perhaps, you may check the Matlab Q&A discussions like this one

https://www.mathworks.com/matlabcentral/answers/489600-the-best-way-to-write-a-large-number-of-constraints-for-an-optimization-problem

Hi Tan, long time no see, how have you been?

x_1^2 + x_2^2 = x_3*x_4 has the form the Rotated Quadratic Cones. You may like to check the MOSEK modeling cookbook.

For the interface, you could use CVXPY in python. Note that, MOSEK is commercial license, for open-sourced license I would suggest SCS or Clarabel.

https://docs.mosek.com/modeling-cookbook/cqo.html

Son Le x_1^2 + x_2^2 = x_3*x_4 is nonconvex, if you relax it by inequality x_1^2 + x_2^2

Tan Pham That's interesting, maybe cutting plane or branching method can be a solution. Have you found the appropriate solver?

Hi Tan Pham , as far as I known, Gurobi has support for this kind of nonconvex equality constraints x_1^2 + x_2^2 = x_3*x_4. In short, they use the bilinear transformation and branching methods.

Hope this help.