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This book explains the theoretical structure of particle swarm optimization (PSO) and focuses on the application of PSO to portfolio optimization problems. The general goal of portfolio optimization is to find a solution that provides the highest expected return at each level of portfolio risk. According to H. Markowitz’s portfolio selection theory, as new assets are added to an investment portfolio, the total risk of the portfolio’s decreases depending on the correlations of asset returns, while the expected return on the portfolio represents the weighted average of the expected returns for each asset.

The book explains PSO in detail and demonstrates how to implement Markowitz’s portfolio optimization approach using PSO. In addition, it expands on the Markowitz model and seeks to improve the solution-finding process with the aid of various algorithms. In short, the book provides researchers, teachers, engineers, managers and practitioners with many tools they need to apply the PSO technique to portfolio optimization.



Applying Particle Swarm Optimization to Portfolio Optimization


Chapter 1. Utility: Theories and Models

The aim of this study is to look at utility theory from a broad perspective. The main hypothesis in the theory of decision is that the person who is in the position of deciding is entitled to the “economic man.” Also, the individual acts rationally. Thus, utility is the ability to satisfy (eliminate) human needs of goods and services. Utility is basically a psychological concept and also is the basis of economics and finance. Three types of utility take place in the economics and finance literature: marginal utility, total utility, and average utility. In addition, two main approaches fall within utility comparison: cardinal utility theory and ordinal utility theory. Furthermore, expected utility theory forms the basis of traditional finance. Expected benefit theory assumes that people choose risky or uncertain opportunities by comparing the expected benefits from them. Allais and Ellsberg paradoxes criticize expected utility theory. Tversky and Kahneman (Econometrica, 47: 263–291, 1979) present that the expected utility axioms are violated for more reasonable lottery alternatives than in the Allais paradox and put a link between finance and psychology. The prospect theory of Tversky and Kahneman forms the basis of behavioral finance.
Murat Akkaya

Chapter 2. Portfolio Optimization

In portfolio management, it is aimed to create a portfolio that gives the best combination of risk and return among the assets in the market. There are different optimization techniques for creating an optimum portfolio depending on the risk and return variable. Particle swarm optimization (PSO) method is one of the important and useful techniques used in portfolio optimization in finance. In this chapter, Markowitz mean-variance model, which is the main model of modern portfolio theory, is explained, and mathematical representations are given. The subject is supported with mathematical notations by mentioning concepts such as portfolio risk and return, efficient frontier, utility theory, asset allocation, indifference curves, Sharpe ratio, and coefficient of variation.
Burcu Adıgüzel Mercangöz

Chapter 3. Behavioral Portfolio Theory

The aim of this study is to explain behavioral portfolio theory in a theoretical way. The study starts with the definition of portfolio which is a financial asset that consists of various securities such as stocks and bonds and derivative products, held by a particular person or group. Also, portfolio management is the management of securities according to investors’ returns and risk targets. There are two basic portfolio management theories in finance literature. The first is the traditional portfolio (simple diversification) approach based on the diversification of securities. The second is the modern portfolio theory, which is based on a more mathematical basis. Modern portfolio theory mathematically shows the measurement of the risk and return of a portfolio of two or more securities and the determination of optimal portfolios. Markowitz’s mean-variance model is the first mathematical explanation of the idea of diversifying investments and is the cornerstone of many risk models developed such as capital asset pricing model and arbitrage pricing model in later years. The empirical studies reveal that investors do not act rationally as financial models assume and anomalies occur. Thus, behavioral finance tries to fill the gap in this area and states that investors should be considered “normal” rather than rational. Prospect theory developed by Kahneman and Tversky (1979), brings psychological explanations to finance issues. Mental accounting in behavioral finance prevents the rule of taking into account the correlation between the returns of an investment. Also, herd behavior of investors distorts the efficiency of the markets and leads to volatility in financial markets. When investors show herd behavior, they do not care about the information received and tend to imitate other investor behavior. Behavioral portfolio theory (BPT) emerged as a descriptive alternative to Markowitz’s mean-variance portfolio theory. BPT connects two issues, the creation of portfolios and the design of securities. There are two versions of the BPT model: single mental accounting (BPT-SA) in which the portfolio is integrated into one mental accounting and multiple mental accounting (BPT-MA).
Murat Akkaya

