Beckmann's mathematical programming formulation

Define a function

$$z(\vec q) := \sum_a \, \int_0^{q_a} \, t_a(\omega) \, d\omega \ .$$ (28.7)

The sum is over all links $a$; for each link, we integrate over the travel time as flow increases, up to the flow $q_a$ actually used on that link.

This is a function which maps high-dimensional space into a scalar number. The number of dimensions is the number of links in the network.

I am not aware of an intuitive motivation for this function. It just turns out that it works: Minimization of this function subject to

$$\sum_p f^{rs,p} = Q^{rs} \ \ , \ \ f^{rs,p} \ge 0$$ (28.8)

and together with the definitions from above gives the desired equilibrium solution. This is actually not too hard to show. However, the derivation does not give any intuitive insight why $z(\vec q)$ is the correct function.

With this transformation, the equilibrium problem is transformed into a constrained optimization problem. Optimization problems are in general much better understood than equilibrium problems.