A Matrix-Based Evolutionary Algorithm for Cardinality-Constrained Portfolio Optimisation

Abstract

This article presents a matrix-based evolutionary algorithm to approximate solutions of the simultaneous multiple portfolio optimisation problem under cardinality constraints, for a selection of indices containing from $n=31$ to $n=493$ assets. This problem is made NP-hard by the requirement to find the best sub-portfolios of k < n assets (in practice, k << n) from the vast number of possibilities and, simultaneously, the efficient frontier (EF) for these sub-portfolios. We study algorithm performance under a spread of cardinality constraint values, finding that there exists a small subset of k < n assets for a given dataset with which it is possible to obtain a close approximation of the unconstrained EF. Computation times can be significantly reduced using this trick. Finally, by pooling results from a number of independent realisations and employing a sifting algorithm to the pooled results, we obtain significantly improved estimates of the EFs for the cardinality-constrained problem.