  We consider the problem of minimizing a sum of clipped convex functions;
applications include clipped empirical risk minimization and clipped control.
While the problem of minimizing the sum of clipped convex functions is NP-hard,
we present some heuristics for approximately solving instances of these
problems. These heuristics can be used to find good, if not global, solutions
and appear to work well in practice. We also describe an alternative
formulation, based on the perspective transformation, which makes the problem
amenable to mixed-integer convex programming and yields computationally
tractable lower bounds. We illustrate one of our heuristic methods by applying
it to various examples and use the perspective transformation to certify that
the solutions are relatively close to the global optimum. This paper is
accompanied by an open-source implementation.
