Solving variational inequalities with a quadratic cut method: a primal–dual, Jacobian-free approach

We extend in two directions the Analytic Center, Cutting Plane Method for Variational Inequalities with quadratic cuts, ACCPM-VI (quadratic cuts), introduced by Denault and Goffin in 1998. First, we define a primal–dual method to find the analytic center at each iteration. Second, the Broyden–Fletcher–Goldfarb–Shanno Jacobian approximation, of quasi-Newton fame, is used in the definition of the cuts, making the algorithm applicable to problems without tractable Jacobians. The algorithm is tested on a variety of variational inequality problems, including one challenging problem of pricing the pollution permits put forward in the Kyoto Protocol.