# Swarm Intelligence Algorithms for Portfolio Optimization Problems: Overview and Recent Advances

Authors: Yinnan Chen; Xinchao Zhao; Jianmei Yuan
Journal: Mobile Information Systems
 (2022)

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## Abstract

Due to the volatility and uncertainty of the financial market, investors often use the form of portfolio to actively manage their assets. Portfolio optimization (PO) is becoming more and more important for investors. However, PO is frequently a kind of NP-hard problem in the field of modern financial optimization, which has gradually attracted the attention and interest of researchers. Some efficient mathematical models were built to describe the return and risk of portfolio. A lot of precise and approximate fast algorithms are used to solve the established PO models. The fundamental purpose is to maximize the return and to minimize the risk of portfolio under certain constraints. In recent years, researchers not only limit the goal of PO to the balance between risk and return, but also pay attention to liquidity, environmental, social, and governance (ESG) controversy level, Sortino ratio, and other indicators. The number of PO targets and constraints is further extended. In the past two decades, swarm intelligence (SI) algorithms have been widely introduced to solve PO problems. SI algorithm is mainly inspired from the daily phenomena in nature or self-organization, self-adaptation, and self-learning of biological population. The existing research results show that SI algorithm has the characteristics of high efficiency and can obtain satisfactory solutions in solving PO problems. The recent advances on the classic portfolio optimization concepts, models, and the usual SI-based solving algorithms are presented. Finally, future potential research directions are presented.


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## Body

## 1. Introduction

Portfolio optimization is an important type of financial optimization. The main goal is to control risk and to increase return. With the in-depth study of PO problems, sharp ratio, Sortino ratio, liquidity, ESG controversy level, etc., are introduced as objective functions or constraints, making the results of PO more robust and competitive. Because the prices of stock, bond, gold, and foreign exchange asset in the financial market fluctuate violently along with the change of risk factors, it is necessary to conduct overall return analysis and risk management in the form of portfolio [1]. In recent years, the issue of PO has aroused widespread attention. Google Scholar© search results on “portfolio optimization” as a keyword are illustrated in Figure 1 with an increasing trend year by year. As shown in Figure 2, the overall trend of the numbers of papers with portfolio optimization as keyword in Web of Science© has been increasing, although there has been a small decline in some years such as 2009 and 2020 since 1998. Both trend figures illustrate PO problem has aroused more and more interest among researchers.Figure 1
Google Scholar© search results on portfolio optimization as keyword.Figure 2
Trend graph of portfolio-related papers in Web of Science©.The milestone research of the portfolio was proposed by Markowitz in 1952 [2]. The established mean-variance model is mainly used to measure the return and risk of the portfolio, where mean is the expected return and variance is the volatility of its returns representing the risk faced by portfolio. With the in-depth study of portfolio risk measure, the value at risk (VaR) model [3] is introduced to study portfolio optimization problems. VaR is mainly used to measure the portfolio loss in a certain confidence level and a certain holding period. Due to the unpredictable financial market, there is likely to be a black swan event [4]. People began to realize that tail risk should not be negligible, but of extreme importance. Therefore, the conditional VaR or expected shortfall (ES) [5] model was introduced to measure the risk. In addition, semi-variance [6], variance and skewness [7], mean absolute deviation [8], and minimax portfolio selection [9] models were introduced to study the return and risk of the portfolio.The establishment of PO model is based on a description of return and risk. The core of PO model is to optimize the two measure functions of return and risk, which obtain the maximum and minimum value under certain constraints.Swarm intelligence is the approximate optimization algorithm. It has the advantages of strong adaptability, easy to accomplish, relatively high accuracy, and low problem-depending property. This research field has become more and more abundant in recent years. As shown in Figure3, the search results of Google Scholar© for keyword about SI algorithm, both PO and SI algorithms show a continuous growth trend year by year. Figure 4 shows the trend graph of the number of papers with PO and SI algorithms as keywords in Web of Science©.Figure 3
Google Scholar© search results with keyword of SI algorithm.Figure 4
Trend graph of portfolio and swarm intelligence algorithm-related papers in Web of Science©.Swarm intelligence algorithms include mainly ant colony algorithm (ACO) [10], particle swarm algorithm (PSO) [11], differential evolution (DE) algorithm [12], bacterial foraging optimization (BFO) [13], artificial bee colony (ABC) [14], cat swarm optimization (CSO) algorithm [15], firefly algorithm (FA) [16], invasive weed optimization (IWO) [17], bat algorithm (BA) [18], fireworks algorithm (FWA) [19], brain storm optimization (BSO) algorithm [20], etc. Whether in single-objective optimization or multi-objective optimization, SI algorithm has a wide range. Figure 4 shows trend graph of portfolio and swarm intelligence algorithm-related papers in Web of Science© of applications in PO. References [21, 22] conducted two valuable investigations on the application of swarm intelligence algorithm in portfolio optimization. These articles summarize many PO-related references and have the characteristics of comprehensive investigation and clear classification. Different from references [21–26], this paper lists representative PO models in different periods and summarizes classical swarm intelligence algorithms which are applied in PO in recent years. It aims to make a reference and prospect for the further application of SI algorithm in PO.The rest of the article is structured as follows. The second section shows the explanation of basic concepts. The third section presents an overview of common portfolio model. The fourth section presents the commonly applications of swarm intelligence algorithms to PO. The fifth section gives a conclusion and several possible directions.


## 2. Explanation of Basic Concepts

### 2.1. Portfolio Optimization

The concept of modern portfolio optimization (PO) was introduced by Markowitz [2] and Sharpe [27]. Simply, a set of different financial assets can be considered as an investment portfolio. The purpose of portfolio optimization is to assume that under certain conditions, reasonably allocating the investment ratio of each asset could achieve the lowest risk under the condition of guaranteeing the expected rate of return. On the other side, it aims to maximize the return of portfolio at a given level of risk.The reason why portfolio is used for investment decision analysis is that although the portfolio may not be able to determine the highest rate of asset return, it could effectively diversify risk. Its essence is similar to that of conducting multiple random tests at the same time. The total return of the portfolio is equal to the linear sum of the expected returns of different financial investment products. Because of the dispersion effect, the total risk of the portfolio is generally lower than the linear sum of the risk of each asset.The core purpose of portfolio optimization research is mainly to reduce nonsystemic risk. By rationally allocating investment ratio of asset in portfolio, PO could maximize the return of the portfolio and minimize the nonsystemic risk. As we can see in Figure5, the area below the red dotted line indicates the systemic risk faced by the portfolio is inherent risk. PO is to minimize the nonsystematic risk above the red dotted line, which can be achieved through reasonable asset allocation. Finally, the overall risk of portfolio is close to dotted line, which is also called systemic risk.Figure 5
Trend graph of portfolio and swarm intelligence algorithm-related papers in Web of Science©.


### 2.2. Value at Risk

Value at risk (VaR) [3] is the worst expected loss over a target horizon within a given confidence interval. Given a confidence level α∈0,1, the VaR of portfolio at the confidence level α is defined by the smallest number l in equation (1). It means that the probability that loss L exceeds l is not larger than 1 − α.(1)VaRα=infl∈R:PL>l⩽1−α.


### 2.3. Expected Shortfall

Expected shortfall (ES) [5] is also called conditional VaR, which refers to the expectation of the tail loss beyond a given VaR such as VaRα. The mathematical expression of ES at the confidence level α is shown as equation (2), in which L also represents the loss of the portfolio.(2)ES=CVaRα=EL|L≥VaRα.


### 2.4. Sharpe Ratio

The Sharpe ratio [28, 29] is a common performance measurement that evaluates the excess portfolio return per unit of risk. Assuming that the portfolio return rate is Rp, Rf and σp represent the risk-free interest rate and the standard deviation of the excess return of the portfolio, and the Sharpe ratio is given as(3)Sharpe Ration=ERP−Rfσp.


### 2.5. Entropy

Entropy [30] is a measurement of uncertainty. Supposing a probability experiment has n results, and each result has discrete probability pi ≥ 0 (i = 1, 2, …, n). Entropy can be defined as(4)Hn=−∑i=1npilnpi.where ∑i=1npi=1. Similarly, supposing px is the density function of a continuous distribution, and entropy can be defined as(5)Hx=−∫−∞+∞pxlnpxdx.


### 2.6. Possibility Measure, Necessity Measure, and Credibility Measure

Given a domain, a sufficient setA is a subset of Γ under any set. Let Pos be a set function defined under the sufficient set A. If the following (6) properties are satisfied, Pos is called the possibility measurement.(6)Pos∅=0;PosΓ=1;Pos∪i∈IAi=supi∈IPosAi,where Ai is any subset of A and I is any subset. Assuming Ac is the complement of A, its necessity measurement is defined as(7)NecA=1−PocAc.Γ,A,Pos is called possibility space. Based on possibility measure and necessity measure, the credibility measure can be defined as(8)CrA=121+PosA−PosAc.According to equations (7) and (8), the credibility measure can be simplified as(9)CrA=12PosA+NecA.


