# Multi-Guide Set-Based Particle Swarm Optimization for Multi-Objective Portfolio Optimization
## 1 Introduction  
Portfolio optimization is a complex problem not only in the depth of the topics that it covers, but also in its breadth. It is the process of determining which assets to include in a portfolio while simultaneously maximizing profit and minimizing risk. To illustrate the investment process, consider the game of Monopoly. In the game, players purchase property. When a player lands on another player's property, that player must then pay the owner a fee. The more expensive the property purchased is, the higher the fees will be. Players must, therefore, strategize about which properties to buy. In some cases, a player might take a risk and purchase an expensive property, but unfortunately, it is hardly ever visited by other players. In this scenario, the risk does not pay off and possibly jeopardizes the player's position in the game. Players must determine what the best possible way to spend their money would be in order to maximize their profits without going bankrupt. The real world is much more complex, with many moving parts, but it is similar to Monopoly in that investing can be a risky venture. However, it is possible that an asset's value increases significantly, making it a worthwhile investment. Identifying which asset or collection of assets--known as a portfolio--would yield an optimal balance between risk and reward is not an easy task. Moreover, there may exist multiple, but equally good, portfolios that have different risk and return characteristics, which further complicates the task. Lastly, when constraints that introduce nonlinearity and non-convexity (such asboundary constraints and cardinality constraints) are added, the problem becomes NP-Hard [1; 2; 3]. Thus, approaches such as quadratic programming cannot be efficiently utilized to obtain solutions. Meta-heuristics are computationally efficient and are effective approaches to obtaining good-quality solutions for a variety of portfolio models [3]. Typically, solutions are represented by fixed-length vectors of floats where the elements in a vector correspond to asset weights. Unfortunately, the performance of fixed-length vector meta-heuristics deteriorate for larger portfolio optimization problems [4]. An alternative approach is to redefine the portfolio problem as a set-based problem where a subset of assets are selected and then the weights of these assets are optimized. For example, hybridization approaches that integrate quadratic programming with genetic algorithms (GAs) have been shown to increase performance for constrained portfolio optimization problems [5; 6; 7; 8; 9]. A new set-based approach, set-based particle swarm optimization (SBPSO) for portfolio optimization, uses particle swarm optimization (PSO) to optimize asset weights instead of quadratic programming and has demonstrated good performance for the portfolio optimization problem [10]. Single-objective portfolio optimization requires a trade-off coefficient to be specified in order to balance the two objectives, i.e., risk and return. A collection of equally good but different solutions can then be obtained by solving the single-objective optimization problem for various trade-off coefficient values. However, a more sophisticated and appropriate approach would be to use a multi-objective optimization (MOO) algorithm to identify an equally spread set of non-dominated solutions, e.g., multi-guide particle swarm optimization (MGPSO). MGPSO is a multi-swarm multi-objective PSO algorithm that uses a shared archive to store non-dominated solutions found by the swarms [11]. This paper proposes a new approach to multi-objective portfolio optimization, multi-guide set-based particle swarm optimization (MGSBPSO), that combines elements of SBPSO with MGPSO. The novelty of the proposed approach is that it can identify multiple but equally good solutions to the portfolio optimization problem with the scaling benefits of a set-based approach.