Cramer's theorem in the large deviations theory gives the rate function $\sup_{\boldsymbol{\lambda}} \left<\boldsymbol{\lambda},\,\mathbf{b}\right>-\log\int_{\mathbb{R}^n} \exp(\left<\boldsymbol{\lambda},\,\mathbf{x}\right>)\,P(dx)$, which is the Legendre transform of $P$'s cumulant generating function.
My problem is how to find the argmax's in the above expression if $P$ is so complicated, such as constructed using Archimedean copulas, that differentiation is almost impossible? Either closed or numerical forms is accepted.
I need this maximizer to solve the minimizer of relative entropy under equality constraint, cf. Altun & Smola, 2006, Unifying divergence minimization and statistical inference via convex duality.
I've searched for R and MATLAB functions capable of dealing with this kind of problem, with no useful results.