Chapter 4. A Comparative Study on PSO with Other Metaheuristic Methods

The research and development of metaheuristic methods are critical issues in computer science. In the past decade, metaheuristic algorithms have been used in many engineering applications such as optimization of engineering problems, telecommunications, information security, and image processing. Many metaheuristic algorithms such as particle swarm optimization (PSO), ant colony optimization (ACO), and genetic algorithm (GA) are recently becoming very popular.
There are many studies conducted in the literature on the comparison of PSO with other metaheuristic algorithms. In this chapter, various studies carried out between the years of 2010 and 2020 about the comparison of PSO with the other metaheuristic algorithms will be examined. The metaheuristic algorithms to be considered are simulated annealing (SA), genetic algorithm (GA), differential evolution (DE), ant colony optimization (ACO), artificial bee colony (ABC) algorithm, particle swarm optimization (PSO), tabu search (TS), harmony search (HS), firefly algorithm (FF), cuckoo search (CS), bat-inspired algorithm (BA), water wave optimization (WWO), clonal selection algorithm (CLONALG), chemical reaction optimization (CRO), sine cosine algorithm (SCA), glowworm swarm optimization (GSO), and grey wolf optimizer (GWO). This study aims to evaluate and analyze the covered papers according to several criteria such as (a) rates of studies according to publishing years, (b) the metaheuristic algorithms that are compared to PSO, (c) performance evaluation of compared algorithms, (d) the metaheuristic algorithms with their inspirational approaches and their initial proposed studies and years, (e) the field of subjects where the algorithms are applied in the reviewed studies, and (f) used databases in the examined studies.
This study is a comprehensive literature review of the comparison of PSO with the most popular metaheuristic algorithms. The intention of this review is to be useful for researchers who want to conduct a survey on this area of the subject as this chapter will cover the essential and helpful analysis of the related research.
Serhat Yarat, Sibel Senan, Zeynep Orman

Chapter 5. Mathematical Model of Particle Swarm Optimization: Numerical Optimization Problems

The Particle Swarm Optimization (PSO) algorithm was put forth by Kennedy and Eberhart in the year 1995. It is widely known for the ease with which it can be implemented and its simple approach. It is a multi-agent parallel search metaheuristic technique aimed at global optimization for numerical optimization problems. It has roots in artificial life techniques like swarm intelligence, fish schooling, etc. This chapter aims to introduce the mathematical bases for the algorithm and illustrates a few pictorial aids to understand the technique better. It is intended to serve as an introduction to spark the interest of the reader. Readers wishing to learn more about the applications of PSO and its variants to multi-objective, constrained, dynamic optimization problems and other advanced topics are recommended to consider the various references at the end of the chapter.
Ashwin A. Kadkol

Chapter 6. Particle Swarm Optimization: The Foundation

Particle swarm optimization (PSO) is a very much popular swarm intelligence algorithm. Since its inception in the year 1995, it is being applied to solve optimization problems in many domains, including portfolio optimization. This chapter lays the basic PSO foundation and introduces existing PSO variants for researchers who want to solve the portfolio optimization problem. It starts with the introduction of PSO, describing the advantages, disadvantages, and applied areas of PSO. Later, the basic PSO procedure and its parameter selection mechanisms are presented. The chapter also presents three popular applications of PSO in finance, including portfolio optimization. Finally, the chapter ends by introducing the existing PSO variants to solve the portfolio optimization problem.
Dadabada Pradeep Kumar

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

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

Chapter 8. A Constrained Portfolio Selection Model Solved by Particle Swarm Optimization Under Different Risk Measures

Portfolio selection has been one of the crucial problems in financial engineering. Investors’ interest is to construct a portfolio having a balance between the investor’s risk-taking and his/her expectations about the portfolio returns. The Markowitz model is a nonlinear constrained multi-objective optimization model that is usually impossible to solve at a good time. In this chapter, the purpose is to examine portfolio optimization models and applications of the particle swarm optimization (PSO) technique in solving these models. A constrained portfolio selection model has been developed, which is solved by the PSO technique as a metaheuristic approach using data from the Tehran Stock Exchange (TSE) to assess the developed model. In this case, the effects of three different risk measures have been analyzed on the constructed portfolios. The numerical results show that conditional value at risk (CVaR) performs better than the other two risk measures, including semivariance and variance. However, from the diversification perspective, the model with the variance risk measure produces a more diversified portfolio compared to the other two risk measures, although the differences are trivial.
Akbar Esfahanipour, Pouya Khodaee

Chapter 9. Optimal Portfolio Selection with Particle Swarm Algorithm: An Application on BIST-30

