This is a distribution of .lp files for instances of the
Mean-Variance portfolio problem with min buy-in constranits and
cardinality constraint. These instances have been pre-processed
with the improved Approximate Projected Perspective Reformulation
(AP$^2$R+) technique, which yields an equivalent MIQP formulation
with a (much) better continuous relaxation bound, although not
as good as the bound of the true Perspective Reformulation, due
to the cardinality constraint, except at the root node. In order
to apply the technique of the optimal dual multiplier \lambda^* of
the cardinality constraint

   \sum_{i \in N} x_i \leq k

(with k = 10) in the Perspective Relaxation has to be used; these
values for the instances at hand are distributed on the web site

http://www.di.unipi.it/optimize/Data/MV.html

In order to apply the Perspective Reformulation (Relaxation), the
nonseparable quadratic objective function of the MV problem has to
be partly diagonalized; doing so changes the PR, and therefore
both the nonseparable quadratic part and the value of the optimal
dual multiplier, hence also the separable one. This is why we
distribute three copies of the instances, one for each of the
"small" (s), "large" (l) and "intermediate" (c = convex combination
of s and l with multipliers 0.5) diagonals also distributed on the
web site. Usually the "l" instances have better bounds and therefore
solve faster, although occasionally "c" ones perform better.

Note that in order to recover the original optimal value of the
problem you have to add to the value you obtain from the AP$^2$R+
reformulation the constant

        + k * \lambda^*

(with k = 10).  For this you have to know the value of \lambda^*,
which is also available on the web site.