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Top1. Introduction
Numerous models have been developed throughout the course of the twentieth century in an attempt to guide investigators toward the right approach to modelling investments. Sector diversity and stock portfolio optimization have evolved into ideas that are used as tools for decision-making and the growth of financial markets. Since Markowitz’s model was released, it has been applied as a useful tool for stock portfolio optimization and has brought about a number of adjustments and advances in the way researchers view investing and stock portfolios. While it may appear straightforward to minimize risk and maximize investment returns, creating an ideal portfolio really involves a number of techniques. Markowitz presented contemporary portfolio theory as a mathematical formula that embodies a classical approach (Markowitz 1952). His mean-variance approach gives information about a portfolio’s risk by displaying the average anticipated return and variance. A lot of individuals have attempted to expand and alter Markowitz’s concept. In the area of stock portfolio optimization, several studies have been conducted using a variety of models as well as numerical and clever techniques. See Kalayci et al. (2019); Ponsich et al. (2012); Thakkar and Chaudhari (2021); Zanjirdar (2020), for an outline of some of these examples. Articles Aranha and Iba (2009); Bazrkar and Hosseini (2023); Chang et al. (2009); Coello et al. (2007); Erwin and Engel- brecht (2023); Lin and Liu (2008); Wright (2006) (also see reviewer articles and books in Goldberg (1989); Gunjan and Bhattacharyya (2022); Zanjirdar (2020)) discuss the use of genetic algorithms, or GA, in stock portfolio optimization. As you shall see below, GA may be used to solve the constrained optimization problem as it is currently integrated into the Matlab toolbox. Furthermore, this software has other algorithms that can address the Markowitz portfolio optimization issue, including pattern-search, Pareto-search, and quadratic programming, or QP for short. However, when it comes to the PSO algorithm, the Matlab toolbox has just one unconstrained optimization toolbox.
When there are no linear restrictions, pattern search automatically searches for a minimum based on an adaptive mesh that is aligned with the coordinate directions. Pattern-search (see the algorithm in Audet and Dennis Jr (2002)) locates a series of locations 
that converge to a minimal point.
The Pareto-search method iteratively looks for non-dominant points by applying pattern-search to a group of points. In every iteration, the Pareto-search meets all linear bounds and restrictions. The method should theoretically converge to locations around the actual Pareto front. See Custódio et al. (2011) for a discussion and proof of convergence, where the proof is used for Lipschitz continuous objective and constraint problems.
Recently, Kennedy and Eberhart (1995) presented a heuristic method called particle swarm optimization. While some PSO research has been done, virtually none of it addresses portfolio optimization for the mean-variance Markowitz model (Cura (2009)). For a complete overview of PSO applications in portfolio optimization, Thakkar and Chaudhari (2021) is a useful resource. Also, refer to recent advancements and applications of PSO in the works Bazrkar and Hosseini (2023); Erwin and Engelbrecht (2023); He and Huang (2014); Kuo and Chiu (2024); Liu and Li (2024); Song et al. (2023).