Portfolio optimization deals with the selection of the best portfolios that maximize the wealth resulting from an investment (Köksalan & Şakar 2016;Masmoudi & Abdelaziz 2018). In the 1950s, Markowitz proposed a way to quantify the return and risk of a security and introduced a mathematical mean-variance (MV) model (Markowitz 1952༛ Markowitz 1959), the modern portfolio theory, to allocate a certain amount of funds among investment alternatives considering the compromise between their returns and risks. The MV model is a bi-objective optimization problem that can be solved by a mathematical programming method.
After the seminal work of Markowitz, portfolio optimization has been a popular research topic in finance. Different portfolio optimization models have been proposed, and most of them are extensions of the classical MV model that have additional or variational objectives, e.g., skewness (Li et al. 2010), volatility in the portfolios (Ehrgott et al. 2004), semivariance (Markowitz 1959), mean absolute deviation (Konno & Yamazaki 1991), minimax (Young 1998), value at risk (Jorion 1996; Cui et al. 2018) and conditional value at risk (Rockafellar & Uryasev 2000), or additional constraints, e.g., boundaries (Speranza 1996), cardinality (Hardoroudi et al. 2017), transaction costs (Paiva et al. 2019), and transaction lots (Golmakani & Fazel 2011). For comprehensive reviews of the MV model and its extensions, the reader is referred to Metaxiotis and Liagkouras (2012), Kolm et al. (2014), and Kalayci et al. (2019).
As additional objectives and constraints are included and the problem size in real applications increases, mathematical programming and exact methods have proven inefficient in acquiring the Pareto optimal front of the problem (Ertenlice & Kalayci 2018; Altinoz & Altinoz 2019), and researchers have paid particular attention to the development of approximation methods. Due to their inherent parallel computation ability, evolutionary algorithms (EAs) have shown high performance in solving multiobjective optimization problems that are very complex; therefore, they have become the method of choice for portfolio optimization problems that have two or more objectives (Kalayci et al. 2019). These methods include classical algorithms such as the genetic algorithm as well as the latest swarm-based algorithms that are inspired by the behaviors of animals living in herds. The representative EAs for solving the MV model and its extensions are presented briefly as follows: the GA was the first EA used to solve portfolio optimization (Metaxiotis & Liagkouras 2012), and afterwards, the well-known EAs for multiobjective optimization problems were adapted for the solution of portfolio optimization, e.g., the vector evaluated genetic algorithm (VEGA) for portfolio optimization with additional cardinality, floor and round-lot constraints (Skolpadungket et al. 2007); the niched Pareto genetic algorithm II (NPGA-II) for additional cardinality-constrained portfolio optimization models (Anagnostopoulos & Mamanis 2011b); the nondominated sorting genetic algorithm II (NSGA-II) for portfolio optimization models with various risk measures (Anagnostopoulos & Mamanis 2011a); the strength Pareto evolutionary algorithm 2 (SPEA2) for classical MV models (García et al. 2011); the Pareto archived evolution strategy (PAES) for MV models extended by four constraints (cardinality, quantity, preassignment and round lot) (Lwin et al. 2014); the multiobjective evolutionary algorithm based on decomposition (MOEA/D) (He & Aranha 2020) for classical MV models; and the Pareto envelope-based selection algorithm (PESA) for portfolio optimization models with three objectives (risk, return and the number of securities) (Anagnostopoulos & Mamanis 2010). Recently, an increasing number of swarm-based algorithms have been adopted, e.g., particle swarm optimization (PSO) for robust multiobjective portfolio models with higher moments (skewness and kurtosis) (Chen & Zhou 2018); ant colony optimization (ACO) for MV models with additional cardinality constraints (Deng & Lin 2010); artificial bee colony (ABC) for fuzzy portfolio selection models (Gao et al. 2019); brain storm optimization (BSO) for portfolio optimization problems considering transaction fees with no short sales (Niu et al. 2016); bacterial foraging optimization (BFO) for portfolio optimization models with the practical constraints of a minimum buy-in threshold and maximum limit, cardinality, etc. (Mishra et al. 2014); the firefly algorithm (FA) for MV models with additional cardinality and bounding constraints (Tuba & Bacanin 2014); the bat algorithm (BA) for cardinality-constrained portfolio optimization (Kamili & Riffi 2016); the fireworks algorithm (FWA) for MV models extended by additional constraints of cardinality, boundaries, transaction lots, etc. (Bacanin & Tuba 2015); cat swarm optimization (CSO) for cardinality-constrained portfolio optimization (Kamili & Riffi 2015); krill herd (KH) for portfolio optimization models with additional constraints of cardinality, boundaries, transaction lots, etc. (Tuba et al. 2014); cuckoo search (CS) for the MV model and its approximation model with cardinality and boundary constraints (El-Bizri & Mansour 2017); and invasive weed optimization (IWO) for MV-based models incorporating the P/E, market experts' recommendation criteria (Pouya et al. 2016), etc.
