Abstract:

This talk explores the geometric foundations of linear complementarity theory, with a focus on the challenging problem of finding a satisfactory characterization of Q-matricity. The latter concept is closely linked to the existence of solutions for Linear Complementarity Problems (LCPs) and their system counterpart, Linear Complementarity Systems (LCSs). At the core lies a covering problem of the Euclidean space by a finite number of structured closed convex cones.  A spatial characterization via a finite local test set will be presented followed by a discussion of the main challenges that arise in higher dimensions. In addition, necessary separation conditions for Q-matricity, inspired by a classical characterization of one its subclasses (P-matricity) will be outlined, highlighting a possible direction for further exploration.