### 2.7. Fuzzy Value at Risk

Assume thatX is a fuzzy variable [31], which represents the future loss of a portfolio. Under the confidence level β, the fuzzy value at risk [32] of X is defined as equation (10). λ is the maximum loss under the confidence level.(10)VaR1−β=supλ:CrL≥λ≥β.


### 2.8. Coherent Risk Measurement

The coherent risk measurement [33] should satisfy four axioms, which are monotonicity, first-order homogeneity, translation invariance, and subadditivity. The specific mathematical definitions are(11)X1≥X2,ρX1≥ρX2:∀λ1>0,ρλ1X=λ1ρX:∀c,ρX+c=ρX−c:ρX1+X2≤ρX1+ρX2.


### 2.9. Uncertainty Theory

Uncertainty theory was invented by Liu [34] as a tool for describing the subjective imprecise quantity. Let K be a nonempty set. A collection Z of subsets of K is called a σ-algebra if(a)
K∈Z;(b)
ifΛ∈Z, then Λc∈Z;(c)
ifΛ1,Λ2,…,∈Z, then Λ1∪Λ2∪,…,∈Z.Each element Λ in theσ-algebra Z is called an event. Uncertain measure is a function from Z to [0, 1]. In order to present an axiomatic definition of uncertain measure, it is necessary to assign each event Λ with a number M {Λ} which indicates the belief degree that Λ will occur. In order to ensure that the number M {Λ} has certain mathematical properties, Liu [34] proposed the following four axioms:Axiom 1. (Normality Axiom)M {K} = 1 for the universal set K.Axiom 2. (Monotonicity Axiom)M {Λ1} ≤ M {Λ2} whenever Λ1⊂Λ2.Axiom 3. (Self-Duality Axiom)M {Λ} + M {Λc} = 1 for any event Λ.Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of event {Λi}, it exists that M∪i=1∞Λi≤∑i=1∞MΛi.The set functionM is called an uncertain measure if it satisfies the normality, monotonicity, self-duality, and countable subadditivity axioms.


## 2.1. Portfolio Optimization

The concept of modern portfolio optimization (PO) was introduced by Markowitz [2] and Sharpe [27]. Simply, a set of different financial assets can be considered as an investment portfolio. The purpose of portfolio optimization is to assume that under certain conditions, reasonably allocating the investment ratio of each asset could achieve the lowest risk under the condition of guaranteeing the expected rate of return. On the other side, it aims to maximize the return of portfolio at a given level of risk.The reason why portfolio is used for investment decision analysis is that although the portfolio may not be able to determine the highest rate of asset return, it could effectively diversify risk. Its essence is similar to that of conducting multiple random tests at the same time. The total return of the portfolio is equal to the linear sum of the expected returns of different financial investment products. Because of the dispersion effect, the total risk of the portfolio is generally lower than the linear sum of the risk of each asset.The core purpose of portfolio optimization research is mainly to reduce nonsystemic risk. By rationally allocating investment ratio of asset in portfolio, PO could maximize the return of the portfolio and minimize the nonsystemic risk. As we can see in Figure5, the area below the red dotted line indicates the systemic risk faced by the portfolio is inherent risk. PO is to minimize the nonsystematic risk above the red dotted line, which can be achieved through reasonable asset allocation. Finally, the overall risk of portfolio is close to dotted line, which is also called systemic risk.Figure 5
Trend graph of portfolio and swarm intelligence algorithm-related papers in Web of Science©.


## 2.2. Value at Risk

Value at risk (VaR) [3] is the worst expected loss over a target horizon within a given confidence interval. Given a confidence level α∈0,1, the VaR of portfolio at the confidence level α is defined by the smallest number l in equation (1). It means that the probability that loss L exceeds l is not larger than 1 − α.(1)VaRα=infl∈R:PL>l⩽1−α.


## 2.3. Expected Shortfall

Expected shortfall (ES) [5] is also called conditional VaR, which refers to the expectation of the tail loss beyond a given VaR such as VaRα. The mathematical expression of ES at the confidence level α is shown as equation (2), in which L also represents the loss of the portfolio.(2)ES=CVaRα=EL|L≥VaRα.


## 2.4. Sharpe Ratio

The Sharpe ratio [28, 29] is a common performance measurement that evaluates the excess portfolio return per unit of risk. Assuming that the portfolio return rate is Rp, Rf and σp represent the risk-free interest rate and the standard deviation of the excess return of the portfolio, and the Sharpe ratio is given as(3)Sharpe Ration=ERP−Rfσp.


## 2.5. Entropy

Entropy [30] is a measurement of uncertainty. Supposing a probability experiment has n results, and each result has discrete probability pi ≥ 0 (i = 1, 2, …, n). Entropy can be defined as(4)Hn=−∑i=1npilnpi.where ∑i=1npi=1. Similarly, supposing px is the density function of a continuous distribution, and entropy can be defined as(5)Hx=−∫−∞+∞pxlnpxdx.


## 2.6. Possibility Measure, Necessity Measure, and Credibility Measure

Given a domain, a sufficient setA is a subset of Γ under any set. Let Pos be a set function defined under the sufficient set A. If the following (6) properties are satisfied, Pos is called the possibility measurement.(6)Pos∅=0;PosΓ=1;Pos∪i∈IAi=supi∈IPosAi,where Ai is any subset of A and I is any subset. Assuming Ac is the complement of A, its necessity measurement is defined as(7)NecA=1−PocAc.Γ,A,Pos is called possibility space. Based on possibility measure and necessity measure, the credibility measure can be defined as(8)CrA=121+PosA−PosAc.According to equations (7) and (8), the credibility measure can be simplified as(9)CrA=12PosA+NecA.


## 2.7. Fuzzy Value at Risk

Assume thatX is a fuzzy variable [31], which represents the future loss of a portfolio. Under the confidence level β, the fuzzy value at risk [32] of X is defined as equation (10). λ is the maximum loss under the confidence level.(10)VaR1−β=supλ:CrL≥λ≥β.


## 2.8. Coherent Risk Measurement

The coherent risk measurement [33] should satisfy four axioms, which are monotonicity, first-order homogeneity, translation invariance, and subadditivity. The specific mathematical definitions are(11)X1≥X2,ρX1≥ρX2:∀λ1>0,ρλ1X=λ1ρX:∀c,ρX+c=ρX−c:ρX1+X2≤ρX1+ρX2.


## 2.9. Uncertainty Theory

Uncertainty theory was invented by Liu [34] as a tool for describing the subjective imprecise quantity. Let K be a nonempty set. A collection Z of subsets of K is called a σ-algebra if(a)
K∈Z;(b)
ifΛ∈Z, then Λc∈Z;(c)
ifΛ1,Λ2,…,∈Z, then Λ1∪Λ2∪,…,∈Z.Each element Λ in theσ-algebra Z is called an event. Uncertain measure is a function from Z to [0, 1]. In order to present an axiomatic definition of uncertain measure, it is necessary to assign each event Λ with a number M {Λ} which indicates the belief degree that Λ will occur. In order to ensure that the number M {Λ} has certain mathematical properties, Liu [34] proposed the following four axioms:Axiom 1. (Normality Axiom)M {K} = 1 for the universal set K.Axiom 2. (Monotonicity Axiom)M {Λ1} ≤ M {Λ2} whenever Λ1⊂Λ2.Axiom 3. (Self-Duality Axiom)M {Λ} + M {Λc} = 1 for any event Λ.Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of event {Λi}, it exists that M∪i=1∞Λi≤∑i=1∞MΛi.The set functionM is called an uncertain measure if it satisfies the normality, monotonicity, self-duality, and countable subadditivity axioms.


## 3. Portfolio Optimization Models

### 3.1. Mean-Variance Model

The mean-variance (M-V) model is proposed by Markowitz in 1952 [2]. The risk is defined as the volatility of the rate of return, and the mathematical statistic method is applied to solve the portfolio selection. This model is based on the following assumptions.(1)
When investors consider each investment choice, it is based on the probability distribution of securities or assets returns within a certain holding time.(2)
The investor estimates the risk of the portfolio based on the variance or standard deviation of the expected return of the securities or assets.(3)
The decisions of investors are based solely on the risk and return of the securities.(4)
At a certain level of risk, investors expect the greatest return. Correspondingly, investors expect the lowest risk at a certain level of return. It is assumed thatn types of financial assets can be invested, the return rate of i-th asset is ri, and its investment ratio is xi. Assuming that the expected return rate is µ, σij is the covariance of the return rate for i-th asset and j-th asset, and then the M-V model can be defined as(12)min,σ2=∑i=1n∑j=1nxixjσijs.t.∑i=1nrixi≥μ,∑i=1nxi=1,xi≥0,i=1,2,…,n.


