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.

Work supported by Grants ONR-N00014-75-C-1183 and NSF ENG-7906332.