Minimizing tracking error while restricting the number of assets

Tracking error minimization is commonly used by the traditional passive fund managers as well as alternative portfolio (for example, hedge fund) managers. We propose a graduated non-convexity method to minimize portfolio tracking error with the total number of assets no greater than a specified integer K. The solution of this tracking error minimization problem is a global minimizer of a minimization problem with a discontinuous counting function in a constraint. We attempt to track the globally minimal tracking error portfolio by approximating the discontinuous counting function with a sequence of continuously differentiable non-convex functions, a graduated non-convexity process. We discuss the advantages of this approach, present numerical results, and compare it with two methods from recent literature.