A Sparsity-Driven Solution Method for the Cardinality Constrained Mean-Variance Portfolio Selection Problem

Cardinality constrained mean-variance (CCMV) portfolio selection problem is commonly formulated as a mixed integer quadratic program (MIQP) that can be solved by a branch-and-bound scheme or metaheuristics. Yet, computational efficiency remains to be a major issue. In this study, we propose a novel metaheuristic method based on the concepts of sparse recovery for solving large-size problems. Using the relationship between investment risk and portfolio sparsity, we develop a "sparsity-driven" binary search framework for solving the CCMV portfolio selection problem through a sequence of quadratically constrained sparse recovery problems. For computational efficiency, a sub-one quasi-norm minimization model is adopted to find high-quality approximate solutions of each sparse recovery problem using a specially designed gradient-descent based algorithm. Then, a final solution to the original CCMV portfolio selection problem is obtained by incorporating the approximate solution information to the binary search framework to solve one MIQP problem in a much more reduced size. Computational experiments using the historical stock data from the S&P's Compustat North America Database provided by Wharton Research Database Service clearly indicate the superior performance of the proposed approach in terms of quality of solutions and speed of computation. It runs orders faster than the commonly used CPLEX solver for solving large-size problems. Comparison with the state-of-the-art perspective reformulation approach in the literature shows that the proposed approach is more efficient for solving large-size CCMV portfolio selection problems, especially for those without a diagonally dominant covariance matrix. Another experiment with a genetic-algorithm-based heuristic method shows that the quality of solutions found by the proposed approach significantly outperforms that of the heuristic method