By Yosef Sheffi
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Additional resources for Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods
Example text
In both cases, it is assumed that the existence and smoothness conditions mentioned above hold (either for any value of the argument, in the case of an unconstrained program, or for the feasible values, in the case of constrained programs). Unconstrained Minimization Programs It is well known from elementary calculus that the necessary condition for a differentiable function in one variable, z(x), to have a minimum at x = x* is that the derivative of z(x) evaluated at x* equals zero. 1, however, the first-order condition is a minimum of z(x).
To solve the traffic assignment problem, it is required that the rule by which motorists choose a route be specified. As explained above, this rule can be viewed as the function or the procedure that specifies the demand for travel over paths. The interaction between the routes chosen between all O-D pairs, on the one hand, and the performance functions on all the network links, on the other, determines the equilibrium flows and corresponding travel times throughout the network. It is reasonable to assume that every motorist will try to minimize his or her own travel time when traveling from origin to destination.
7]. The sufficient conditions include strict convexity in the vicinity of the minimum point (this guarantees that it is a local minimum) and convexity of both the objective function and the feasible region (this guarantees that there are no other local minima). This concludes the discussion of single-variable programs. The concepts introduced in this section are expanded in the following section to the multivariable case. 5 Two local minima for a strictly convex function bounded by a constraint set that is not convex.