Maximizing the Sum of Cubes I have ten variables, $x_1$ through $x_{10}$. $-1 \leq x_i \leq 1$, and $\sum x_i = 0$. What is the maximum of $\sum x_i^3$? I've tried to use Lagrange multipliers, but writing the restricted domain as a constraint seems, if not impossible, then very difficult.

You can write it in a standard form 
\begin{equation}\notag
\begin{split}
\min & -\sum_{i=1}^{10} x_i^3 \\
\mathrm{s.t.} \hspace{1ex}& x_i - 1 \leq 0 \\
& -1 -x_i \leq 0 \hspace{1ex} \forall i \in [1, 10]\\
& \sum_{i=1}^{10} x_i = 0 \\
\end{split}
\end{equation}
Then the Lagrange Dual will be
\begin{equation}
L(x,\lambda, \mu)=-\sum_{i=1}^{10} x_i^3+\sum_{i=1}^{10} \lambda_i (x_i-1) - \sum_{i=1}^{10} \mu_i (x_i+1) + \gamma \sum_{i=1}^{10} x_i \hspace{2ex} \lambda_i, \mu_i \geq 0 
\end{equation}
