I am looking for a way to qualify the difficulty of the resolution of a nonliner programming. In other words, what let us say that, for two nonlinear programmings involving the same nature of function, a problem is more difficult than the other.
For example in solving linear systems, the conditionning number of the matrix is a good indicator about the difficulty.
I am looking for indicators involving for exapmple: the magnitude of Lagrange multipliers, magnitudes of active constraints gradients, Hessian matrices...
Thank you in advance.
The real difficulty of (non-convex) nonlinear programming tends to be global rather than local in nature. That is, is often quite easy for a solver to converge to a local optimum, but this might not be the global optimum. One might consider how many local optima there are and how high and wide are the barriers between them. Unfortunately, for interesting problems it tends to be at least as difficult to measure those quantities as it is to find the optimal solution.