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Complementarity problems and variational inequalities arise in a wide varietyof areas, including machine learning, planning, game theory, and physicalsimulation. In all of these areas, to handle large-scale problem instances, weneed fast approximate solution methods. One promising idea is Galerkinapproximation, in which we search for the best answer within the span of agiven set of basis functions. Bertsekas proposed one possible Galerkin methodfor variational inequalities. However, this method can exhibit two problems inpractice: its approximation error is worse than might be expected based on theability of the basis to represent the desired solution, and each iterationrequires a projection step that is not always easy to implement efficiently.So, in this paper, we present a new Galerkin method with improved behavior: ournew error bounds depend directly on the distance from the true solution to thesubspace spanned by our basis, and the only projections we require are onto thefeasible region or onto the span of our basis.

Galerkin Methods for Complementarity Problems and Variational Inequalities