I solve a nonlinear optimization problem of the form
\begin{align} &\max x_0 \text{ such that } \\ &\left[ \sum_{j=0}^N \left(\alpha_j x_0^{j} \prod_{k=3}^j x_k \right)\right]^2 + \left[ \sum_{j=1}^N \left(\beta_j x_0^{j} \prod_{k=3}^j x_k \right)\right]^2 \overset{!}{\leq} 1 \end{align}
where I know relatively tight bounds of the optimization variables $\boldsymbol x$, something between $0.1\% - 30 \%$ depending on the component. These stem from solutions for a different value of $N$. I can observe that for increasing the dimensionality $N \to N+1$, the values change only merely (this is where the bounds come from). However, the optimizer I am currently using (IpOpt
) takes quite some time even for relatively small allowed deviations.
Thus my question: Is there a well-known way how to treat those long products of optimization variables, or are these just notoriously difficult problems?
I tested setting some variables as constant, but the constraint is very sensitive and even small deviations matter (this is also probably why this problem is hard). In that case, the optimization was often found infeasible.