With the different EA approaches, a set of nondominated solutions that are close to the Pareto optimal front of the portfolio optimization problem and are evenly distributed are produced. However, confronted with such a large number of portfolios, an investor is usually at a loss regarding how to select a subset or only one that is best in his/her interests (Köksalan & Phelps 2007). Thus, to obtain an investor’s most satisfactory portfolio, an evolutionary multiobjective optimization (EMO), which can incorporate investor’s preference information into the solution process, is appropriate (Köksalan & Şakar 2016). There are three principal methods for the incorporation of a decision maker’s (DM's) preference information into the solution process (Branke et al. 2008), the a priori method, a posteriori method and interactive method, in which the DM articulates his/her preferences before, after and during the solution process, respectively. By the a priori method, the DM gives his/her global preferences in advance, which is very difficult when he/she knows little about the problem (Battiti & Passerini 2010;Guo et al. 2020). The a posteriori method requires a DM to make a selection in terms of his/her true preferences from a large number of Pareto optimal points. However, this is often impractical in reality while the human mind is limited to handling a small number of information pieces simultaneously (Cruz-Reyes et al. 2014; Tomczyk & Kadzinski 2020a). In contrast, the interactive method allows the users to learn from the solution process and adjust their preferences progressively, which makes it more suitable in capturing a DM’s preferences (Branke et al. 2008; Cruz-Reyes et al. 2020). Another advantage of interactive method is that it can greatly reduce the computational time since it focuses on the search of a certain region of interest on the Pareto optimal front of an optimization problem (Ojalehto et al. 2016; Ruiz et al. 2020).
An interactive method has three essential aspects (Xin et al. 2018): a search engine, preference information and preference model. The search engine is used to produce a set of nondominated solutions to a multiobjective optimization problem. The preference information is articulated by the DM in light of the set of solutions produced. The preference model is built from the DM’s preference information and is used to guide the search engine. Many of the aforementioned EAs have been used as search engines due to their excellence in parallel computation (Deb 2020; Meignan et al. 2015). The DM’s preference information can be given in the form of quantification or qualification. Reference points (Deb et al. 2006; Filatovas et al. 2020), reference directions (Li et al. 2017), trade-offs (Branke et al. 2001), weights (Ruiz et al. 2009), preference regions (Hu et al. 2017), quantitative evaluations (Li et al. 2019), etc., have the form of quantification. Extreme point selection (Fowler et al. 2010; Koksalan & Karahan 2010) and pairwise comparison (Phelps & Köksalan 2003; Battiti & Passerini 2010; Branke et al. 2015; Branke et al. 2016) are examples of qualification forms. Because of the great cognitive effort required, preference information articulated by a DM in quantified form may contain a high percentage of noise and thus may have a counterproductive effect in aiding decisions (Wang et al. 2015; Branke et al. 2016). In contrast, preference information in qualified form, especially in the form of holistic pairwise comparisons, requires much less cognitive effort from a DM and hence has become prevalent (Ciomek et al. 2017; Tomczyk & Kadzinski 2020b). There are two main types of preference models used in interactive methods, utility function-based and outranking-based models, which are in agreement with the ideas of the American and the European school of thought on multicriteria decision making (MCDM), respectively (Braun et al. 2017). With the utility function-based preference model, the Pareto optimal solutions are ordered in a complete rank that represents the DM’s preferences. The utility functions include achievement scalarizing functions (ASFs) (Ruiz et al. 2015; Luque et al. 2020), linear functions (Phelps & Köksalan 2003), polynomial functions (Deb et al. 2010), general polynomial functions (Mukhlisullina et al. 2013), Tchebycheff functions (Ozbey & Karwan 2014) and additive functions (Branke et al. 2015; Branke et al. 2016). The outranking-based preference model is expressed in a relational form that enables the comparison of the solutions according to reference profiles from each category (Doumpos et al. 2009). This model has been widely used in interactive methods (Govindan & Jepsen 2016; Cruz-Reyes et al. 2020).