### 3.2. Mean-Semi-Variance Model

In actual investment, risk is often thought as loss. That is, when the actual rate of return of an investment is higher than its expected rate of return, it does not consider it to be a risk. And only when the actual rate of return is lower than the expected rate of return, it is considered to be a risk. Therefore, the mean-semi-variance (M-SV) model was proposed in [35], which is an extension of the standard Markowitz’ mean-variance model.The return on asseti in the portfolio is(13)ri−rit−=0,ri≤rit,ri−rit,ri>rit,where rit represents the rate of return of the i-th risk asset in period t. ri represents the expected return of the i-th risk asset, which is defined as(14)ri=1T∑t=1Triti=1,2,…,n.The variance of thei-th security in period t is defined as(15)σij−=Eri−rit−rj−rjt−.The half variance is defined as(16)Vx−=∑i=1N∑j=1Nxixjσij−.The M-SV model is presented as(17)min,Vx−,s.t.∑i=1nxirit≥rf,∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where rf is the risk-free rate of return.


### 3.3. Mean-Semi-Variance Entropy Model

In the portfolio, the rate of return of various assets is uncertain because of different allocations. Therefore, entropy is introduced to measure the uncertainty of asset allocation. Since the return rateri of each asset is a random variable, the investment allocation ratio is also a random variable. The entropy principle can be used to measure the uncertainty of investment allocation [36]. The entropy of portfolio asset allocation is defined as(18)−∑i=1nxilnxi.Thus, the following mean-semi-variance entropy model is established as(19)min,Vx−=∑i=1N∑j=1Nxixjσij−,max,−∑i=1nxilnxi,s.t.∑i=1nrixi≥rf,∑i=1nxi=1,xi≥0,i=1,2,…,n.


### 3.4. Mean Absolute Deviation Model

Absolute deviation can also be used to measure the risk of portfolio. Its advantage is to avoid the influence of the estimation error of the covariance matrix in the mean-variance model. Assuming that the portfolio containsn assets, ri is the expected rate of return of the i-th asset, and the investment period is divided into T time periods. rit is the historical rate of return of the i-th asset at time t(t = 1, 2, …, T), xi represents the proportion of investment of the i-th asset, r0 is the minimum expected return of investors on the portfolio, and xu is the maximum proportion of investment of the i-th asset. The mean absolute deviation (MAD) portfolio model in [8] can be described as(20)min,1T∑t=1T∑i=1nxirit−ris.t.∑i=1nrixi≥r0,∑i=1nxi=1,0≤xi≤xu,i=1,2,…,n.


### 3.5. Single-Index Model

The single-index model (SIM) was proposed by Sharpe in the 1960s [23]. The model assumes that the return rate of each asset only has a linear relationship with the market return rate, which can be expressed as(21)rit=αit+βidrdt+eit.rit is the rate of return i in a given period of t. rdt is the rate of return of specific asset market index d during this period. βid represents the slope or sensitivity coefficient of the return rate of asset i relative to a specific market index d. αit is the excess return of security i when the excess return in a specific market is 0. eit is the residual between the actual rate of return and the estimated value.Taking the expectation from equation (21), the expected return of the asset is shown as(22)Eri=αi+βiErd.In the single-index model, risk is still measured by variance as(23)σi2=βi2σd2+σ2ei,where βi2σd2 is called system risk and σ2ei is nonsystemic risk.


### 3.6. Value at Risk Model

Value at risk (VaR) is commonly used to measure the overall risk of a portfolio. There are three kinds of approaches to calculate VaR.


#### 3.6.1. Historical Simulation Method

It is based on the assumption that the future fluctuations of market risk factors are not much different from the historical fluctuations. So, the future return and loss distribution of the portfolio can be simulated according to the changes of historical data. The quantile can be used to describe VaR.


#### 3.6.2. Monte Carlo Simulation Method

It is to establish a random process model of risk factors based on the historical volatility of market risk factors. By simulating possible future volatility paths, it acquires a simulated distribution close to the true distribution of the portfolio. The Monte Carlo simulation method does not require the assumption of distribution normality, and it can be applied to various distributions of market factors. However, a large amount of simulation calculations is required to calculate, and the efficiency is relatively low.


#### 3.6.3. Variance-Covariance Method

It is required to calculate the standard deviation and correlation coefficient of assets based on the historical data of asset returns. Assuming that it follows a certain distribution, the standard deviation of the portfolio is calculated based on the variance and covariance.Finally, the corresponding VaR is obtained.Supposing that there aren kinds of assets in portfolio, the vectors of investment return rate and weight are r = (r1, …, rn) and X = (x1, …, xn). Obviously, the return of the portfolio is calculated as(24)R=∑i=1nxiri.Assuming that the return distribution of the portfolio is normally distributed, the expected return isER. The variance is σ2=XTΣX, in which Σ is the covariance matrix. It is easy to know that when a confidence level α is given, VaR can be expressed as(25)VaR=ϕ−1ασ−ER,where is the quantile with a normal distribution of confidence level.The mean-VaR model can be described as(26)max,ER,s.t.PR<VaR≤1−α,∑i=1nxi=1,0≤xi≤xu,i=1,2,…,n.However, VaR model has its shortcomings and limitations. For example, it fails to satisfy consistent risk measurement. Namely, it is contrary to the original intention of portfolio diversification.


### 3.7. Expected Shortfall Model

Expected shortfall (ES) is used to measure the extreme loss of the portfolio under a certain confidence level. It is defined as the average of extreme loss that exceeds the VaR and is a coherent risk measurement.SupposeL(X, Y) is the loss function of the portfolio, in which X is the vector of decision variable and represents the investment weight of asset in portfolio. Y represents the uncertainty factor that causes loss to the portfolio such as asset price, rate of return, and stock index. For each X, the loss L(X, Y) caused by Y is a random variable which subjects to a certain distribution. Assuming that the density function of Y is p(y), then the probability that L(X, Y) does not exceed a certain threshold τ is(27)ψX,τ=∫LX,Y<τpydy.WhenX is determined, equation (27) is a function of τ. ψX,τ is the cumulative distribution function of loss corresponding to X. The VaRα and ESα under the given confidence level α are ταx and ϕαx, and then(28)ταx=minτ∈R:ψx,τ≥α,ϕαx=1−α−1∫Lx,y≥ταxLx,ypydy.In order to facilitate the calculation of ES, Rockafellar and Uryasev [5, 37, 38] introduced auxiliary functions,(29)ψX,τ=∫LX,Y<τpydy,where x+=maxx,0.The mean-ES model can be described as(30)max,ER,s.t.ϕαx≤L0,∑i=1nxi=1.It can also be described as(31)min,ϕαx,s.t.ER=R0,∑i=1nxi=1.


### 3.8. Fuzzy Value at Risk Model

Based on the mean-variance model and the concept of fuzzy VaR, a multi-objective portfolio optimization model based on fuzzy VaR [39] is presented as(32)max,E∑i=1nrixi,min,V∑i=1nrixi,min,VaRβ=infλ|CrL>λ≤β,s.t.∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where L represents the future loss of a portfolio and λ is the quantile of loss L under the confidence level β.


### 3.9. Uncertainty Portfolio Selection Model

It also assumes that there aren types of financial assets that can be invested, the return rate of the i-th asset is ri, and its investment ratio is xi. A product uncertainty measure M is defined by Liu in [34]. According to articles [36–42], the uncertain portfolio selection model is described as equation (33), in which β′ is a given confidence level.(33)max,E∑i=1nrixi,min,V∑i=1nrixi,s.t.M∑i=1nrixi≥R0≥β′,∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where R0 is a given risk-free rate of return.


