Particle Swarm Optimization with non-smooth penalty reformulation, for a complex portfolio selection problem

In the classical model for portfolio selection the risk is measured by the variance of returns. It is well known that, if returns are not elliptically distributed, this may cause inaccurate investment decisions. To address this issue, several alternative measures of risk have been proposed. In this contribution we focus on a class of measures that uses information contained both in lower and in upper tail of the distribution of the returns. We consider a nonlinear mixed-integer portfolio selection model which takes into account several constraints used in fund management practice. The latter problem is NP-hard in general, and exact algorithms for its minimization, which are both effective and efficient, are still sought at present. Thus, to approximately solve this model we experience the heuristics Particle Swarm Optimization (PSO). Since PSO was originally conceived for unconstrained global optimization problems, we apply it to a novel reformulation of our mixed-integer model, where a standard exact penalty function is introduced.