  This paper addresses the robust counterparts of optimization problems
containing sums of maxima of linear functions. These problems include many
practical problems, e.g.~problems with sums of absolute values, and arise when
taking the robust counterpart of a linear inequality that is affine in the
decision variables, affine in a parameter with box uncertainty, and affine in a
parameter with general uncertainty.
  In the literature, often the reformulation is used that is exact when there
is no uncertainty. However, in robust optimization this reformulation gives an
inferior solution and provides a pessimistic view. We observe that in many
papers this conservatism is not mentioned. Some papers have recognized this
problem, but existing solutions are either conservative or their performance
for different uncertainty regions is not known, a comparison between them is
not available, and they are restricted to specific problems. We describe
techniques for general problems and compare them with numerical examples in
inventory management, regression and brachytherapy. Based on these examples, we
give recommendations for reducing the conservatism.