### 3.10. Other Important and Effective Models

Using the worst possible return of the portfolio as a risk measure, Young [9] proposed a minimax model in 1998. When the return meets certain condition, the minimax model can maximize the worst return to obtain a better investment portfolio. Roy [43] proposed safety first model, which takes the probability of return below a certain catastrophic level as a measurement of risk. Pyle and Turnovsky [44] analyzed the relationship between the safety first criterion and the mean-standard deviation utility function in the presence or absence of risky assets. Chen et al. [45] established higher-order moment portfolio model. They solved the portfolio selection problem with skewness and kurtosis, and it includes not only the multi-objective optimization but also the data-driven asset selection and return prediction. The techniques of two-stage clustering, radial basis function neural network, and genetic algorithm are employed. Gong et al. [46] presented an operational model for portfolio selection under cumulative prospect theory and proposed a real-coded genetic algorithm to solve the problem of portfolio choice.Lu et al. [47] proposed a new uncertain multi-objective multi-period mean-semientropy-skewness portfolio optimization model, in which several real-world constraints such as cardinality and liquidity are considered simultaneously. A novel hybrid MFA-SOS algorithm which combines the features of the firefly algorithm (FA) and symbiotic organism search (SOS) algorithm is designed to solve the corresponding proposed model. Several numerical examples were given to illustrate the validity of the proposed model and approach.Based on the credibility theory, Garcia et al. proposed a multi-objective credibilistic return-risk-liquidity portfolio selection model in [48]. In this model, semi-variance and CVaR are introduced simultaneously to measure the risk of portfolio. Trapezoidal fuzzy numbers are introduced to quantify the uncertainty of the future returns and the liquidity of each risky asset. Sufficient experiments have demonstrated the effectiveness of this model and its good application prospect for researchers and investors interested in PO strategies. Oliver [49] proposed a multi-objective PO model using ESG controversies. This model is mainly to help investors find socially responsible companies to invest when building their portfolio after considering the balance of risk and return.Combining option and uncertainty theory, Wang et al. proposed an uncertain mean-chance PO model in [50]. Experiments show that the portfolio with option has more advantages in expected return and sensitivity analysis results than the portfolio without option.


## 3.1. Mean-Variance Model

The mean-variance (M-V) model is proposed by Markowitz in 1952 [2]. The risk is defined as the volatility of the rate of return, and the mathematical statistic method is applied to solve the portfolio selection. This model is based on the following assumptions.(1)
When investors consider each investment choice, it is based on the probability distribution of securities or assets returns within a certain holding time.(2)
The investor estimates the risk of the portfolio based on the variance or standard deviation of the expected return of the securities or assets.(3)
The decisions of investors are based solely on the risk and return of the securities.(4)
At a certain level of risk, investors expect the greatest return. Correspondingly, investors expect the lowest risk at a certain level of return. It is assumed thatn types of financial assets can be invested, the return rate of i-th asset is ri, and its investment ratio is xi. Assuming that the expected return rate is µ, σij is the covariance of the return rate for i-th asset and j-th asset, and then the M-V model can be defined as(12)min,σ2=∑i=1n∑j=1nxixjσijs.t.∑i=1nrixi≥μ,∑i=1nxi=1,xi≥0,i=1,2,…,n.


## 3.2. Mean-Semi-Variance Model

In actual investment, risk is often thought as loss. That is, when the actual rate of return of an investment is higher than its expected rate of return, it does not consider it to be a risk. And only when the actual rate of return is lower than the expected rate of return, it is considered to be a risk. Therefore, the mean-semi-variance (M-SV) model was proposed in [35], which is an extension of the standard Markowitz’ mean-variance model.The return on asseti in the portfolio is(13)ri−rit−=0,ri≤rit,ri−rit,ri>rit,where rit represents the rate of return of the i-th risk asset in period t. ri represents the expected return of the i-th risk asset, which is defined as(14)ri=1T∑t=1Triti=1,2,…,n.The variance of thei-th security in period t is defined as(15)σij−=Eri−rit−rj−rjt−.The half variance is defined as(16)Vx−=∑i=1N∑j=1Nxixjσij−.The M-SV model is presented as(17)min,Vx−,s.t.∑i=1nxirit≥rf,∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where rf is the risk-free rate of return.


## 3.3. Mean-Semi-Variance Entropy Model

In the portfolio, the rate of return of various assets is uncertain because of different allocations. Therefore, entropy is introduced to measure the uncertainty of asset allocation. Since the return rateri of each asset is a random variable, the investment allocation ratio is also a random variable. The entropy principle can be used to measure the uncertainty of investment allocation [36]. The entropy of portfolio asset allocation is defined as(18)−∑i=1nxilnxi.Thus, the following mean-semi-variance entropy model is established as(19)min,Vx−=∑i=1N∑j=1Nxixjσij−,max,−∑i=1nxilnxi,s.t.∑i=1nrixi≥rf,∑i=1nxi=1,xi≥0,i=1,2,…,n.


## 3.4. Mean Absolute Deviation Model

Absolute deviation can also be used to measure the risk of portfolio. Its advantage is to avoid the influence of the estimation error of the covariance matrix in the mean-variance model. Assuming that the portfolio containsn assets, ri is the expected rate of return of the i-th asset, and the investment period is divided into T time periods. rit is the historical rate of return of the i-th asset at time t(t = 1, 2, …, T), xi represents the proportion of investment of the i-th asset, r0 is the minimum expected return of investors on the portfolio, and xu is the maximum proportion of investment of the i-th asset. The mean absolute deviation (MAD) portfolio model in [8] can be described as(20)min,1T∑t=1T∑i=1nxirit−ris.t.∑i=1nrixi≥r0,∑i=1nxi=1,0≤xi≤xu,i=1,2,…,n.


## 3.5. Single-Index Model

The single-index model (SIM) was proposed by Sharpe in the 1960s [23]. The model assumes that the return rate of each asset only has a linear relationship with the market return rate, which can be expressed as(21)rit=αit+βidrdt+eit.rit is the rate of return i in a given period of t. rdt is the rate of return of specific asset market index d during this period. βid represents the slope or sensitivity coefficient of the return rate of asset i relative to a specific market index d. αit is the excess return of security i when the excess return in a specific market is 0. eit is the residual between the actual rate of return and the estimated value.Taking the expectation from equation (21), the expected return of the asset is shown as(22)Eri=αi+βiErd.In the single-index model, risk is still measured by variance as(23)σi2=βi2σd2+σ2ei,where βi2σd2 is called system risk and σ2ei is nonsystemic risk.


## 3.6. Value at Risk Model

Value at risk (VaR) is commonly used to measure the overall risk of a portfolio. There are three kinds of approaches to calculate VaR.


### 3.6.1. Historical Simulation Method

It is based on the assumption that the future fluctuations of market risk factors are not much different from the historical fluctuations. So, the future return and loss distribution of the portfolio can be simulated according to the changes of historical data. The quantile can be used to describe VaR.


### 3.6.2. Monte Carlo Simulation Method

It is to establish a random process model of risk factors based on the historical volatility of market risk factors. By simulating possible future volatility paths, it acquires a simulated distribution close to the true distribution of the portfolio. The Monte Carlo simulation method does not require the assumption of distribution normality, and it can be applied to various distributions of market factors. However, a large amount of simulation calculations is required to calculate, and the efficiency is relatively low.


### 3.6.3. Variance-Covariance Method

It is required to calculate the standard deviation and correlation coefficient of assets based on the historical data of asset returns. Assuming that it follows a certain distribution, the standard deviation of the portfolio is calculated based on the variance and covariance.Finally, the corresponding VaR is obtained.Supposing that there aren kinds of assets in portfolio, the vectors of investment return rate and weight are r = (r1, …, rn) and X = (x1, …, xn). Obviously, the return of the portfolio is calculated as(24)R=∑i=1nxiri.Assuming that the return distribution of the portfolio is normally distributed, the expected return isER. The variance is σ2=XTΣX, in which Σ is the covariance matrix. It is easy to know that when a confidence level α is given, VaR can be expressed as(25)VaR=ϕ−1ασ−ER,where is the quantile with a normal distribution of confidence level.The mean-VaR model can be described as(26)max,ER,s.t.PR<VaR≤1−α,∑i=1nxi=1,0≤xi≤xu,i=1,2,…,n.However, VaR model has its shortcomings and limitations. For example, it fails to satisfy consistent risk measurement. Namely, it is contrary to the original intention of portfolio diversification.


## 3.6.1. Historical Simulation Method

It is based on the assumption that the future fluctuations of market risk factors are not much different from the historical fluctuations. So, the future return and loss distribution of the portfolio can be simulated according to the changes of historical data. The quantile can be used to describe VaR.


## 3.6.2. Monte Carlo Simulation Method

It is to establish a random process model of risk factors based on the historical volatility of market risk factors. By simulating possible future volatility paths, it acquires a simulated distribution close to the true distribution of the portfolio. The Monte Carlo simulation method does not require the assumption of distribution normality, and it can be applied to various distributions of market factors. However, a large amount of simulation calculations is required to calculate, and the efficiency is relatively low.


