Decomposition and Approximation Methods for Variational Inequalities, with Applications to Deterministic and Stochastic Energy Markets
Juan Pablo Luna
Decomposition | variational inequality | Energy Market.
We present contributions in decomposition algorithms for variational inequalities (VI) and energy market equilibrium problems. Concerning the first issue, there were studied two families of decomposition algorithms for VI that extend the Dantzig-Wolfe and Benders decomposition algorithms for linear programming. The generalization was made in two aspects: significantly weaker hypotheses for ensuring convergence, and the possibility of solving approximately some subproblems that arise in the algorithms. The computational tests show the interests on the decomposition algorithms when the size of the problems considered increases. For the second issue, a new approach through Generalized Nash Games allows to find variational equilibria (VE) in energy markets, in both deterministic and stochastic contexts. It turns out that, if the market does not present risk aversion, the new model provides the same solutions as the usual model based on Mixed Complementarity Problems (MCP). The new model leads to a VI where separable structures, hidden in the MCP approach, are evident. Thanks to this separability, the decomposition methods studied can be applied for computing the VE. Yet, if the agents are risk averse, the stochastic equilibrium obtained by the two models no longer coincide and leads to different economic interpretations. When the market is risk averse, the VE results from solving a multivalued VI. We propose a new smoothing technique adapted to the structure of games that solves the problem via a sequence of singlevalued VI.