This paper provides a refined a posteriori error control for
the obstacle problem with an affine obstacle which allows
for a proof of optimal complexity of an adaptive algorithm.
This is the first adaptive mesh-refining
finite element method known to be of optimal
complexity for some variational inequality.
The result holds for first-order conforming finite element methods in
any spacial dimension
based on shape-regular triangulation into simplices for an affine obstacle.
The key contribution is the discrete reliability of the a
posteriori error estimator from
[Numer. Math. 107 (2007), 455–471]
in an edge-oriented modification which circumvents the difficulties caused
by the non-existence of a positive second-order approximation
[Math. Comp. 71 (2002), 1405–1419].