## 3.6.3. Variance-Covariance Method

It is required to calculate the standard deviation and correlation coefficient of assets based on the historical data of asset returns. Assuming that it follows a certain distribution, the standard deviation of the portfolio is calculated based on the variance and covariance.Finally, the corresponding VaR is obtained.Supposing that there aren kinds of assets in portfolio, the vectors of investment return rate and weight are r = (r1, …, rn) and X = (x1, …, xn). Obviously, the return of the portfolio is calculated as(24)R=∑i=1nxiri.Assuming that the return distribution of the portfolio is normally distributed, the expected return isER. The variance is σ2=XTΣX, in which Σ is the covariance matrix. It is easy to know that when a confidence level α is given, VaR can be expressed as(25)VaR=ϕ−1ασ−ER,where is the quantile with a normal distribution of confidence level.The mean-VaR model can be described as(26)max,ER,s.t.PR<VaR≤1−α,∑i=1nxi=1,0≤xi≤xu,i=1,2,…,n.However, VaR model has its shortcomings and limitations. For example, it fails to satisfy consistent risk measurement. Namely, it is contrary to the original intention of portfolio diversification.


## 3.7. Expected Shortfall Model

Expected shortfall (ES) is used to measure the extreme loss of the portfolio under a certain confidence level. It is defined as the average of extreme loss that exceeds the VaR and is a coherent risk measurement.SupposeL(X, Y) is the loss function of the portfolio, in which X is the vector of decision variable and represents the investment weight of asset in portfolio. Y represents the uncertainty factor that causes loss to the portfolio such as asset price, rate of return, and stock index. For each X, the loss L(X, Y) caused by Y is a random variable which subjects to a certain distribution. Assuming that the density function of Y is p(y), then the probability that L(X, Y) does not exceed a certain threshold τ is(27)ψX,τ=∫LX,Y<τpydy.WhenX is determined, equation (27) is a function of τ. ψX,τ is the cumulative distribution function of loss corresponding to X. The VaRα and ESα under the given confidence level α are ταx and ϕαx, and then(28)ταx=minτ∈R:ψx,τ≥α,ϕαx=1−α−1∫Lx,y≥ταxLx,ypydy.In order to facilitate the calculation of ES, Rockafellar and Uryasev [5, 37, 38] introduced auxiliary functions,(29)ψX,τ=∫LX,Y<τpydy,where x+=maxx,0.The mean-ES model can be described as(30)max,ER,s.t.ϕαx≤L0,∑i=1nxi=1.It can also be described as(31)min,ϕαx,s.t.ER=R0,∑i=1nxi=1.


## 3.8. Fuzzy Value at Risk Model

Based on the mean-variance model and the concept of fuzzy VaR, a multi-objective portfolio optimization model based on fuzzy VaR [39] is presented as(32)max,E∑i=1nrixi,min,V∑i=1nrixi,min,VaRβ=infλ|CrL>λ≤β,s.t.∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where L represents the future loss of a portfolio and λ is the quantile of loss L under the confidence level β.


## 3.9. Uncertainty Portfolio Selection Model

It also assumes that there aren types of financial assets that can be invested, the return rate of the i-th asset is ri, and its investment ratio is xi. A product uncertainty measure M is defined by Liu in [34]. According to articles [36–42], the uncertain portfolio selection model is described as equation (33), in which β′ is a given confidence level.(33)max,E∑i=1nrixi,min,V∑i=1nrixi,s.t.M∑i=1nrixi≥R0≥β′,∑i=1nxi=1,0≤xi≤1,i=1,2,…,n,where R0 is a given risk-free rate of return.


## 3.10. Other Important and Effective Models

Using the worst possible return of the portfolio as a risk measure, Young [9] proposed a minimax model in 1998. When the return meets certain condition, the minimax model can maximize the worst return to obtain a better investment portfolio. Roy [43] proposed safety first model, which takes the probability of return below a certain catastrophic level as a measurement of risk. Pyle and Turnovsky [44] analyzed the relationship between the safety first criterion and the mean-standard deviation utility function in the presence or absence of risky assets. Chen et al. [45] established higher-order moment portfolio model. They solved the portfolio selection problem with skewness and kurtosis, and it includes not only the multi-objective optimization but also the data-driven asset selection and return prediction. The techniques of two-stage clustering, radial basis function neural network, and genetic algorithm are employed. Gong et al. [46] presented an operational model for portfolio selection under cumulative prospect theory and proposed a real-coded genetic algorithm to solve the problem of portfolio choice.Lu et al. [47] proposed a new uncertain multi-objective multi-period mean-semientropy-skewness portfolio optimization model, in which several real-world constraints such as cardinality and liquidity are considered simultaneously. A novel hybrid MFA-SOS algorithm which combines the features of the firefly algorithm (FA) and symbiotic organism search (SOS) algorithm is designed to solve the corresponding proposed model. Several numerical examples were given to illustrate the validity of the proposed model and approach.Based on the credibility theory, Garcia et al. proposed a multi-objective credibilistic return-risk-liquidity portfolio selection model in [48]. In this model, semi-variance and CVaR are introduced simultaneously to measure the risk of portfolio. Trapezoidal fuzzy numbers are introduced to quantify the uncertainty of the future returns and the liquidity of each risky asset. Sufficient experiments have demonstrated the effectiveness of this model and its good application prospect for researchers and investors interested in PO strategies. Oliver [49] proposed a multi-objective PO model using ESG controversies. This model is mainly to help investors find socially responsible companies to invest when building their portfolio after considering the balance of risk and return.Combining option and uncertainty theory, Wang et al. proposed an uncertain mean-chance PO model in [50]. Experiments show that the portfolio with option has more advantages in expected return and sensitivity analysis results than the portfolio without option.


## 4. Swarm Intelligence Algorithms for PO

Swarm intelligence algorithm originates from imitating the behavior of animal population, such as birds, fishes, ants, bees, bats, and behaviors of bacteria. It has been widely used in approximate calculation field to solve practical application problems, mainly due to human observations of related social phenomena in nature. More and more swarm intelligence algorithms have been proposed by researchers. They are successfully applied to solve PO problems which are genetic algorithm (GA), PSO, FA, GWO, MOEA/D, NSGA-II algorithms, etc. Table1 shows examples of the usual single-objective and multi-objective swarm intelligence algorithms.Table 1
Usual swam intelligence algorithms.
Single-objective SI algorithmsMulti-objective SI algorithms(1) Genetic algorithm(1) NSGA-II(2) Differential evolution algorithm(2) NPGA-II(3) Particle swarm optimization algorithm(3) VEGA(4) Artificial bee colony(4) SPEA-II(5) Ant colony algorithm(5) MOGA(6) Fireworks algorithm(6) MOEA/D(7) Firefly algorithm(7) MOPSO(8) Bacterial foraging optimization(8) MOBFO(9) Bat algorithm(9) Coevolutionary multi-swarm PSO(10) Gray wolf optimizer(10) Multi-objective ACS(11) Brain storm optimization algorithm(11) Multiple objective framework-based GAConsidering the characteristic of the decision variable of the PO problem, there are mainly two kinds of methods: approximation algorithm and exact algorithm. In recent years, the most popular approximation solving methods are mainly heuristic algorithms and machine learning algorithms. The exact solving method mainly includes quadratic programming, nonlinear programming, Lagrangian method, and so on.


### 4.1. Genetic Algorithm

GA was firstly proposed by Holland [51]. The original motivation for GA approach was a biological analogy. In selective breeding of plants or animals, people will seek offspring with some ideal characteristics, which are determined at the genetic level by the parents’ chromosome combination [52].For an optimization problem, a candidate solution population is firstly constructed, and each candidate solution is called as an individual. According to Darwin’s natural selection of the survival of the fittest, well-adapted solutions have greater probabilities of being chosen and reproduced. After generations of replacement, the population becomes more and more promising, and the approximate results finally enhance the solution. Shoaf and Foster [53] applied GA to Markowitz’ portfolio selection problem and found that the time complexity of GA approximates O(nlogn), which is better than that of quadratic programming. Lin and Liu [54] presented and compared three possible decision models to solve the portfolio selection problem with their corresponding genetic algorithms. Table 2 shows the model and objective function type established by genetic algorithm in solving PO problem.Table 2
GA to solve PO problem.
Proposed byModelObjective typeShoaf and Foster [53]M-VSingle objectiveLin and Liu [54]Improved M-VSingle objective


