There Are Capacity Constraints Of The Network Equilibrium Theory

Comparisons are made among four generalized traffic network equilibrium with capacity constraints. Some relationships among them are obtained. When the ca-pacity constraints are only on the paths, these equilibrium definitions are equivalent except Generalized Wardrop Equilibrium(GWE) and are the subset of GWE. When the capacity constraints are only on the arcs, there does not exist inclusion rela-tionships between GWE and Capacitated User Equilibrium(CUE). However, there exists nonempty intersection between GWE and CUE and the "best" equilibrium in it. Frequently used solutions to the variational inequality with path or arc ca-pacity constraints(VI-PC or VI-AC) are the "best" equilibrium. These solutions are not only expressed by succinct model, but the Prices Of Anarchy(POA) do not depend on the capacity constraints. Generally speaking, the POA of GWE may be unbounded.Based on the analysis on the definitions of the generalized traffic network equi-librium in the preceding chapter, several definitions of generalized traffic equilibrium in multiclass network under general form capacity constraints g(x)< 0 are pre-sented. Firstly, Generalized Wardrop Equilibrium in Multiclass network(MGWE) and the stabilities of the equilibrium solutions are investigated. The stabilities embrace:uniqueness of the sets of generalized shortest routes and uniqueness of generalized equilibrium travel costs. Wardrop-type results are also presented. Sec-ond, in finitely multiclass traffic network, another definition on generalized traf-fic equilibrium under capacity constraints g(x)< 0 is Extended Wardrop Equilib-rium(MEWE). Since A(v) in this definition can be looked as a kind of toll, from another standpoint, the research on MEWE can be thought of as the existence of tolls in corresponding network. The definition and property of Extended Wardrop Equilibrium in Infinitely Multiclass network(IMEWE) are investigated at the end of Chapter 3.When we take the capacity constraint functions g(x) = x - xS, where xS is a system optimal solution vector, Slater condition does not hold. In fact, the con-straints are/∫0amax x(a)da=xs. The conditionΩv(a)∩intDv(a)≠φ(in Chapter 3) isn't satisfied too. Infinitely multiclass traffic network equilibrium problem with this kind of constraints can be rewrite as an infinite linear programming problem(ILPP) with capacity constraints. The purpose of Chapter 4 is to investigate this kind of ILPP. In two cases:only one origin-destination(OD) pair and multiple OD pairs, we construct special feasible solutions to the ILPP. By virtue of the nature of net-work, we prove that the solutions are optimal. In 2009, Marcotte and Zhu proved the existence of the optimal tolls in infinitely multiclass traffic network equilibrium problem. Based on this conclusion, we analyze the property of these tolls by us-ing the optimal solutions constructed. We also investigate the optimal solutions to the ILPP in the general network where OD pairs may be differ in their probability density functions.For a multiclass network equilibrium problem, the multiplier vectors corre-sponding to capacity constraints in an auxiliary problem (some linear programming with implicit variables) are one kind of valid tolls. In order to compute the multi-plier vectors in this linear programming problem, we consider the computation of saddle points of the corresponding (augmented) Lagrangian function. In this linear programming problem with formal variables, the feasible set cannot be expressed explicitly. But if we replace the formal variables with actual ones, the new difficulty comes:the actual variables are infinitely dimensional. We first get that there exists some subsequence (in nonexact solutions sequence from Uzawa algorithm) which converges to some saddle point of the Lagrangian function. By discretizing the augmented Lagrangian function with actual variables, we prove that there exists a nonexact solutions sequence in the corresponding Uzawa algorithm. So we can ob-tain some approximated saddle point. Simultaneously, we obtain one approximated valid toll.