Stability of the linear complementarity problem properties under interval uncertainty

TL;DR

This paper investigates how key properties of the linear complementarity problem are preserved under interval uncertainty, providing characterizations and efficient checks for robust properties of interval matrices.

Contribution

It introduces a framework for analyzing the stability of LCP properties under interval data, including characterizations and subclasses for robust properties.

Findings

Characterization of robust properties of interval matrices

Efficient recognition of subclasses with stable properties

Extension of LCP property analysis to uncertain data environments

Abstract

We consider the linear complementarity problem with uncertain data modeled by intervals, representing the range of possible values. Many properties of the linear complementarity problem (such as solvability, uniqueness, convexity, finite number of solutions etc.) are reflected by the properties of the constraint matrix. In order that the problem has desired properties even in the uncertain environment, we have to be able to check them for all possible realizations of interval data. This leads us to the robust properties of interval matrices. In particular, we will discuss $S$-matrix, $Z$-matrix, copositivity, semimonotonicity, column sufficiency, principal nondegeneracy, $R_0$-matrix and $R$-matrix. We characterize the robust properties and also suggest efficiently recognizable subclasses.