# 2-Phase NSGA II: An Optimized Reward and Risk Measurements Algorithm in Portfolio Optimization
## 2 Related Work  
Portfolio optimization has been used in several works with multiple objectives, and many heuristic algorithms have been applied for solving this problem. For example, authors in [17] proposed a decision making model for portfolio selection aimed at minimizing transaction lots. They solved the model with genetic algorithm (GA). Their method found the solution in a short reasonable time, but the current case that is the market is the only uncertainty in reality and some other risk cases must be added. Moreover, authors in [4] used simulated annealing (SA) for solving portfolio selection problem. The objective function of their model was to reduce portfolio risk, and expected return of investor was assigned and set as a constraint. The method is an efficient solution in the scope, however, it is still a complicated solution to manage due to the space of feasible portfolios that is simplified in our algorithm. A heuristic model based on neural network was developed by [18] and was handled for tracing efficiency curve. They compared their results with tabu search (TS), simulated annealing (SA), and genetic algorithm (GA). The presented model is indeed a combination of quadratic programming model and integer programming which none of exact algorithms could not solve this problem efficiently and consequently, the necessity of utilization of evolutionary algorithms (EAs) seems worthwhile for solving the problem. Moreover, authors in [19] solved portfolio optimization problem in terms of various risk criteria containing semi-variance, variance with skewness, and genetic algorithm. On the other hand, authors in [20] utilized a new neural network algorithm for selecting optimized portfolio based on investor preferences. The given model was based on mean, return, and skewness. They showed that the presented model led to the result in shorter time as compared to other models. The big drawback of the method is its inefficacy when face with the dynamic and uncertain environment and the risk can be tuned fairly. The paper [21] investigated particle swarm optimization (PSO) for solving portfolio selection problem. This method instead of finding the global optimal solution converges to a close optimal solution and won't be a useful option for the uncertain dynamic risk-aware systems. Even though, [22] suggested an improved particle swarm optimization algorithm for the problem of selecting optimized portfolio with the assumption that there was some admissible error for both risk and return. Although this method resolves several raised limitations for the previous solutions it still faces problem in scalable uncertain markets. Authors in [23] considered a new factor called market capitalization in addition to transaction cost and the number of stocks in the portfolio constraints. They used genetic algorithm for solving the proposed model. This method seems could be salient and an efficient method applied in portfolio problem but we would that it has had some limitations for the various markets with hard boundaries and still has a problem in scalable uncertain markets. Besides, authors in [5] exploit threshold policy to control portfolio optimization and applied VaR, ES, mean absolute semi deviation and semi variance for risk measurements. In addition, [24] used quadratic programming for portfolio selection. Besides, the paper [25] applied greedy search, simulated annealing and ant colony optimization for portfolio problem. On the other hand, [26] used NSGA II, PESA and SPEA 2 for solving mean-variance portfolio optimization problem and [27] integrated evolutionary computations and linear programming to suggest a hybrid multi-objective optimization approach. Moreover, authors in [12] used multi-objective evolutionary algorithm presentation on an ordered basis. Reference [28] suggested a hybrid algorithm that integrated critical line algorithm and NSGA II. Furthermore, NSGA II, PESA and SPEA 2 in portfolio optimization problem was compared by [27].