A class of Dantzig–Wolfe type decomposition methods for variational inequality problems

Abstract

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig-Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems may be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretic models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise intractable if the widely used PATH solver is applied directly, without decomposition.

On error bounds and Newton-type methods for generalized Nash equilibrium problems

“…This is a simpler but important case, which indeed occurs in various practical models [14]; see also [28] for an example. Observe that imposing μ 1 = μ 2 , (2) becomes an instance of the mixed complementarity problem [15] whose solutions can be expected to be isolated.…”

“…Assuming now that u(σ ) solves (28) for a given σ , first note that for the part F of Φ, defined in (19), (28) implies that F (u(σ )) = (a 1 , a 2 ). It thus remains to show that the norm of the complementarity residuals…”

An approximation scheme for a class of risk-averse stochastic equilibrium problems

“…Regarding the merits of the two modelling approaches, it is sometimes argued that the MCP formulation provides an adequate framework for imperfect markets. Nevertheless, at least in a deterministic environment, it is shown in [31] that equivalent equilibria can be obtained by GNEP and by MCP. Since the two models are eventually cast in this paper as related stochastic VIs, we shall carry on our discussion for both of them in parallel.…”

“…At least for our test problems, the single VI for the AVaR reformulation will become unsolvable much earlier than VIs in our approach. Note also that our approach allows for decomposition, and even parallelization (see [31]), thus pushing the boundary even further. In fact, one of the subjects of future research would be precisely combining Benders decomposition [14] along scenarios with Dantzig-Wolfe decomposition for the players [31].…”

“…Section 5 contains numerical results investigating the effects or risk aversion, volatility, and other issues, for part of the real-life European gas network. In addition, it is shown how the Dantzig-Wolfe decomposition [22,31] of VIs can improve solution times in this context.…”