# Maximizing the sum of nonnegative functions.

I was trying to solve the problem A maximization problem when I ask myself if the general problem

$$\begin{array}{c} maximize\hspace{1cm} f(\mathbf{X})^p +g(\mathbf{X})^p \\ s.t. \hspace{1cm} \mathbf{X} \in K \subseteq \mathbb{R}^{m \times n}, \end{array}$$ is equivalent to

$$\begin{array}{c} maximize\hspace{1cm} f(\mathbf{X}) +g(\mathbf{X}) \\ s.t. \hspace{1cm} \mathbf{X} \in K \subseteq \mathbb{R}^{m \times n}, \end{array}$$ when the scalar functions $f(\mathbf{X})$ and $g(\mathbf{X})$ are nonnegative on $K$, and $p > 0$.

Is this true? If not, how to find a counterexample?

Take $f(x)=(2-x)$ and $g(x)=\sqrt x$ for $x \in [0,2]$

Take $p=2$

$h_1(x)=f+g$

$h_2(x)=f^2+g^2$

$h_1$ is maximum for $x=\dfrac{1}{4}$ and $h_2$ is maximum for $x=0$ (edited after comment...), with its minimum for $\dfrac{3}{2}$

• $x=3/2$ is the minimum of $h_2$, not the maximum. – Alex Silva Jan 5 '15 at 13:29
• @AlexSilva $h_1(0)=2$ while $h_1(1/4)=9/4>2$. – Martigan Jan 5 '15 at 13:38
• Yes! You're right. I have seen it. Thanks for the answer. :) – Alex Silva Jan 5 '15 at 13:39