A novel hybrid PSO-based metaheuristic for costly portfolio selection problems

Annals of Operations Research (2021) 304:109–137 https://doi.org/10.1007/s10479-021-04075-3 ORIGINAL RESEARCH A novel hybrid PSO-based metaheuristic for costly portfolio selection problems Marco Corazza1 · Giacomo di Tollo1 · Giovanni Fasano2,3 · Raffaele Pesenti2 Accepted: 7 April 2021 / Published online: 21 April 2021 © The Author(s) 2021 Abstract In this paper we propose a hybrid metaheuristic based on Particle Swarm Optimization, which we tailor on a portfolio selection problem. To motivate and apply our hybrid metaheuristic, we reformulate the portfolio selection problem as an unconstrained problem, by means of penalty functions in the framework of the exact penalty methods. Our metaheuristic is hybrid as it adaptively updates the penalty parameters of the unconstrained model during the optimization process. In addition, it iteratively refines its solutions to reduce possible infeasibilities. We report also a numerical case study. Our hybrid metaheuristic appears to perform better than the corresponding Particle Swarm Optimization solver with constant penalty parameters. It performs similarly to two corresponding Particle Swarm Optimization solvers with penalty parameters respectively determined by a REVAC-based tuning procedure and an irace-based one, but on average it just needs less than 4% of the computational time requested by the latter procedures. Keywords Hybrid metaheuristics · Particle Swarm Optimization · Global optimization · Portfolio selection problems · Exact penalty functions · REVAC · irace B Giacomo di Tollo Marco Corazza Giovanni Fasano Raffaele Pesenti 1 Department of Economics, Università Ca’ Foscari, Venezia, Sestiere di Cannaregio 873, 30121 Venezia, Italy 2 Department of Management, Ca’ Foscari University of Venice, Sestiere di Cannaregio 873, 30121 Venezia, Italy 3 National Research Council – Maritime Technology Research Institute (CNR – INSEAN), Via di Vallerano 139, 00128 Rome, Italy 123 110 Annals of Operations Research (2021) 304:109–137 1 Introduction Setting the parameters used within an algorithm is a key-point to insure its reliability, performances, robustness, and scalability. Although many approaches resort to experts’ judgement to determine the algorithm’s parameter values (see Kotthoff et al. 2019), the literature proposes a great number of parameter setting procedures (Lobo et al. 2007). As in Eiben et al. (1999), we can partition these approaches in parameter tuning techniques (also referred to as off-line configuration), which determine the algorithm parameters values before the algorithm execution, and parameter control techniques (also referred to as on-line control), which continuously update the parameter values during the algorithm execution. On this guideline, also Particle Swarm Optimization (PSO) has been used to assess the parameters of other algorithms. In this regard we have for instance: (a) (Hong 2009), where parameters value for a Support Vector Regression model are determined, using chaotic PSO, (b) (Lin et al. 2008), where PSO is used to set parameters for Support Vector Machines, (c) (Si et al. 2012) that uses PSO to tune Differential Evolution parameters. Conversely, several approaches have also been proposed in the literature to determine PSO parameters value. These approaches get started from extensive studies on PSO parameters (inertia weight and coefficients), since the early PSO related research (Clerc and Kennedy 2002; Eberhart and Shi 2001; Shi and Eberhart 1998a, b). In this context, Trelea (2003), Campana et al. (2010) study the possible range for PSO parameters in order to evaluate their impact on convergence. Methodologies and concepts to determine PSO parameter values can be partitioned in tuning and control methods. Our contribution can be framed in this latter class of methods that in the PSO jargon are also referred as to adaptive. Amongst parameter tuning procedures, Dai et al. (2011) proposes the idea of using an additional PSO scheme that analyses the impact of each PSO parameter, while Wang et al. (2014) proposes to use Taguchi method. In addition, other general purpose procedures of this type could also be applied to PSO such as: (1) statistical procedures to evaluate parameter settings and to eliminate candidate parameters configurations that are dominated by others (Trujillo et al. 2020; Birattari et al. 2010); (2) meta-heuristic methods to explore the candidate configurations space (Nannen et al. 2008; Hutter et al. 2007); (3) sequential model-based optimisation in order to define both a correlation between parameter settings and algorithm performance, and to identify high-performing parameter values (Hutter et al. 2011); (4) other approaches, including Bayesian Optimization (Eggensperger et al. 2013), jointly used with Gaussian process (Snoek et al. 2012), Random Forests (Hutter et al. 2011), and Tree Parzen Estimator (Bergstra et al. 2011) (see Huang et al. 2019 for a detailed overview of parameter tuning approaches). Generally speaking, parameter tuning may be time consuming: this is why tuning is often done by using cheap synthetic test functions that may turn to be rather different from the real benchmarks, or by using cheap-to-evaluate surrogates of real hyperparameter optimization benchmarks (Eggensperger et al. 2015). Amongst control procedures we find: Shi and Obaiahnahatti (1998), which presents a basic adaptive procedure for the assessment of PSO parameters that makes the inertia weight decrease linearly over time; Zhan and Zhang (2008), which introduces the Adaptive Particle Swarm Optimization (APSO) that defines four evolutionary states to control the inertia weight and the acceleration coefficients (along with other parameters); Hsieh et al. (2009), which proposes an adaptive population management procedure to automatically determine the population size; Winner et al. (2009), which employs non-explicit control parameters that 123 Annals of Operations Research (2021) 304:109–137 111 describe self-organizing systems at an abstract level; Tang et al. (2011), which uses the search history collected by particles to determine acceleration coefficients; time-varying acceleration coefficients are considered also in Ratnaweera et al. (2004a). Stemming algorithms derived from Genetic and Evolutionary Algorithms can be also seen as control procedures for PSO. As an example, this is the case when mutation operators are introduced to avoid premature convergence, as suggested by many contributions (Si et al. 2011; Sharma and Chhabra 2019; Jana et al. 2019; Wang et al. 2019). Recently, a mechanism to control the balance between exploration and exploitation has been detailed in Xia et al. (2020) (Dynamic Multi-Swarm Global Particle Swarm Optimization), and a great attention to define learning strategies to increase swarm diversity was given in Zhang et al. (2020). The interested reader can find a comparative analysis among PSO schemes in, e.g., Harrison et al (...truncated)