We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions of all the agents. An important characteristic of our approach is that the agents hedge risk in the objective functions (on costs or profits) of their optimization problems, which has a clear economic interpretation. This feature is obviously desirable, but in the risk-averse case it leads to variational inequalities with set-valued operators -- a class of problems for which no established software is currently available. To overcome this difficulty, we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents' problems, such as average or conditional value-at-risk. The smoothed variational inequalities can be tackled by the PATH solver, for example. The approximation scheme is shown to converge, including the case when smoothed problems are solved approximately. An interesting by-product of our proposal is that smoothing the average value-at-risk yields another risk measure (differentiable but not coherent). To assess the interest of our approach, numerical results are presented. The first set of experiments is on randomly generated equilibrium problems, for which we show the advantages of our approach when compared to the standard smooth reformulation of minimization involving the max-functions (such as the average value-at-risk). The second set of experiments deals with part of the real-life European gas network, for which Dantzig-Wolfe decomposition is combined with the smoothing approach.
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View An approximation scheme for a class of risk-averse stochastic equilibrium problems