The application of interactive methods to portfolio optimization is a challenging task due to the complexity of various aspects of the optimization problem that are considered simultaneously, and published papers addressing this challenge are still scarce (Fernandez et al. 2019). Recently, Fernandez et al. (2019, 2020) investigated an interactive approach to solving a portfolio optimization problem considering the criteria of expected returns and financial and technical indicators. This approach uses an outranking-based preference model and a differential evolution algorithm. The values of the parameters of the preference model are elicited from the DM directly and indirectly through his/her holistic judgments (pairwise comparisons on a set of solutions). The interactive method for a portfolio optimization problem with four objectives (return on investment, risks as measured by the Sharp and Treynor indices and maximum investment) in Zhou-Kangas & Miettinen (2019) utilizes an ASF as a preference model, by which a reference point specified by the DM is incorporated. Ruiz et al. (2020) applied three interactive methods—the weighting achievement scalarizing function genetic algorithm (WASF-GA), gradient-based NSGA-II (g-NSGA-II) and the parallel multiobjective genetic algorithm (P-MOGA)—to a portfolio optimization problem with three objectives (expected return, below-mean absolute semiderivation and loss aversion), in which the DM’s preference information is represented by a reference point and the preference models are represented by an ASF function, g-dominance relationship and Pareto-dominance relationship. These methods have demonstrated prominent capabilities in helping a DM find his/her satisfactory portfolios. However, considering the large number of MV models together with their extended versions resulting from different scenarios in real applications as well as the many effective interactive methods proposed in the literature, the research on portfolio optimization approaches is far beyond mature.
Since the MV model has had a major impact on academic research and the financial industry as a whole (Kolm et al. 2014) and the latest findings indicate that it is a more robust bi-objective model than other well-known models (Pavlou et al. 2019), it is employed as the optimization problem in our study on interactive methods of portfolio optimization. For the EA, we select the NSGA-II due to its outstanding performance in a wide range of applications for solving multiobjective optimization problems by interactive and noninteractive means (Xin et al. 2018; Banzhaf et al. 2020). The aforementioned approaches involved in the other two aspects of an interactive method, the preference information and preference model, may be adapted for use in solving the MV model. However, considering that it is not often realistic to suppose a DM with high cognitive capability is participating in interactive portfolio optimization, we assume a DM with limited cognition. In addition to a proposed learning-based preference model that is compatible with this DM, we concentrate on reducing his/her cognitive burden and pioneering innovative studies in the area of preference information. Our contributions can be attributed to the following:
First, the DM’s cooperative preference articulation regarding the primary and auxiliary factors is investigated. As a human being is becoming involved in an interactive decision process, whether he/she can articulate his/her preferences correctly is the key to the success of the interactive method. In general, a DM can learn from interactions and adjust his/her preferences (Xin et al. 2018). However, a complex question that a typical DM is required to answer may cause a high percentage of noise in the preference information he/she gives in response; therefore, a large number of interactions may be needed for the adjustment, which is a heavy cognitive burden beyond his/her bounded rationality (Battiti & Passerini 2010). Even though qualitative pairwise comparisons have been adopted as easy questions in many interactive methods, due to the DM’s cognitive limitations, noise can also be induced in his/her responses; thus, corrections are needed through interaction (Goulart & Campelo 2016; Li et al. 2019). Another fact that has been ignored by the researches in the literature