# Commercial solvers for non-convex optimization problems

Is there any recommended commercial non-convex solver for bilinear equality constraints? E.g.,

\begin{aligned} x_1 x_2 &= x_3 \\ x_1 x_3 &= x_4 \\ &\vdots \\ x_1 x_n &= x_{n+1} \end{aligned}

Non-commercial solver IPOPT can solve it efficiently, and I would like to try some commercial solvers and comparing them.

• Why not use Lagrange multipliers? Jul 17, 2023 at 12:21
• Could you explain more? I didn't get your point..... thanks Jul 19, 2023 at 11:10
• What kind of objective function do you have? Please provide an example. You might consider using Lagrange multipliers, one multiplier per equality constraint. It would also be nice to know what motivated this question. Where do these constraints come from? Jul 19, 2023 at 11:14
• I tried to estimate the decay parameter in Exponentially Weighted Moving Average (EWMA) model: $y_{t} = \sum_{k=1}^K \frac{\lambda^k}{\sum_{j=1}^K \lambda^j } y_{t - k} + e_t$, where $K<N$. $x_1 = \lambda$, $x_{k>1} = \frac{\lambda^k}{\sum_{j=1}^K \lambda^j }$ Jul 19, 2023 at 11:31
• The objective function is the standard quadratic function for linear least squares and $\lambda^K \neq 0$ Jul 19, 2023 at 11:34