- This paper focuses on the description and computation of the B-differential of the
componentwise minimum of two affine vector functions. This issue arises in the
reformulation of the linear complementarity problem with the Min C-function. The
question has many equivalent formulations and we identify some of them in linear
algebra, convex analysis and discrete geometry. These formulations are used to state
some properties of the B-differential, like its symmetry, condition for its complete-
ness, its connectivity, bounds on its cardinality, etc. The set to specify has a finite
number of elements, which may grow exponentially with the range space dimension
of the functions, so that its description is most often algorithmic. We first present
an incremental-recursive approach avoiding to solve any optimization subproblem,
unlike several previous approaches. It is based on the notion of matroid circuit and the
related introduced concept of stem vector. Next, we propose modifications, adapted
to the problem at stake, of an algorithm introduced by Rada and ˇCerný (SIAM J Dis-
cret Math 32(1):455-473, 2018, https://doi.org/10.1137/15M1027930) to determine
the cells of an arrangement in the space of hyperplanes having a point in common.
Measured in CPU time on the considered test-problems, the mean acceleration ratios
of the proposed algorithms, with respect to the one of Rada and ˇCerný, are in the range
15..31, and this speed-up can exceed 100, depending on the problem, the approach
and the chosen linear optimization and matroid solvers.