Projection methods for variational inequalities with application to the traffic assignment problem

Abstract

It is well known [2, 3, 16] that if \(\bar T:R^n \to R^n\) is a Lipschitz continuous, strongly monotone operator and X is a closed convex set, then a solution x *∈X of the variational inequality \((x - x^ * )'\bar T(x^ * ) \geqslant 0\), ∨x∈X can be found iteratively by means of the projection method \(x_{k - 1} = Px[x_k - \alpha \bar T(x_k )]\), x 0∈X, provided the stepsize α is sufficiently small. We show that the same is true if \(\bar T\) is of the form \(\bar T = A'TA\), where A:R n→R m is a linear mapping, provided T:R m→R m is Lipschitz continuous and strongly monotone, and the set X is polyhedral. This fact is used to construct an effective algorithm for finding a network flow which satisfies given demand constraints and is positive only on paths of minimum marginal delay or travel time.