| This thesis focuses on the stochastic variational inequalities (VI) with linear constraints. We present a new residual function defined by the gap function in Chapter 2. The expected residual minimization (ERM) formulation is a nonsmooth optimization problem with linear constraints. We prove the Lipschitz continuity and semismoothness of the objective function and the existence of minimizers of the ERM formulation.;In Chapter 3, we propose a globally convergent (a.s.) smoothing sample average approximation (SSAA) method for finding a minimizer of the ERM formulation. We show that the SSAA problems of the ERM formulation have minimizers, and any cluster point of minimizers (stationary points) of the SSAA problems is a minimizer (stationary point) of the ERM formulation (a.s.) as the sample size goes to infinity and the smoothing parameter goes to zero.;We discuss the ERM formulation for the stochastic linear VI in Chapter 4, which is convex under some mild conditions. We apply the Moreau-Yosida regularization to present an equivalent smooth convex minimization problem. To have the convexity of the SAA problems of the ERM formulation, we adopt the Tikhonov regularization. We show the convergence results of the regularized SAA problems, and prove the semismoothness of the gradients of the regularized SAA problems.;In Chapter 5, we discuss the distributionally robust stochastic linear VI. We introduce the CVaR formulation defined by the ERM formulation and establish the relationship between the CVaR and the ERM formulations. For lots of cases, we show that the two formulations have the same minimizers. We employ the sublinear expectation to consider the distributionally robust CVaR formulation, and prove the existence of minimizers of it.;In Chapter 6, we show the assumptions imposed in this thesis hold in traffic flow problems. Moreover, numerical results illustrate that solutions of the ERM formulation have desirable properties. |