is that a DM with limited cognition may be in trouble in giving a certain answer to even an easy question in cases, e.g., his/her shortage of knowledge about the optimization problem at the early stage of interaction, the solutions provided for his/her references becoming too similar at the later stage of interaction, etc. A human’s cognition is affected by both objective and subjective factors (Settles 2010; Frenay & Verleysen 2014; Greco et al. 2016): The former includes the complexity of the question and the lack of information to support a decision, and the latter are the DM’s cognitive limitations. So based on qualitative pairwise comparison questions, our study aims to provide additional valuable information to support a DM in articulating his/her preferences as soon and precisely as possible. In the present interactive methods, a DM’s preferences are articulated only on primary factors, i.e., the objective functions of a multiobjective optimization problem; auxiliary factors, even those closely correlated with the primary ones, are not utilized during the decision process. However, these correlated auxiliary factors can help a DM give a more reliable answer, while he/she may not be confident in the judgment of a pair of solutions with very similar objective values. For example, in the optimization of a bi-objective traveling salesman problem (Wang et al. 2015), when a DM cannot select a better itinerary considering only cost and time objectives with similar values, if he/she is provided with additional city information (denoted as variables of the problem), he/she may select an itinerary with cities that are preferable to him/her. A DM’s cognitive burden can be measured by the total number of pairwise comparisons made during the interactive process, and his/her comparative results are then used to learn a preference model. From the perspective of concept learning (Sheng et al. 2008; Ipeirotis et al. 2014), the lower the percentage of noise in the results, the fewer comparisons are needed to learn the correct preference model. The utility of valuable information such as auxiliary factors in addition to the adoption of an easy question in our interactive method can greatly reduce the noise in a DM’s preference feedback, thus leading to a lower cognitive burden required to obtain a portfolio in his/her best interests.
Second, DM decision behavior supported by primary and auxiliary factors during the interactive process is simulated. There are two types of DMs employed in the interactive methods, real and artificial ones. The inconsistency of human nature and variability among humans make it very hard to conduct good quality experiments (Ojalehto et al. 2016; Chen et al. 2017). So we will use an artificial DM in our interactive method. At present, a DM’s true preference is emulated by a value function of objective functions (Deb et al. 2010; Chen et al. 2017). This DM-emulating value function is only used to give his/her preferred solutions at interaction, however, a DM’s complex decision behavior supported by primary factors is not simulated. In our study, a DM-emulating value function based on primary factors is also formulated for portfolio optimization. This function is used to give his/her preferred solutions in the case that the DM is able to make a selection. Furthermore, the DM’s decision behavior is simulated in the case that he/she is not able to make a selection, showing how he/she becomes able to decide when assisted by auxiliary factors. This simulation is the first attempt to provide an interpretation of a DM’s complex cognitive activity in the interactive decision process.
The remainder of this paper is organized according to the human-machine interaction system depicted in Fig. 1. In section 2, we define auxiliary factors based on the MV model and investigate how they are presented in a cooperative manner to a DM for preference articulation. The behavior of a DM in making a judgment supported by auxiliary factors is simulated in section 3. In section 4, we first give a learning-based preference model compatible with a cognition-limited DM and then describe the implementation steps of our proposed interactive method. The effectiveness of our innovative approaches is demonstrated in section 5. Finally, we conclude this research and suggest future development.