Many Portfolio optimization problems deal with allocation of assets which carry a relatively high market price. Therefore, it is necessary to determine the integer value of assets when we deal with Portfolio optimization. In addition, one of the main concerns with most Portfolio optimization is associated with the type of constraints considered in different models. In many cases, the resulted problem formulations do not yield in practical solutions. Therefore, it is necessary to apply some managerial decisions in order to make the results more practical. This paper presents a Portfolio optimization based on an improved knapsack problem with the cardinality, floor and ceiling, budget, class, class limit and pre-assignment constraints for asset allocation. To handle the uncertainty associated with different parameters of the proposed model, we use robust optimization techniques. The model is also applied using some realistic data from US stock market. Genetic algorithm is also provided to solve the problem for some instances.

One of the key issues for investors is the issue of creating an optimal stock Portfolio. In the issue of choosing an Portfolio, the decision maker faces different and sometimes conflicting goals such as rate of return, liquidity, dividend, and risk. In Portfolio optimization, the main issue is the optimal choice of assets and securities that can be made with a certain amount of capital, but on the one hand, the uncertainties associated with each share, and, on the other hand, the multiplicity of the optimal Portfolio selection model, on the complexity of the problem increases. In this paper, the Portfolio optimization under uncertainty has been studied. A randomized approach to converting uncertainty into a state of definiteness and agreeing to plan for a single objective is used in combination. Information about 20 pharmaceutical companies from the Tehran Stock Exchange has been used and the validity of the model has been investigated. The results show that the stock Portfolio offered has a high performance.

The Markowitz-based Portfolio selection turns to an NP-hard problem when considering cardinality constraints. In this case, existing exact solutions like quadratic programming may not be efficient to solve the problem. Many researchers, therefore, used heuristic and metaheuristic approaches in order to deal with the problem. This work presents Asexual Reproduction Optimization (ARO), a model-free metaheuristic algorithm inspired by the asexual reproduction, in order to solve the Portfolio optimization problem including cardinality constraint to ensure the investment in a given number of different assets and bounding constraint to limit the proportions of fund invested in each asset. This is the first time that this relatively new metaheuristic is applied in the field of Portfolio optimization, and we show that ARO results in better quality solutions in comparison with some of the well-known metaheuristics stated in the literature. To validate our proposed algorithm, we measured the deviation of the obtained results from the standard efficient frontier. We report our computational results on a set of publicly available benchmark test problems relating to five main market indices containing 31, 85, 89, 98, and 225 assets. These results are used in order to test the efficiency of our proposed method in comparison to other existing metaheuristic solutions. The experimental results indicate that ARO outperforms Genetic Algorithm (GA), Tabu Search (TS), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) in most of test problems. In terms of the obtained error, by using ARO, the average error of the aforementioned test problems is reduced by approximately 20 percent of the minimum average error calculated for the above-mentioned algorithms.

The optimal Portfolio selection problem to find an optimal way to allocate a fixed amount of capital to a set of available asset swhich aims to maximize expected returns and minimize risk at the same time, to take place. In this Study is shown that an investor with n risky share, how to reach certain profits with minimal risk. Such a Portfolio, efficient Portfolio is called. For this purpose, the study of evolutionary algorithms, Genetic Algorithm, Imperialist Competitive Algorithm and Particle Swarm Optimization algorithm, also with regard to the basic constraints on the investment, we use these practical methods to solve the Portfolio optimization problem. Practical results for the Portfolio optimization problem in the Tehran Stock Exchange, of the30 company' sactivein the industry with the selection of20companies along with their validation, is obtained. Aims to help investors better and more practical to select different stocks and thus is an effective investment.

The optimal Portfolio selection problem to find an optimal way to allocate a fixed amount of capital to a set of available assets which aims to maximize expected returns and minimize risk at the same time, to take place. In this Study is shown that an investor with n risky share, how to reach certain profits with minimal risk. Such a Portfolio, efficient Portfolio is called. For this purpose, the study of evolutionary algorithms, Genetic Algorithm, Imperialist Competitive Algorithm and Particle Swarm Optimization algorithm, also with regard to the basic constraints on the investment, we use these practical methods to solve the Portfolio optimization problem. Practical results for the Portfolio optimization problem in the Tehran Stock Exchange, of the 30 company's active in the industry with the selection of 20 companies along with their validation, is obtained. Aims to help investors better and more practical to select different stocks and thus is an effective investment.

Portfolio selection always has been one of the interesting subjects in financial problems and markets. In this paper, a novel logical and useful model for optimizing Portfolio is proposed that this model have some constraints such as flexibility in stock's weight that don't determinate a fixed and solid bounds for stock's weights in Portfolio optimization problem and also, cardinality constraint is applied for Portfolio problem. Then fuzzy programming applied to handle uncertainty of stock returns and with flexible and possibilistic programming, that both of these methods are categories of fuzzy programming, proposed model converts to a crisp one. The proposed model for evaluating, performance testing and logicality approved, is applied to some monthly return of companies stock of Tehran Stock Exchange and the results is reported. The results showed that in lower values of confidence level in proposed Portfolio problem, it's possible to obtain a higher profit with low risk.

The purpose of this study is to determine the efficiency limit and evaluate the performance of the stock Portfolio in issues with numerical limitations in the Tehran Stock Exchange and IRAN Farabors in year 1398, using data envelopment analysis method. The efficient frontier in the issue of stock Portfolio with numerical constraints is discrete and asymmetric. It is not possible to use the data envelopment analysis model. In this study, the efficient boundary is categorized into several continuous boundaries and the "data envelopment analysis" model is used. For this purpose, first the beginning and end points of each continuous boundary are determined as "data segment points" based on the proposed algorithm, and then the data envelopment analysis model is used for each continuous interval. 188 stock exchange and OTC symbols listed in Tehran Stock Exchange and IRAN Farabors were examined in 1398, and with the limitation of selecting 5 to 10 stocks from 188 stocks based on different weights (total stock weight equal to 1), the efficient limit of the stock Portfolio was determined. Due to the lack of fast and accurate computational methods in determining the efficiency limit of stock Portfolio with numerical constraints, data envelopment analysis method to determine the efficiency limit and evaluate stock Portfolio performance of particle swarm algorithm, genetic algorithm and imperialist competition algorithm And the efficient border can be drawn and used.

Cardinality Constrained Portfolio optimization problems are widely used Portfolio optimization models which incorporate restriction on the number of assets in the Portfolio. Being mixed-integer programming problems make them NP-hard thus computationally challenging, specially for large number of assets. In this paper, we consider cardinality Constrained mean-variance (CCMV) and cardinality Constrained mean-CVaR (CCMCVaR) models and propose a hybrid algorithm to solve them. At first, it solves the relaxed model by replacing L_0-norm, which bounds the number of assets, by L_1-norm. Then it removes those assets that do not significantly contribute on the Portfolio and apply the original CCMV or CCMCVaR model to the remaining subset of assets. To deal with the large number of scenarios in the CCMCVaR model, conditional scenario reduction technique is applied. Computational experiments on 3 large data sets show that the proposed approach is competitive with the original models from risk, return and Sharpe ratio perspective while being significantly faster.