Stability of the linear complementarity problem properties under interval uncertainty
Milan Hladík
TL;DR
This work addresses the linear complementarity problem (LCP) under interval uncertainty by developing robust (strong) characterizations for several matrix classes relevant to LCP solvability and solution structure. It derives concrete, efficiently checkable criteria that guarantee properties such as solvability, convexity of the solution set, and finiteness of solutions for all realizations within an interval matrix ${\boldsymbol A}$. The paper covers $S$-, $Z$-, copositive, semimonotone, principally nondegenerate, column sufficient, $R_0$-, and $R$-matrices, detailing both general robust conditions and tractable special cases, supplemented by an illustrative example. The findings provide practical tools for ensuring LCP properties in the presence of data uncertainty, with implications for robust optimization and interval analysis.
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.