Optimization is to find the best-performing solution under the constraints given. It can be something better by optimization process. Heuristic algorithm is an optimization algorithm which depends on natural events. The algorithms are simple and easy to implement for the researcher. The portfolio optimization is a process to find a solution to select the most appropriate combination between all financial assets under certain expectations and constraints. While solving portfolio optimization problems, the aim is to create portfolios by selecting the assets that provide the highest return from huge numbers of financial assets at a certain risk level or provide the lowest risk at a certain level of return. This chapter aims to examine the optimum portfolio with minimum risk by using the particle swarm optimization (PSO) technique, for the stocks in the BIST-30 index. Logarithmic returns are calculated using the price data of the stocks. By using these returns, the optimum portfolio with minimum risk is created with PSO and nonlinear GRG (generalized reduced gradient) techniques. The empirical results obtained indicate that both methods give similar results.
Burcu Adıgüzel Mercangöz, Altaf Q. H. Badar

Chapter 10. Cardinality-Constrained Higher-Order Moment Portfolios Using Particle Swarm Optimization

Particle swarm optimization (PSO) is often used for solving cardinality-constrained portfolio optimization problems. The system invests in at most k out of N possible assets using a binary mapping that enforces compliance with the cardinality constraint. This may lead to sparse solution vectors driving the velocity in PSO algorithm. This sparse-velocity mapping leads to early stagnation in mean-variance-skewness-kurtosis expected utility optimization when k is small compared to N. A continuous-velocity driver addresses this issue. We propose to combine both the continuous- and the sparse-velocity transformation methods so that it updates local and global best positions based on both the drivers. We document the performance gains when k is small compared to N in the case of mean-variance-skewness-kurtosis expected utility optimization of the portfolio.
Mulazim-Ali Khokhar, Kris Boudt, Chunlin Wan

Different Applications of PSO


Chapter 11. Different Applications of PSO

Particle swarm optimization (PSO) is an evolutionary optimization algorithm. PSO is a robust and well researched optimization technique. There are a large number of applications of PSO. “Applications of PSO” chapter tries to present a classified literature review for the applications of PSO in different fields. The applications are classified into different sections based on the area of implementation.
The chapter also presents a table with references to multiple other applications over and above those covered in the chapter. References of some largely cited review papers dealing with the applications of PSO are also mentioned at the end of the chapter.
Altaf Q. H. Badar

Chapter 12. Particle Swarm Optimization in Global Path Planning for Swarm of Robots

Planning of the optimal path of an autonomous swarm of mobile robots is quite challenging since they may need to meet multiple targets while avoiding obstacles. This chapter addresses the problem using a method of global navigation based on particle swarm optimization technique. Since it is a meta-heuristic search technique, the path can be found following any optimization criteria such as shortest distance or minimum time. The chapter explores different traditional path planning approaches in contrast with the evolutionary algorithms. The PSO algorithm is tested in different scenarios. Several modifications were implemented in the algorithm for optimization improvements and faster convergence, leading to better results. Geometrical illustrations were used to explain the changes in position of the particles with respect to the environment and obstacles. Consequently, the experiments are conducted on a simulated robot, and the visualizations demonstrated the feasibility of the technique to solve global path planning problem.
Ritesh Kumar Halder

Chapter 13. Training Multi-layer Perceptron Using Hybridization of Chaotic Gravitational Search Algorithm and Particle Swarm Optimization

A novel amalgamation strategy, namely, chaotic gravitational search algorithm (CGSA) and particle swarm optimization (PSO), has been employed for training multi-layer perceptron (MLP) neural network. It is called CGSAPSO. In CGSAPSO, exploration is carried out by CGSA, and exploitation is performed using PSO. The sigmoid activation function is utilized for training MLP. Besides, a matrix encoding strategy has been used for providing a synergy between neural biases, weights, and CGSAPSO searcher agents. To validate the effectiveness of the hybrid framework, CGSAPSO is applied to three different classification datasets, namely, XOR, Iris, and Balloon. The investigation of results is carried out through various performance metrics like average, standard deviation, median, convergence speed, execution time, and classification rate analysis. Besides, a pair-wise non-parametric signed Wilcoxon rank-sum test has also been conducted for statistical verification of simulation results. In addition, the numerical outcomes of CGSAPSO are also compared with standard GSA, PSO, and hybrid PSOGSA. The experimental results indicate that CGSAPSO provides better results in the form of recognition accuracy and global optima as compared to competing algorithms.
Sajad Ahmad Rather, P. Shanthi Bala, Pillai Lekshmi Ashokan