### 4.2. Particle Swarm Optimization Algorithm

PSO is one of the SI algorithms that was firstly introduced by Kennedy and Eberhart in 1995 [11], inspired by swarm behaviors of birds flocking and fishes schooling. Since its inception in 1995, PSO has been rapidly developing and improving, with lots of successful applications to real-world problems [55]. In PSO, a swarm of particles is introduced to represent the solutions. Each particle is associated with two vectors, the velocity vector and the position vector. During an iteration, the fitness of particle will be evaluated at its current position. If the fitness is better than the personal best position that the particle has achieved so far, the personal best position will be replaced by the current solution. Updating the velocity and position of every particle, then the algorithm goes to next iteration, until a stop condition is met [56].PSO was firstly applied to the portfolio optimization problem using M-V model by Chen et al. [57]. References [58–66] presented some enhanced PSO algorithms for solving portfolio optimization. Wang and Watada [67] used VaR as the risk measurement and solved the multi-objective fuzzy portfolio selection problem. Liang and Qu [68] proposed multi-objective dynamic multi-swarm PSO (DMPSO) to solve the large-scale PO problems and carried out experiments with data sets that include 200 and 500 assets. Table 3 shows the model and objective function types established by PSO in solving PO.Table 3
PSO to solve PO problem.
Proposed byModelObjective typeChen et al. [57]Improved M-VSingle objectiveXu and Chen [58]Stochastic M-VSingle objectiveLiu and Wang [59]Entropy modelSingle objectiveWang et al. [60]Improved M-VSingle objectiveNiu et al. [61]Improved M-VMulti-objectiveYaakob and Watada [62]Improved M-VSingle objectiveMozafari et al. [63]Improved M-VSingle objectiveDeng et al. [64]Improved M-VSingle objectiveCorazza et al. [65]Mixed-integerSingle objectiveYin et al. [66]Improved M-VSingle objectiveWang and Watada [67]M-V-VaRMulti-objectiveLiang and Qu [68]M-VMulti-objective


### 4.3. Fireworks Algorithm

FWA is a novel intelligent algorithm which is proposed by Tan et al. in 2010 [19]. By imitating the fireworks explosion process, the fireworks algorithm can effectively search for the best solution. Due to its powerful ability to solve optimization problems, fireworks algorithm has been applied to different optimization problems in different fields, and it has performed great potential. The main application areas include knapsack problem, image recognition, filter designing [69–71], etc. Zhang and Liu [72] applied FWA into the field of portfolio optimization to solve the mean-VaR/CVaR model. Compared with the classical genetic algorithm, fireworks algorithm not only improves the optimization accuracy and the optimizing speed, but also makes the optimal solution more stable. Bacanin and Tuba [73] adapted fireworks algorithm for solving constrained portfolio selection problem that extends classical mean-variance portfolio model by adding additional constraints. Table 4 shows the model and objective type which are established to solve the PO problem by FWA.Table 4
FWA to solve PO problem.
Proposed byModelObjective typeZhang and Liu [72]Mean-VaR/CVaRSingle objectiveBacanin and Tuba [73]Improved M-VSingle objective


### 4.4. Gray Wolf Optimizer

Gray wolf optimizer (GWO) is a population-based metaheuristic algorithm introduced by Mirjalili et al. [74], and its inspiration comes from the unique hierarchical hunting behavior of gray wolves in nature. Gray wolves hunt in packs, and the pack is distributed into alpha wolf (α, the fittest and their leader), beta wolf (β, second in line of leadership), and delta wolf (δ, third fittest), and all other remaining are omega wolves. The hunting (optimization) of prey is guided by α, β, and δ wolves. Mazumdar et al. [75] proposed GWO technique for portfolio construction and unsystematic risk optimization. They offered practical insights into portfolio selection problem using swarm-based algorithms to construct equity portfolio and manage unsystematic risk. The model and objective type are shown in Table 5 which are established to solve the PO problem by GWO.Table 5
GWO to solve PO problem.
Proposed byModelObjective typeMazumdar et al. [75]Improved ratioSingle objective


### 4.5. MOEA/D Algorithm

MOEA/D algorithm [76] is a powerful multi-objective optimization framework. The main idea of MOEA/D lies in decomposing multiple objective optimization problem into multiple scalar sub-problems and optimizing them simultaneously [77]. Every sub-problem utilizes the information of its neighboring sub-problems, which reduces its computational complexity of each generation. Scalarizing functions, which are provided with uniformly distributed weight vectors, are the fitness evaluation condition of MOEA/D [78]. In MOEA/D, the choice of reference point is very important [79]. In the standard MOEA/D, the reference point is usually the optimal solution for each sub-objective. For the study of reference points, Wang et al. [80] discussed the influence of reference points on the MOEA/D and divided the effects of reference points into three categories, which are pessimistic, optimistic, and dynamic. Ma et al. [81] proposed an improved MOEA/D algorithm based on opposition-based learning with faster convergence speed.Zhang et al. [82] summarized the importance of weight vector in MOEA/D and analyzed its effect on the algorithm performance. The model and objective type are established to solve PO problem by MOEA/D algorithm which is listed in Table 6. The Pareto front of multi-objective optimization (MOP) in reality is usually complex, and it has proved that a MOEA/D-CP method can get more excellent solutions on four portfolio optimization problems.Table 6
MOEA/D algorithm to solve PO problem.
Proposed byModelObjective typeZhang et al. [82]Improved M-VMulti-objective


### 4.6. NSGA-II Algorithm

In [83], the nondominated sorting genetic algorithm (NSGA) was proposed by Srinivas and Deb. NSGA approach has three main disadvantages, which are high computational complexity of nondominated sorting, lack of elitism, and needing to specify the sharing parameter σ share. Deb et al. [84] addressed all of these issues and proposed an improved version of NSGA, which is called NSGA-II. This approach has a computational complexity with MN2, where M is the number of objectives and N is the population size. They also presented a selection operator which creates a mating pool by combining the parents and offspring populations and selecting the best (with respect to fitness and spread) N solutions. They modified the definition of dominance to solve constrained multi-objective problems efficiently. With the properties of a fast nondominated sorting procedure, an elitist strategy and a parameterless approach, NSGA-II has been widely used to solve multi-objective optimization problems in various fields. Eftekharian et al. [85] developed a new efficient multi-objective portfolio optimization algorithm, 2-phase NSGA-II algorithm, which is used to optimize the reward and risk in Portfolio. Anagnostopoulos and Mamanis [86] analyzed and compared NSGA-II, PESA, and SPEA2 for finding an approximation of the best possible tradeoffs between return, risk, and the cardinality of the portfolio. The model and objective type are established to solve PO problem by NSGA-II which are listed in Table 7.Table 7
NSGA-II algorithm to solve PO problem.
Proposed byModelObjective typeEftekharian et al. [85]Mean-VaR/CVaRMulti-objectiveAnagnostopoulos and Mamanis [86]Improved M-VMulti-objective


### 4.7. MPMO-Based Algorithms

Multiple populations for multiple objectives algorithm (MPMO) is new kind of efficient multi-objective optimization framework, which can not only improve the diversity of population, but also improve the search efficiency of algorithm. Zhan et al. [87] developed a novel MPMO to solve multi-objective optimization problems by letting each population correspond with only one objective. Chen et al. [88] proposed a multiple population ant colony system approach to solve cloud workflow scheduling problem. The experimental results show that the algorithm has significant advantages in solving large-scale cloud workflow scheduling through effective design of pheromone update rule, complementary heuristic, and elite study strategies. A coevolutionary particle swarm optimization with a bottleneck objective learning strategy for many-objective optimization was proposed by Liu et al. in [89]. Bottleneck objective learning strategy, elitist learning strategy, and juncture learning strategy are applied simultaneously to improve the performance and efficiency of the algorithm. Liu et al. [90] proposed a MPMO framework-based GA to efficiently solve five objectives job-shop scheduling problem. Archive sharing technique and archive update strategy were designed to guide the coevolution of all the populations and enhance the quality of the elite solutions. By combining the ant colony system optimizer and the MPMO framework, MOACS was proposed to solve the airline crew rostering problem (CRP) by Zhou et al. in [91]. A hybrid complementary heuristic strategy and a local search strategy were designed in MOACS to help explore the global Pareto front sufficiently of CRP. A coevolutionary PSO algorithm was proposed by Zhao et al. in [92], which was applied in solving multi-objective cardinality constrained PO problem. Bidirectional local search strategy and hybrid elite competition strategy were designed to improve the solution accuracy and bring diversity to avoid local Pareto front. The model and objective type are established to solve PO problem by coevolutionary PSO algorithm which is listed in Table 8.Table 8
MPMO-based algorithm to solve PO problem.
Proposed byModelObjective typeZhao et al. [92]Cardinality constrained M-VMulti-objective


