<t>How nonlinear is the constraint? Did you try non-feasible interior point methods? For these ones you don't need a feasible starting point. If the nonlinear constraint is quadratic or involves norms/distances then you might be able to formulate your problem in a 'Conic Programming' form and solve ...
I am a PhD student in Operations Research (Linear, Semidefinite and Stochastic Programming). I am curious to know if there are any financial institutions (especially from NYC) interested in people with such a background.