  Linear complementarity problems are a powerful tool for modeling many
practically relevant situations such as market equilibria. They also connect
many sub-areas of mathematics like game theory, optimization, and matrix
theory. Despite their close relation to optimization, the protection of LCPs
against uncertainties -- especially in the sense of robust optimization -- is
still in its infancy. During the last years, robust LCPs have only been studied
using the notions of strict and $\Gamma$-robustness. Unfortunately, both
concepts lead to the problem that the existence of robust solutions cannot be
guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the
latter, a part of the LCP solution is allowed to adjust via a function that is
affine in the uncertainty. We show that this notion of robustness allows to
establish strong characterizations of solutions for the cases of uncertain
matrix and vector, separately, from which existence results can be derived.
  Our main results are valid for the case of an uncertain LCP vector. Here, we
additionally provide sufficient conditions on the LCP matrix for the uniqueness
of a solution. Moreover, based on characterizations of the affinely adjustable
robust solutions, we derive a mixed-integer programming formulation that allows
to solve the corresponding robust counterpart. If, in addition, the certain LCP
matrix is positive semidefinite, we prove polynomial-time solvability and
uniqueness of robust solutions. If the LCP matrix is uncertain,
characterizations of solutions are developed for every nominal matrix, i.e.,
these characterizations are, in particular, independent of the definiteness of
the nominal matrix. Robust solutions are also shown to be unique for positive
definite LCP matrix but both uniqueness and mixed-integer programming
formulations still remain open problems if the nominal LCP matrix is not psd.
