Hybridising Local Search With Branch-And-Bound For Constrained Portfolio Selection Problems

In this paper, we investigate a constrained portfolio selection problem with cardinality constraint, minimum size and position constraints, and non-convex transaction cost. A hybrid method named Local Search Branch-and-Bound (LS-B&B) which integrates local search with B&B is proposed based on the property of the problem, i.e. cardinality constraint. To eliminate the computational burden which is mainly due to the cardinality constraint, the corresponding set of binary variables is identified as core variables. Variable fixing (Bixby, Fenelon et al. 2000) is applied on the core variables, together with a local search, to generate a sequence of simplified sub-problems. The default B&B search then solves these restricted and simplified subproblems optimally due to their reduced size comparing to the original one. Due to the inherent similar structures in the sub-problems, the solution information is reused to evoke the repairing heuristics and thus accelerate the solving procedure of the subproblems in B&B. The tight upper bound identified at early stage of the search can discard more subproblems to speed up the LS-B&B search to the optimal solution to the original problem. Our study is performed on a set of portfolio selection problems with non-convex transaction costs and a number of trading constraints based on the extended mean-variance model. Computational experiments demonstrate the effectiveness of the algorithm by using less computational time.