# Minimum of the sum of two functions

I want to show that trying to find the minimum of the sum of two or more functions of two different groups is a not convex problem. For example: $\min\limits_{Y,Z} f(X,Y,Z)=...$. Moreover the values $X,Y,Z$ are matrices. My idea is to show that the Hesse-Matrix of the sum of those added functions is not always positive semidefinit by finding a point x, where it is not psd?
$\operatorname{H}_f({Y},{Z})= \left(\frac{\partial^2f}{\partial c_i\partial c_j}({Y},{Z})\right) \begin{pmatrix} \frac{\partial^2 f}{\partial c_1\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_1\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_1\partial c_n}({X},{Y})\\[0.5em] \frac{\partial^2 f}{\partial c_2\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_2\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_2\partial c_n}({X},{Y})\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial^2 f}{\partial c_n\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_n\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_n\partial c_n}({X},{Y}) \end{pmatrix}$
with $c_i \in X,Y$, for example $c_i=x_{11}$ Then I pick one $X,Y,Z$ and show the the eigenvalues are $<0$. P.S. log is elementwise

• Why should the sum of non-convex functions be convex in general? I don't understand your question. Aug 21 '15 at 15:08
• I just want to proof that there is no combination for the sum of my non convex function and some of my other functions so that the result is convex. (I think it can happen that the sum of a non convex and another non convex or convex function might be convex) Aug 21 '15 at 15:30
• if you only want to show that your example is not convex, please edit your question accordingly. Aug 21 '15 at 15:33
• No, it is not only for the example, my functions are much more difficults. I deleted it in the question ;) Aug 21 '15 at 15:38
• Well, but for your question, it is only about your example isn't it? Notice that there are some sums of non convex function which are convex, and some which are non convex. So in full generality your question makes little sense to me. Aug 21 '15 at 15:44