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So my problem is non-quadratic, but I want to use the usual SQP procedure of minimising the quadratic approximation (obtained from a Taylor expansion) to obtain the next iterate. Do I need to formulate this myself? If so, I'd like to use a finite differences approximation for the gradient and hessian. Does alglib expose any functions that will approximate the gradient and hessian for me?

Hi!

Unfortunately, no - you have to do numerical differentiation and QP problem formulation yourself. Buy you can use ALGLIB QP solvers to solve QP problems after you prepared them.