I am struggling to understand the Concavity of function in higher dimensions (more than one).
For instance, $f(x,y)=-x^2-y^2$, is a cup-down paraboloid which looks pretty much concave, as shown below:
But when proven mathematically it is confusing to apply and understand the mathematical result.
- Let us apply the known fact that sum of concave functions results in another concave function. Let's say $-x^2$ is concave in $\mathbb{R}^2$; similarly $-y^2$ is concave in $\mathbb{R}^2$ too. Therefore, $-x^2-y^2$ which is a sum of two concave functions is concave too in $\mathbb{R}^2$.
I don't know whether the argument of "sum of two concave functions is another concave function" is equally valid in dimensions more than one. But here at least it is consistent with the intuitive picture of the plot. Any comments on this please.
- Secondly, let's find concavity from another way: determinant of Hessian matrix. The determinant of Hessian matrix of $-x^2-y^2$ turns out to be 4, which shows positive definiteness.
This indicates convexity instead of concavity and which certainly is not true. How the definition of determinant of hessian matrix got changed in higher dimensions? Reference to any literature/book would be highly appreciated.
You have some confusion in point 2. The Hessian of $f$ is $$ Hf(x,y) = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix}. $$ The Hessian matrix $Hf(x,y)$ is negative semidefinite for all points $(x,y)$. This shows that $f$ is concave.
(Why are you computing the determinant of the Hessian? There are several different ways to check that $Hf(x,y)$ is negative semidefinite, but perhaps the easiest way is to note that its eigenvalues are all negative.)