### 4.8. Other Algorithms

ABC algorithm was proposed by Karaboga [14] based on intelligent behaviors of bee swarms. Liu et al. [93] firstly applied ABC to PO problem and compared with GA. BFO algorithm was inspired by the organized activities in bacterial foraging and the survival of bacteria in different environments [13]. A BFO algorithm consists of a swarm of bacteria, which represent solutions in the optimization problem and consists of three processes: chemotaxis, reproduction, and elimination-dispersal. Mishra et al. [94] employed the multi-objective BFO algorithm to solve constrained portfolio asset selection problem. Chen et al. [95] developed a multi-objective firefly algorithm (MOFA) to solve fuzzy portfolio selection based on data envelopment analysis cross-efficiency model. Strumberger et al. [96] applied the hybridized BA to the constrained portfolio optimization problem which shows a great potential. Shi [20] proposed BSO algorithm in 2011, which is inspired by the brainstorming process. Hou et al. [97] reviewed the historical development, different state-of-the-art variants, and the real-world applications of BSO algorithm. Ma et al. [98] proposed an orthogonal learning framework to improve BSO, and the experimental results have shown that the new improved approach is very powerful in optimizing complex functions. Niu et al. [99] employed original BSO, simplified BSO (SBSO) [100], and BSO in objective space (BSO-OS) [101] algorithms to solve portfolio optimization (PO) problem with transaction fee and no short sales, which not only saves the computation time but also finds the optimal value. It shows an extraordinary performance in all set of test data. DE is proposed by Price et al. in [12] which is a simple and efficient heuristic algorithm for global optimization over continuous spaces. A Gaussian perturbation-based differential evolution with opposition-based learning was proposed by Zhao et al. in [102] which has a good ability to balance between exploitation and exploration. By combining with differential mutation, periodic learning, and linear population size reduction strategy for global optimization, Zhao et al. [103] proposed a new adaptive harmony search algorithm (aHSDE) which has a very competitive performance on CEC 2014 benchmark among well-known HS variants and several state-of-the-art evolutionary algorithms. Sun et al. [104] proposed a relatively simple and direct method using turning-based mutation that is aimed to solve the problem of premature convergence of algorithms based on success-history-based adaptive differential evolution (SHADE) in high-dimensional search space. Hu and Li [105] proposed an enhancing differential evolution algorithm with a rank-up selection (RUSDE) which is applied to the real-world optimization problem of the four-bar linkages, where the performance of RUSDE is better than other algorithms. Krink and Paterlini [106] proposed a new multi-objective evolutionary algorithm based on differential evolution (DEMPO) for portfolio optimization, which can obtain very satisfying results in reasonable runtime. Table 9 shows the model and objective type of PO which are solved by these algorithms.Table 9
Other algorithms to solve PO problem.
AlgorithmsModelObjective typeABC [93]Semi-varianceSingle objectiveMOBFO [94]Improved M-VMulti-objectiveMOFA [95]Mean-semi-varianceMulti-objectiveBA [96]Improved M-VSingle objectiveBSO [97]Improved M-VSingle objectiveDE [106]M-V, M-VaR, mean-ESMulti-objective


## 4.1. Genetic Algorithm

GA was firstly proposed by Holland [51]. The original motivation for GA approach was a biological analogy. In selective breeding of plants or animals, people will seek offspring with some ideal characteristics, which are determined at the genetic level by the parents’ chromosome combination [52].For an optimization problem, a candidate solution population is firstly constructed, and each candidate solution is called as an individual. According to Darwin’s natural selection of the survival of the fittest, well-adapted solutions have greater probabilities of being chosen and reproduced. After generations of replacement, the population becomes more and more promising, and the approximate results finally enhance the solution. Shoaf and Foster [53] applied GA to Markowitz’ portfolio selection problem and found that the time complexity of GA approximates O(nlogn), which is better than that of quadratic programming. Lin and Liu [54] presented and compared three possible decision models to solve the portfolio selection problem with their corresponding genetic algorithms. Table 2 shows the model and objective function type established by genetic algorithm in solving PO problem.Table 2
GA to solve PO problem.
Proposed byModelObjective typeShoaf and Foster [53]M-VSingle objectiveLin and Liu [54]Improved M-VSingle objective


## 4.2. Particle Swarm Optimization Algorithm

PSO is one of the SI algorithms that was firstly introduced by Kennedy and Eberhart in 1995 [11], inspired by swarm behaviors of birds flocking and fishes schooling. Since its inception in 1995, PSO has been rapidly developing and improving, with lots of successful applications to real-world problems [55]. In PSO, a swarm of particles is introduced to represent the solutions. Each particle is associated with two vectors, the velocity vector and the position vector. During an iteration, the fitness of particle will be evaluated at its current position. If the fitness is better than the personal best position that the particle has achieved so far, the personal best position will be replaced by the current solution. Updating the velocity and position of every particle, then the algorithm goes to next iteration, until a stop condition is met [56].PSO was firstly applied to the portfolio optimization problem using M-V model by Chen et al. [57]. References [58–66] presented some enhanced PSO algorithms for solving portfolio optimization. Wang and Watada [67] used VaR as the risk measurement and solved the multi-objective fuzzy portfolio selection problem. Liang and Qu [68] proposed multi-objective dynamic multi-swarm PSO (DMPSO) to solve the large-scale PO problems and carried out experiments with data sets that include 200 and 500 assets. Table 3 shows the model and objective function types established by PSO in solving PO.Table 3
PSO to solve PO problem.
Proposed byModelObjective typeChen et al. [57]Improved M-VSingle objectiveXu and Chen [58]Stochastic M-VSingle objectiveLiu and Wang [59]Entropy modelSingle objectiveWang et al. [60]Improved M-VSingle objectiveNiu et al. [61]Improved M-VMulti-objectiveYaakob and Watada [62]Improved M-VSingle objectiveMozafari et al. [63]Improved M-VSingle objectiveDeng et al. [64]Improved M-VSingle objectiveCorazza et al. [65]Mixed-integerSingle objectiveYin et al. [66]Improved M-VSingle objectiveWang and Watada [67]M-V-VaRMulti-objectiveLiang and Qu [68]M-VMulti-objective


## 4.3. Fireworks Algorithm

FWA is a novel intelligent algorithm which is proposed by Tan et al. in 2010 [19]. By imitating the fireworks explosion process, the fireworks algorithm can effectively search for the best solution. Due to its powerful ability to solve optimization problems, fireworks algorithm has been applied to different optimization problems in different fields, and it has performed great potential. The main application areas include knapsack problem, image recognition, filter designing [69–71], etc. Zhang and Liu [72] applied FWA into the field of portfolio optimization to solve the mean-VaR/CVaR model. Compared with the classical genetic algorithm, fireworks algorithm not only improves the optimization accuracy and the optimizing speed, but also makes the optimal solution more stable. Bacanin and Tuba [73] adapted fireworks algorithm for solving constrained portfolio selection problem that extends classical mean-variance portfolio model by adding additional constraints. Table 4 shows the model and objective type which are established to solve the PO problem by FWA.Table 4
FWA to solve PO problem.
Proposed byModelObjective typeZhang and Liu [72]Mean-VaR/CVaRSingle objectiveBacanin and Tuba [73]Improved M-VSingle objective


## 4.4. Gray Wolf Optimizer

Gray wolf optimizer (GWO) is a population-based metaheuristic algorithm introduced by Mirjalili et al. [74], and its inspiration comes from the unique hierarchical hunting behavior of gray wolves in nature. Gray wolves hunt in packs, and the pack is distributed into alpha wolf (α, the fittest and their leader), beta wolf (β, second in line of leadership), and delta wolf (δ, third fittest), and all other remaining are omega wolves. The hunting (optimization) of prey is guided by α, β, and δ wolves. Mazumdar et al. [75] proposed GWO technique for portfolio construction and unsystematic risk optimization. They offered practical insights into portfolio selection problem using swarm-based algorithms to construct equity portfolio and manage unsystematic risk. The model and objective type are shown in Table 5 which are established to solve the PO problem by GWO.Table 5
GWO to solve PO problem.
Proposed byModelObjective typeMazumdar et al. [75]Improved ratioSingle objective


## 4.5. MOEA/D Algorithm

MOEA/D algorithm [76] is a powerful multi-objective optimization framework. The main idea of MOEA/D lies in decomposing multiple objective optimization problem into multiple scalar sub-problems and optimizing them simultaneously [77]. Every sub-problem utilizes the information of its neighboring sub-problems, which reduces its computational complexity of each generation. Scalarizing functions, which are provided with uniformly distributed weight vectors, are the fitness evaluation condition of MOEA/D [78]. In MOEA/D, the choice of reference point is very important [79]. In the standard MOEA/D, the reference point is usually the optimal solution for each sub-objective. For the study of reference points, Wang et al. [80] discussed the influence of reference points on the MOEA/D and divided the effects of reference points into three categories, which are pessimistic, optimistic, and dynamic. Ma et al. [81] proposed an improved MOEA/D algorithm based on opposition-based learning with faster convergence speed.Zhang et al. [82] summarized the importance of weight vector in MOEA/D and analyzed its effect on the algorithm performance. The model and objective type are established to solve PO problem by MOEA/D algorithm which is listed in Table 6. The Pareto front of multi-objective optimization (MOP) in reality is usually complex, and it has proved that a MOEA/D-CP method can get more excellent solutions on four portfolio optimization problems.Table 6
MOEA/D algorithm to solve PO problem.
Proposed byModelObjective typeZhang et al. [82]Improved M-VMulti-objective


