Hi Fedja - just updated with a description of the algorithm. Thanks!

Hi fedja, thanks for spending some time on it. As mentioned, I currently solve this problem (well, find some local minimum I suppose) using the barrier method. I remain hopeful that there is something provable about the optimality of my solutions, given that the solutions i get are simply very good and much better than competing methods. Furthermore, the algorithm always achieves the correct solutions for toy problems to which I know the answer a priori.

Hi Fedja - there isn't sufficient space in the comments to write the whole problem out, so I will make a new question... Thanks!

Ok, I have fixed a bug in my Monte-Carlo sim and now see the non-convexity there as well. Thanks for all the responses! This was very helpful. This question is certainly answered, however, I now need to pose the next logical question...

Hi fedja - this function is a component of the objective in an optimization problem, in which I seek to find the vector $\mathbf{x}$ which minimizes the maximum squared error between the function in the norm and a target, e.g., $\vert x_1 + x_2 + x_3 - x_1 x_2 x_3 - d\vert^2$, where $d$ is a constant.

Hi Denis, thanks for your thoughtful reply. I think I understand your reasoning, but when I calculate the second derivative of the restricted function and substitute $(-1,\frac{1}{2})$, I get 1.375 as the second derivative with respect to $x_1$. Here is the expression for that second derivative that I get: \begin{equation} 6x_1^2 x_2^2 + x_2^4 + 6x_1x_2^3 \end{equation} What am I missing here?

Thanks - I removed the background info and tried to add the full expression for general N, but the LateX processing seems to fail on \array's? So there should be a newline between the $-x_N$ and the $x_N$, and also between the $-x_1$ and the $x_1$, and between the right 1 and 0.