Abstract
In this paper, we consider stochastic vector variational inequality problems (SVVIPs). Because of the existence of stochastic variable, the SVVIP may have no solutions generally. For solving this problem, we employ the regularized gap function of SVVIP to the loss function and then give a low-risk conditional value-at-risk (CVaR) model. However, this low-risk CVaR model is difficult to solve by the general constraint optimization algorithm. This is because the objective function is nonsmoothing function, and the objective function contains expectation, which is not easy to be computed. By using the sample average approximation technique and smoothing function, we present the corresponding approximation problems of the low-risk CVaR model to deal with these two difficulties related to the low-risk CVaR model. In addition, for the given approximation problems, we prove the convergence results of global optimal solutions and the convergence results of stationary points, respectively. Finally, a numerical experiment is given.
Highlights
Variational inequality problems (VIPs) are a class of equilibrium optimization problems. ey are powerful tools to solve large-scale optimization problems and equilibrium problems, and they have widely used in economic equilibrium, optimal control, countermeasure theory, and transportation planning
Based on the needs of such problems in practical applications, we urgently need to give a reasonable solution which can be regarded as the solution of problem (3) or (5). erefore, we will present the conditional value-at-risk (CVaR) model for solving stochastic vector variational inequality problems
We introduce the definition of value-at-risk (VaR) firstly before giving the low-risk CVaR model of (3)
Summary
Variational inequality problems (VIPs) are a class of equilibrium optimization problems. ey are powerful tools to solve large-scale optimization problems and equilibrium problems, and they have widely used in economic equilibrium, optimal control, countermeasure theory, and transportation planning. In order to meet the needs in practice, many authors began to consider the following stochastic variational inequality problems, denoted by SVI(X, f), which requires an x∗ ∈ X such that x − x∗Tf x∗, ξ(ω) ≥ 0, ∀x ∈ X, ξ(ω) ∈ Ω, a.s.,. Scholars have obtained many theoretical results in the study of stochastic variational inequality problems. In order to make these evaluation indicators comprehensively optimal, scholars began to study the vector variational inequality problems. This paper considers stochastic vector variational inequality problems (SVVIPs). M, ξ(ω): Ω ⟶ Q is a stochastic vector defined on probability space (Ω, F, P) with support set Q ⊂ Rr, and a.s. is the abbreviation for “almost surely” under the given probability measure.
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