Chapter 14. Solving Optimization Problem with Particle Swarm Optimization: Solving Hybrid Flow Shop Scheduling Problem with Particle Swarm Optimization Algorithm

The flow shop scheduling problem is widely discussed in the literature since it is frequently applied in real industry. This paper presents a variant of flow shop scheduling problem with parallel machines. The proposed problem includes multistage and identical parallel machines at each stage, and the sequence-dependent setup time and transportation time are considered. The objective function is minimization of makespan. The particle swarm optimization algorithm (PSO) is addressed to solve the problem and compared with genetic algorithm and heuristics. The benchmark instances are generated to demonstrate the performance of the PSO. The numerical results show that the PSO significantly outperforms the comparison set.
Fatma Selen Madenoğlu

Chapter 15. Constriction Coefficient-Based Particle Swarm Optimization and Gravitational Search Algorithm for Image Segmentation

Image segmentation is one of the pivotal steps in image processing. Actually, it deals with the partitioning of the image into different classes based on pixel intensities. In this work, a new image segmentation method has been introduced based on the constriction coefficient-based particle swarm optimization and gravitational search algorithm (CPSOGSA). The random samples of the image histogram act as searcher agents of the CPSOGSA. Besides, the optimal number of thresholds is determined using Kapur’s entropy method. The effectiveness and applicability of CPSOGSA have been accomplished by applying it to four standard images from the USC-SIPI image database including airplane, cameraman, clock, and truck. Various performance metrics have been employed to investigate the simulation outcomes including optimal thresholds, standard deviation, mean, run-time analysis, PSNR (peak signal-to-noise ratio), best fitness value calculation, convergence maps, and box plot analysis. In addition, the experimental results of CPSOGSA are compared with standard PSO and GSA. The simulation results clearly indicate that hybrid CPSOGSA takes less computational time in finding the best threshold values of the benchmark images.
Sajad Ahmad Rather, P. Shanthi Bala

Chapter 16. An Overview of the Performance of PSO Algorithm in Renewable Energy Systems

An increase in the penetration of renewable energy sources in the electrical production has been matched by the emergence of many and varied challenges and problems. Among the most important challenges is finding the smart technologies and algorithms which are capable of achieving efficient solutions. This chapter provides an expanded view of the uses of the particle swarm optimization (PSO) algorithm in the renewable energy systems field. Additionally, it describes how the algorithm can be developed to cope with problems related to renewable energies to achieve desired goals. The PSO algorithm was used to solve many problems in the renewable energy systems, such as in optimal hybrid power systems, optimal sizing, and optimal net present cost, among others, where the PSO algorithm showed its high adaptability in problem-solving. Further, many researchers proceeded with the study and development of the PSO algorithm. In contrast, other researchers tried to hybridize it with different algorithms to be more efficient and convenient to overcome some of the problems and challenges that they encountered. The renewable energy systems have several issues to discuss, such as the cost of investment, the feasible technical criteria, optimal control, and the ecological problems as well as the social effect. Overall, studies and research have proven that the PSO algorithm is one of the best algorithms used in the field of renewable energy. This is attributed to the algorithm’s simplicity, high efficiency, and effectiveness compared to other algorithms and optimization methods.
Omar Hazem Mohammed, Mohammed Kharrich

Chapter 17. Application of PSO in Distribution Power Systems: Operation and Planning Optimization

Being an engineering field, power systems provide an extensive subject for optimization to be applied upon. Modern power systems have evolved in an increasingly highly complex system. The liberalization of the energy market and the introduction of distributed generation and, in particular, distributed renewable energy resources (DRES) have raised both opportunities and challenges that need to be tackled. Thus, complex issues related to the operation and planning of the distribution systems have emerged. Such issues involve many variables and refer to nonlinear objectives; thus their optimization is significantly based on heuristic techniques, such as particle swarm optimization (PSO). In this chapter, the implementation of PSO when contemplating various problems in power systems is presented. In particular, the utilization of PSO is demonstrated in the optimal distributed generation placement problem (ODGP), also known as optimal siting and sizing of distributed generation problem, and in the network reconfiguration problem. Finally, PSO is implemented in an optimal schedule of electric vehicles (EVs) charging, providing an apt example of the variety of problems for which PSO can be utilized and providing useful aid to important decisions, in the field of power systems.
Paschalis A. Gkaidatzis, Aggelos S. Bouhouras, Dimitris P. Labridis
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