Introduction to Tensor Variational Inequalities

TL;DR

This paper introduces tensor variational inequalities (TVI), extending classical variational inequalities with tensor-based functions, and explores their properties, solvability, and applications to game theory.

Contribution

It defines TVI, links it to game theory, and establishes conditions for its global uniqueness and solvability, advancing the understanding of tensor-based variational inequalities.

Findings

TVI generalizes affine variational inequalities and tensor complementarity problems.

Under certain conditions, TVI has unique solutions.

Structured tensors play a key role in solvability analysis.

Abstract

In this paper, we introduce a class of variational inequalities, where the involved function is the sum of an arbitrary given vector and a homogeneous polynomial defined by a tensor; and we call it the tensor variational inequality (TVI). The TVI is a natural extension of the affine variational inequality and the tensor complementarity problem. We show that a class of multi-person noncooperative games can be formulated as a TVI. In particular, we investigate the global uniqueness and solvability of the TVI. To this end, we first introduce two classes of structured tensors and discuss some related properties; and then, we show that the TVI has the property of global uniqueness and solvability under some assumptions, which is different from the existed result for the general variational inequality.