I often meet the initialization problem shown below, it says that some kind of non-linear solver will attempt to handle this problem. As I know it, Dymola uses Newton Method to solve the equation after getting the BLT form of the original DAE system, so what is this nonlinear solver exactly?
In general, a Modelica simulation environment needs to do something more than just relying on a known solver that is based on a derivative-based optimization algorithm. It could be based on already known solvers with adaptive capabilities to switch between solvers (line-search or trust-region approaches) and attempting different solvers parameters. Moreover, in advanced instances of the initialization problem, global optimization strategies are needed to avoid local optima of bad quality or divergence.
Reasoning: There are serious obstacles that may hinder identifying a semi-optimal and not necessarily the optimal solution of the initialization problem:
The need of good initial guesses in order to converge to a correct optimum with a residual cost function numerically near to zero
The Jacobian (or the Hessian) computed for the underlying optimization cost function would be usually singular. Here pseudo inversion is rather used (not necessarily provided by common solvers)
Due to the potentially large-scale nature of the underlying problem, values of different variables and initial conditions would lie within a large numerical range of values. This would cause numerical instability of known standard solvers. Here, scaling techniques are followed (which converts all optimization variables under consideration to the value 1 or smaller range of values by logarithmic scaling techniques). Such techniques are usually not provided by common solvers and need to be externally applied to the formulation of the cost function.
The provided initial conditions by the modeler might be incomplete and inconsistent and the simulation environment needs to implement rigorous heuristics for guessing a good start values for the solver.
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