Hybrid Adaptive Large Neighborhood Search for the Optimal Statistic Median Problem

In this paper, the problem of maximizing the median of a convex combination of vectors having important applications in finance is considered. The objective function is a highly nonlinear, nondifferentiable function with many local minima and the problem was shown to be APX hard. We present two hybrid Large Neighborhood Search algorithms that are based on mixed-integer programs and include a time limit for their running times. We have tested the algorithms on three testbeds and showed their superiority compared to other state-of-the-art heuristics for the considered problem. Furthermore, we achieved a significant reduction in running time for large instances compared to solving it exactly while retaining high quality of the solutions returned.