Maximization of non-convex function over a convex set.

Question

I have ${n\choose 2}$ non negative variables $0\leq v_{ij}\leq \frac{1}{2}$, $1\leq i<j\leq n$, and I'm trying to solve the maximization $$ \max_{v_{ij}}\sum_{\begin{subarray}{l}|S|=r\\ S\subseteq [n]\end{subarray}}\prod_{i<j\in S}(\frac{1}{2}+v_{ij})\\s.t\;\; \sum_{1\leq i<j\leq n}v_{ij}=c\\\;\;\;\;\;\;\;\;\;\;0\leq v_{ij}\leq \frac{1}{2}$$ I was able to show that $$v_{ij}=\frac{c}{{n\choose 2}}$$ is a local maximum, and I wonder if I can prove that it is a global one as well? I'll appreciate any pointers.