# 2-Phase NSGA II: An Optimized Reward and Risk Measurements Algorithm in Portfolio Optimization
## 1 Introduction  
Portfolio optimization is a bi-objective optimization problem aimed at maximizing the reward and at the same time, minimizing the risk. The reward and the risk of portfolio are estimated by the mean of return and the variance of return, respectively [1; 2]. It is unlikely to come across an optimal solution in a multi-objective optimization problem since the employed objective functions often conflict with each other and it is impossible to optimize all objective functions at the same time. Instead, a set of best solutions which came to be known as efficient frontier is obtained and authorized many alternatives to decision maker who choose the solution that is most suited to a particular application. Apart from the constraint of ensuring that all money has been invested (budget constraint), there are some other constraints in the real-world [3]. One of these constraints is cardinality constraint that limits the number of assets in portfolio [4]. This constraint leads to introducing integer variables. Therefore, the results of a mixed integer optimization problem are multiple local exterma and discontinuities [5; 6; 7]. Indeed, authors in [7] present GAMCC as a genetic-aware credit crunch constraint applied in an intelligent model in bank lending decisions. It maximizes the bank profit and minimizes the probability of bank default in a search for a dynamic lending decision. The big limitation of the method compared our model is time and global optimization which is covered in the proposed method and has been confirmed in the simulation. Because of a complex problem as a result of additional constraints, classical optimization methods do not work and so, heuristic optimization techniques need to be applied in order to find optimal or near optimal solutions. Authors in [8] exemplifies one of the first uses of heuristics toportfolio selection. They applied semi-variance as a risk measure and utilized threshold accepting in a portfolio selection problem. The big gap they did not address the risk constrained raised in portfolio selection problem. So their threshold needs to be tuned to be applied in scale case which is covered in our solution. Also, the reference [3] introduced cardinality constraint and quantity constraint for portfolio selection problem and solved the new model by three heuristic algorithms, i.e., genetic algorithms, tabu search and simulated annealing. Authors in [9; 10] studied the capability of heuristic techniques, namely NSGA II, PESA and SPEA, for overcoming complex portfolio optimization problem. They replaced variance as an index of risk with value at risk and expected shortfall. The aforementioned solution did not address the mutual portfolio risk effect in the portfolio selection problem that is accounted in the proposed method. A new objective function related to the number of securities in the portfolio was added to two former objectives of risk and return of the portfolio by [11]. Moreover, quantity and class limitations are imposed to the model and they solved it by three evolutionary multi-objective algorithms, namely NSGA II, PESA and SPEA2. On the other hand, authors in [12] suggested an order-based representation for the integration of realistic constraints with the portfolio optimization problem model [13] and included preference criterion in the optimization search process and addressed evolutionary multi-objective portfolio optimization for both of them. In [14] authors presented two intelligent trading systems in which exploits fuzzy logic techniques to enhance the power of genetic procedures and attempts to improve the performances of fuzzy system through Neural Networks. Although the method is interesting, compared with the proposed solution, it suffers from some realistic constraints. Later on, the same authors in [15] design a joint fuzzy-GSA system to tune suitable parameters of GSA, by impacting on the exploration, the exploitation capabilities and get ride of local optima using the fuzzy-manner choice of adjusting, that use fuzzy input and output altogether.