On the unconstrained optimization reformulations for a class of stochastic vector variational inequality problems

Abstract

Abstract In this paper, a class of stochastic vector variational inequality (SVVI) problems are considered. By employing the idea of a D-gap function, the SVVI problem is reformulated as a deterministic model, which is an unconstrained expected residual minimization (UERM) problem, while it is reformulated as a constrained expected residual minimization problem in the work of Zhao et al. Then, the properties of the objective function are investigated and a sample average approximation approach is proposed for solving the UERM problem. Convergence of the proposed approach for global optimal solutions and stationary points is analyzed. Moreover, we consider another deterministic formulation, i.e., the expected value (EV) formulation for an SVVI problem, and the global error bound of a D-gap function based on the EV formulation is given.

Keywords

• Stochastic vector variational inequality
• D-gap function
• Sample average approximation
• Convergence
• Error bound