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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.

Rodrigo de Azevedo
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Stephen Ge
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  • Why not use Lagrange multipliers? – Rodrigo de Azevedo Jul 17 '23 at 12:21
  • Could you explain more? I didn't get your point..... thanks – Stephen Ge Jul 19 '23 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? – Rodrigo de Azevedo Jul 19 '23 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 } $ – Stephen Ge Jul 19 '23 at 11:31
  • The objective function is the standard quadratic function for linear least squares and $\lambda^K \neq 0$ – Stephen Ge Jul 19 '23 at 11:34
  • If the objective and constraints are quadratic, why not use symbolic methods instead? Take a look at this – Rodrigo de Azevedo Jul 19 '23 at 11:40
  • Thank you, I have a look and test the speed, BTW, the ipopt can solve it within 0.5 sec with K = 500, N = 5000. – Stephen Ge Jul 19 '23 at 11:48

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