In my engineering problem, there are about 30 variables and 30 nonlinear equations, and there can be more than 100 solutions. Only one unique solution is what I need. The following system is a simple one: $$x_1+x_2+x_3x_4=0$$ $$x_5+x_3+x2x_6=0$$ $$x_6+x_4+x_1x_5=0$$ $$-9054x_5=0$$ $$1406x_2-1507x_5+2449x_6+4768x_3+2449x_5x_2-5370x_5x_3+9802x_2x_6+1406x_6x_3-5370=0$$ $$-8207x_3-4794=0$$ When there is a perturbation on one of the coefficients(I mean the coefficients which are not 1), the system of equation becomes a different one. For this new problem, is it possible to track the previous unique solution without solving all of the solutions?
I have tried some local optimization method like gradient descent, but it's easy to fall into local minimum and it cannot be proved that the obtained solution is exactly corresponding to the previous one.