I have a nonlinearly-constrained model and wonder if a nonlinear solver like Ipopt or Knitro can solve the problem.
Briefly, my objective function is linear. I have the following variables with their associated domains: $d^\tau_i, d^\rho_o, y_{ij}, \eta_{ij}\in \mathbb{R}_{\geq 0}$ and $x_i, z_{oo^\prime}\in \{0,1\}$. The used sets are $\mathcal{I}$, $\mathcal{O}$, and $\mathcal{V}=\mathcal{I}\cup\mathcal{O}$. You may assume all variables are created based on $\mathcal{V}$ to sort of ease the interpretability. All symbols other than the aforementioned are parameters in $\mathbb{R}_{\geq 0}$. I have a bunch of linear constraints listed below (big M has a tight bound):
\begin{equation} \sum_{o\in\mathcal{O}}y_{io} \leq K_i \qquad \forall i\in\mathcal{I}. \end{equation}
\begin{equation} K_i x_i \geq y_{io} \qquad \forall i\in\mathcal{I}, o\in\mathcal{O}. \end{equation}
\begin{equation} L_i^\tau \leq d^\tau_i \leq U_i^\tau \qquad \forall i\in\mathcal{I}. \end{equation}
\begin{equation} L_o^\rho \leq d^\rho_o \leq U_o^\rho \qquad \forall o\in\mathcal{O}. \end{equation}
\begin{equation} y_{o o^\prime} \leq \mathbb{M} z_{oo^\prime}, z_{oo^\prime}+z_{o^\prime o} \leq 1 \qquad \forall o,o^\prime \in \mathcal{O} \end{equation}
and the following nonlinear constraints:
\begin{equation} \sum_{o^\prime\in\mathcal{O}\land o^\prime\neq o}y_{oo^\prime} \leq \sum_{j\in\mathcal{V}\land j\neq o}\eta_{jo}y_{jo} \qquad \forall o\in\mathcal{O}. \end{equation}
\begin{equation} \sum_{j\in\mathcal{V}\land j\neq o}\eta_{jo}y_{jo} \geq E_o \qquad \forall o\in\mathcal{O}. \end{equation}
and the following additional nonlinear constraints including the exponential function:
\begin{equation} \eta_{io} = 1-e^{\displaystyle d^\tau_i d^\rho_o} \qquad \forall i\in\mathcal{I}, o\in\mathcal{O}. \end{equation}
\begin{equation} \eta_{oo^\prime} = 1-e^{\displaystyle d^\rho_o d^\rho_{o^\prime}} \qquad \forall o,o^\prime\in\mathcal{O}, o\neq o^\prime. \end{equation}
As you may notice, the above are the sources of the nonlinearity. As a person living in the linear world, I am not sure (aware) if Knitro or Ipopt can handle such a model as there exists a chain of continuous variable multiplications. If they can do so, what do you think the magnitude of $|\mathcal{V}|$ it can tackle?
I guess, I should also clarify that I am not chasing the global optimality when I ask if the above nonlinear solvers can handle the problem. A locally optimal solution (if one exists) could be satisfactory.
