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When you browse the source, you can see exactly what algorithm is being used--perhaps this can help answer some of your performance questions? For minimize_constrained, Sage calls the multivariate constrained optimization functions from scipy.optimize: either fmin_tnc (truncated Newton's method) or fmin_cobyla (Constrained Optimization BY Linear) or, if requested, fmin_l_bfgs_b (L-BFGS-B algorithm).

If there are any other optimization methods, I would expect them to be in this same source file. Also, you might find some of the other functionality of scipy.optimize useful; in that case, the source here could be a template for how to use it.