Stochastic $R_0$ Tensors to Stochastic Tensor Complementarity Problems

TL;DR

This paper introduces the stochastic tensor complementarity problem, demonstrating that the solution set is nonempty and bounded when the tensor is an $R_0$ tensor, and establishing conditions for stochastic $R_0$ tensors.

Contribution

It defines the stochastic tensor complementarity problem and characterizes the solution set properties using stochastic $R_0$ tensors, providing necessary and sufficient conditions.

Findings

Solution set is nonempty and bounded for $R_0$ tensors.

Stochastic $R_0$ tensor condition is necessary and sufficient.

Expected residual minimization is used to analyze solutions.

Abstract

The main purpose of this paper is devoted to an introduction of the stochastic tensor complementarity problem. We consider the expected residual minimization formulation of the stochastic tensor complementarity problem. We show that the solution set of the expected residual minimization problem is nonempty and bounded, if the associated tensor is an $R_0$ tensor. We also prove that the associated tensor being a stochastic $R_0$ tensor is a necessary and sufficient condition for the solution set of the expected residual minimization problem to be nonempty and bounded.