I am working on a function with $11$ input parameters and would like to know if the function is convex. I am aware of the general definition that the Hessian matrix of the function should be positive semidefinite. However, given the complexity of my function, it is problematic to (at least analytically) calculate the Hessian matrix.
So I am wondering if there is any way to get a feeling for the convexity of my function. Similar to the univariate case where the following can be checked:
$$ f:X \rightarrow R \\ \forall x_1,x_2\in X, \forall t \in [0,1] : f(t x_1+(1-t)x_2)\le tf(x_1)+(1-t)f(x_2) $$
How should I extend this to an $11$-dimensional case? Or should I rather aim for a numerical approximation of my Hessian matrix?