ imply that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?
$|2^x-3^y|=1$ has only three natural pairs as solutions
Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.