7. The PSO Family: Application to the Portfolio Optimization Problem

Nonlinear high-dimensional optimization problems are generally ill-posed and ill-conditioned, with different sets of models located in one or different disconnected valleys of the cost function landscape with similar values. This situation generates uncertainty in the identification of the optimum model parameters that should be translated to the decision that has to be made. Therefore, the analysis of the uncertainty space of this type of problems is required in order to adopt robust decisions. This is done by sampling the cost function topography within the intricate solution set valleys that belong to the nonlinear equivalence region and validating the existence of different scenarios. This is also the case of the portfolio optimization problem, which admits multiple solutions depending on the expected return-risk ratio. As the popular wisdom says, you cannot have the butter and the money of having sold it. This mathematical situation is known as a Pareto front, which aims to show the boundary of the nonlinear equivalence region of the corresponding decision problem. In this chapter, we introduce the concept of uncertainty in high-dimensional problems, proposing the particle swarm optimization family as a parameter free-tuning global algorithm, capable of sampling the nonlinear equivalent region in parallel with the optimization. For that purpose, these algorithms should be exploratory. This feature is related to the automatic tuning of the PSO parameters in the neighborhood of the stochastic second-order stability region of the particle trajectories. These algorithms are faster than Monte Carlo methods. The chapter concludes with the application of PSO to the portfolio optimization problem of the IBEX-35, which is the main stock market index of the Bolsa de Madrid. In this case, the cost function is constructed in a way that the investors seek to maximize the portfolio’s expected return subjected to a given risk. The optimization of the portfolio composition follows a previous selection of the stocks.