Decomposition and sampling methods for stochastic equilibrium problems

In this thesis, we consider three equilibrium problems in which agents solve continuous stochastic nonlinear programs.; In Chapter 2, we consider stochastic Nash games in which agents solve two-stage stochastic quadratic programs (QPs). We show that the resulting equilibrium point may be obtained by solving a larger stochastic QP. An inexact-cut L-shaped method for two-stage QPs is presented. Convergence of the resulting algorithm is proved. We also present a cut-sampling method that uses a sample of the cuts to construct probabilistic bounds on the optimal value. The sampling method results in solutions that are within 3% of the optimal solution when a fixed sample size of 5000 is used at each iteration.; Interior point methods for mathematical programs with complementarity constraints (MPCC) have been studied extensively over the past decade. In Chapter 3, we present a method that ensures that the iterates converge to a second-order KKT point. Convergence theory for the method is provided and demonstrate the performance of the algorithm on a test problem set.; We consider the generalization of the MPCC to the two-stage case under uncertainty: a stochastic MPCC. Such problems arise in the study of Stackelberg equilibria under uncertainty. In Chapter 4, a new primal-dual method is described for this class of ill-posed stochastic nonlinear programs. The method relies on sampling to construct the linearized KKT system, which is subsequently solved using a scenario-based decomposition. Computational results from a test set of stochastic MPCCs are provided.; In Chapter 5, we construct a two-period spot-forward market under uncertainty. Such games may be formulated using a Nash-Stackelberg framework. Instead, we propose a simultaneous stochastic Nash game that requires the solution of a stochastic complementarity problem. We prove that under certain conditions, the simultaneous stochastic Nash equilibrium (SSNE) is a Nash Stackelberg equilibrium. We present a sampling-based iterative-decomposition method for solving such problems efficiently and provide convergence theory for the method. Scalability is demonstrated on a class of stochastic complementarity problems. We also provide some policy-based insights using a 6-node model of an electricity market.