Joint work with Youngdae Kim and Andy Philpott

We introduce an extended mathematical programming framework for specifying equilibrium

problems and their variational representations, such as generalized Nash equilibrium, multiple optimiza-

tion problems with equilibrium constraints, or quasi-variational inequalities, and computing solutions of

them from modeling languages. We dene a new set of constructs with which users annotate variables and

equations of the model to describe the equilibrium and variational problems. Our constructs 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 sophisticated structures such as shared constraints and additional constraints on

their solutions. We dene a new concept, shared variables, and demonstrate its uses for sparse reformulation,

equilibrium problems with equilibrium constraints, a mixed pricing behavior of agents, and so on. The shared

variables also enable back-end solvers to exploit the problem structure for improved performance. We give

some equilibrium and variational examples from the literature and describe how to formulate them using

our framework. Experimental results comparing performance of various complementarity formulations for

shared variables combined with structure exploitation of the back-end solver are given. Our framework has

been implemented and is available within GAMS/EMP. Agents problems will be formulated to allow risk

preferences and some computational results related to energy systems and other applications will be given.