I think this is possibly a very deep question.

Could I suggest that we agree on some terms? If have (possibly non convex) constrained optimization problem, we can certainly approach it in a very general way -- perhaps exploring dual formulations. Many assignment problems however descend from an artificial objective function, chosen so that its first order conditions for an optimum provide the user equilibrium "rules" that we have come to expect (see great text and explanation of Prof Sheffi for example). So, ... when we get these conditions, we can then perhaps embed them in an iterative scheme, or a set of successive convergent steps. I think the answer to your problem is that the careful implementation of these convergent algorithms beats doing the problem as a monolithic optimization task. I also recommend works of Prof Nagurney for the cases when you do not have just a simple static case. I'll be very interested to see if you can get some other answers.

We do not know the limit to how big networks or how accurate solutions we can obtain, but it is definitely the case that the standard case of the problem - positive and strictly increasing travel costs as functions only of the volume of traffic on the link itself, and perhaps a likewise strictly decreasing but positive demand function - we can solve as big problems as we want, to a very good accuracy. (The best codes are TAPAS and a few more.) So great is the accuracy from these solvers in fact that it in some sense is absurd, as the data is not very accurate to begin with (and often simply made up), and the model itself is too simple to catch real phenomena in real traffic networks. 

Solving the problem using Lagrangean relaxation is something I have some experience with - see my paper http://link.springer.com/article/10.1007%2Fs10107-014-0772-2#page-1 - or download it from my RG page! The perhaps best one sofar is the one we invented in this paper. But it does not really compete with the very best primal methods - although if you want to extend the model somewhat with additional constraints the dual scheme might win.

Here is the local RG link: 

https://www.researchgate.net/publication/262008873_Primal_convergence_from_dual_subgradient_methods_for_convex_optimization

Article Primal convergence from dual subgradient methods for convex ...