A comparative study of heuristic methods for cardinality constrained portfolio optimization

The cardinality constrained mean–variance (CCMV) portfolio selection model aims to identify a subset of the candidate assets such that the constructed portfolio has a guaranteed expected return and minimum variance. By formulating this model as the mixed-integer quadratic program (MIQP), the exact solution can be solved by a branch-and-bound algorithm. However, computational efficiency is the central issue in the time-sensitive portfolio investment due to its NP-hardness properties. To accelerate the solution speeds to CCMV portfolio optimization problems, we develop various heuristic methods based on techniques such as continuous relaxation, $l_1$-norm approximation, integer optimization, and relaxation of semi-definite programming (SDP). We evaluate our heuristic methods by applying them to the US equity market dataset. The experimental results show that our SDP-based method is effective in terms of the computation time and the approximation ratio. Our SDP-based method performs even better than a commercial MIQP solver when the computational time is limited. In addition, several investment companies in China have adopted our methods, gaining good returns. This paper sheds light on the computation optimization for financial investments.