Minimizing a sum of clipped convex functions

摘要

We consider the problem of minimizing a sum of clipped convex functions. Applications of this problem 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, that makes the problem amenable to mixed-integer convex programming and yields computationally tractable lower bounds. We illustrate our heuristic methods by applying them 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.