We introduce interfaces and algorithms for (quasi-) variational inequalities and equilibrium problems, such as generalized Nash equilibrium and multiple optimization problems with equilibrium constraints. For interfaces, we present an extended mathematical programming framework through which users can specify the variational and equilibrium problems and compute their solutions from modeling languages. The framework defines a new set of constructs that enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of complicated structures, such as shared constraints and shared variables. For algorithms, we present PATHAVI and SELKIE.

PATHAVI is a structure-preserving pivotal method for affine variational inequalities, and SELKIE is a generalization of the diagonalization method. These algorithms enable various solution methods to be employed under a single problem specification. Our interfaces and algorithms have been implemented and are available within GAMS.