On the R$_0$-tensors and the solution map of tensor complementarity problems

TL;DR

This paper explores the properties of the solution map in tensor complementarity problems, emphasizing the role of R0-tensors and employing semi-algebraic geometry techniques to analyze stability and continuity.

Contribution

It introduces new insights into the stability and semicontinuity of solution maps for tensor complementarity problems, focusing on the significance of R0-tensors.

Findings

R0-tensors are crucial in understanding solution map properties

The solution map exhibits finite-valuedness under certain conditions

Semi-algebraic geometry techniques reveal lower semicontinuity results

Abstract

Our purpose is to investigate the local boundedness, the upper semicontinuity, and the stability of the solution map of tensor complementarity problems. To do this, we focus on the set of R$_0$--tensors and show that this set plays an important role in the investigation. Furthermore, by using a technique in semi-algebraic geometry, we obtain some results on the finite-valuedness and the lower semicontinuity of the solution map.