## 4.6. NSGA-II Algorithm

In [83], the nondominated sorting genetic algorithm (NSGA) was proposed by Srinivas and Deb. NSGA approach has three main disadvantages, which are high computational complexity of nondominated sorting, lack of elitism, and needing to specify the sharing parameter σ share. Deb et al. [84] addressed all of these issues and proposed an improved version of NSGA, which is called NSGA-II. This approach has a computational complexity with MN2, where M is the number of objectives and N is the population size. They also presented a selection operator which creates a mating pool by combining the parents and offspring populations and selecting the best (with respect to fitness and spread) N solutions. They modified the definition of dominance to solve constrained multi-objective problems efficiently. With the properties of a fast nondominated sorting procedure, an elitist strategy and a parameterless approach, NSGA-II has been widely used to solve multi-objective optimization problems in various fields. Eftekharian et al. [85] developed a new efficient multi-objective portfolio optimization algorithm, 2-phase NSGA-II algorithm, which is used to optimize the reward and risk in Portfolio. Anagnostopoulos and Mamanis [86] analyzed and compared NSGA-II, PESA, and SPEA2 for finding an approximation of the best possible tradeoffs between return, risk, and the cardinality of the portfolio. The model and objective type are established to solve PO problem by NSGA-II which are listed in Table 7.Table 7
NSGA-II algorithm to solve PO problem.
Proposed byModelObjective typeEftekharian et al. [85]Mean-VaR/CVaRMulti-objectiveAnagnostopoulos and Mamanis [86]Improved M-VMulti-objective


## 4.7. MPMO-Based Algorithms

Multiple populations for multiple objectives algorithm (MPMO) is new kind of efficient multi-objective optimization framework, which can not only improve the diversity of population, but also improve the search efficiency of algorithm. Zhan et al. [87] developed a novel MPMO to solve multi-objective optimization problems by letting each population correspond with only one objective. Chen et al. [88] proposed a multiple population ant colony system approach to solve cloud workflow scheduling problem. The experimental results show that the algorithm has significant advantages in solving large-scale cloud workflow scheduling through effective design of pheromone update rule, complementary heuristic, and elite study strategies. A coevolutionary particle swarm optimization with a bottleneck objective learning strategy for many-objective optimization was proposed by Liu et al. in [89]. Bottleneck objective learning strategy, elitist learning strategy, and juncture learning strategy are applied simultaneously to improve the performance and efficiency of the algorithm. Liu et al. [90] proposed a MPMO framework-based GA to efficiently solve five objectives job-shop scheduling problem. Archive sharing technique and archive update strategy were designed to guide the coevolution of all the populations and enhance the quality of the elite solutions. By combining the ant colony system optimizer and the MPMO framework, MOACS was proposed to solve the airline crew rostering problem (CRP) by Zhou et al. in [91]. A hybrid complementary heuristic strategy and a local search strategy were designed in MOACS to help explore the global Pareto front sufficiently of CRP. A coevolutionary PSO algorithm was proposed by Zhao et al. in [92], which was applied in solving multi-objective cardinality constrained PO problem. Bidirectional local search strategy and hybrid elite competition strategy were designed to improve the solution accuracy and bring diversity to avoid local Pareto front. The model and objective type are established to solve PO problem by coevolutionary PSO algorithm which is listed in Table 8.Table 8
MPMO-based algorithm to solve PO problem.
Proposed byModelObjective typeZhao et al. [92]Cardinality constrained M-VMulti-objective


## 4.8. Other Algorithms

ABC algorithm was proposed by Karaboga [14] based on intelligent behaviors of bee swarms. Liu et al. [93] firstly applied ABC to PO problem and compared with GA. BFO algorithm was inspired by the organized activities in bacterial foraging and the survival of bacteria in different environments [13]. A BFO algorithm consists of a swarm of bacteria, which represent solutions in the optimization problem and consists of three processes: chemotaxis, reproduction, and elimination-dispersal. Mishra et al. [94] employed the multi-objective BFO algorithm to solve constrained portfolio asset selection problem. Chen et al. [95] developed a multi-objective firefly algorithm (MOFA) to solve fuzzy portfolio selection based on data envelopment analysis cross-efficiency model. Strumberger et al. [96] applied the hybridized BA to the constrained portfolio optimization problem which shows a great potential. Shi [20] proposed BSO algorithm in 2011, which is inspired by the brainstorming process. Hou et al. [97] reviewed the historical development, different state-of-the-art variants, and the real-world applications of BSO algorithm. Ma et al. [98] proposed an orthogonal learning framework to improve BSO, and the experimental results have shown that the new improved approach is very powerful in optimizing complex functions. Niu et al. [99] employed original BSO, simplified BSO (SBSO) [100], and BSO in objective space (BSO-OS) [101] algorithms to solve portfolio optimization (PO) problem with transaction fee and no short sales, which not only saves the computation time but also finds the optimal value. It shows an extraordinary performance in all set of test data. DE is proposed by Price et al. in [12] which is a simple and efficient heuristic algorithm for global optimization over continuous spaces. A Gaussian perturbation-based differential evolution with opposition-based learning was proposed by Zhao et al. in [102] which has a good ability to balance between exploitation and exploration. By combining with differential mutation, periodic learning, and linear population size reduction strategy for global optimization, Zhao et al. [103] proposed a new adaptive harmony search algorithm (aHSDE) which has a very competitive performance on CEC 2014 benchmark among well-known HS variants and several state-of-the-art evolutionary algorithms. Sun et al. [104] proposed a relatively simple and direct method using turning-based mutation that is aimed to solve the problem of premature convergence of algorithms based on success-history-based adaptive differential evolution (SHADE) in high-dimensional search space. Hu and Li [105] proposed an enhancing differential evolution algorithm with a rank-up selection (RUSDE) which is applied to the real-world optimization problem of the four-bar linkages, where the performance of RUSDE is better than other algorithms. Krink and Paterlini [106] proposed a new multi-objective evolutionary algorithm based on differential evolution (DEMPO) for portfolio optimization, which can obtain very satisfying results in reasonable runtime. Table 9 shows the model and objective type of PO which are solved by these algorithms.Table 9
Other algorithms to solve PO problem.
AlgorithmsModelObjective typeABC [93]Semi-varianceSingle objectiveMOBFO [94]Improved M-VMulti-objectiveMOFA [95]Mean-semi-varianceMulti-objectiveBA [96]Improved M-VSingle objectiveBSO [97]Improved M-VSingle objectiveDE [106]M-V, M-VaR, mean-ESMulti-objective


## 5. Conclusion and Possible Research Topics

As the financial market volatility is intensifying, the rate of financial asset return becomes more and more uncertain. Rationally allocating the ratio of different investment assets to maximize the portfolio overall return and to minimize the risk becomes more and more important. Recently, SI algorithms have been widely used to solve these complex PO problems which are very popular in scientific and engineering fields. In this paper, we present a concise overview mainly for PO models and SI algorithms. Although many publications have appeared on the application of SI algorithms in PO by many scholars, many important problems remain unsolved and new application areas of SI keep continually emerging.Three possible research directions for solving PO problem with SI are discussed below.


### 5.1. Establishment of Dynamic Multi-Period PO Model

The investment decision in reality is actually a dynamic change process, which needs to continuously purchase and sale asset according to market conditions. Dynamic multi-period PO model is in line with real investment strategies. Therefore, how to achieve the maximum benefit and minimum risk of dynamic multi-period portfolio is a worthy study problem in the future.


### 5.2. Obtaining the Reliable Portfolio Preference Function

Due to the complexity of investor preferences, preference selection function is often difficult to acquire. How to choose the preference function will be also a valuable research topic.


### 5.3. Evaluation of Model and Algorithm Validity

Because the return and risk of different portfolios vary greatly, it is necessary to evaluate the effectiveness of the established PO model. Corresponding standard test sets should be established to evaluate the models and algorithms, which is also an interesting research topic.


## 5.1. Establishment of Dynamic Multi-Period PO Model

The investment decision in reality is actually a dynamic change process, which needs to continuously purchase and sale asset according to market conditions. Dynamic multi-period PO model is in line with real investment strategies. Therefore, how to achieve the maximum benefit and minimum risk of dynamic multi-period portfolio is a worthy study problem in the future.


## 5.2. Obtaining the Reliable Portfolio Preference Function

Due to the complexity of investor preferences, preference selection function is often difficult to acquire. How to choose the preference function will be also a valuable research topic.


## 5.3. Evaluation of Model and Algorithm Validity

Because the return and risk of different portfolios vary greatly, it is necessary to evaluate the effectiveness of the established PO model. Corresponding standard test sets should be established to evaluate the models and algorithms, which is also an interesting research topic.



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*Source: 4241049-2022-07-15.xml*