## 2 Background and Literature Review
### Quantitative Portfolio Management
#### 2.2.2 Portfolio Optimization  
In general, portfolio optimization techniques are proposed to optimal asset allocation in order to maximize a portfolio return and minimize its risk. Modern portfolio theory was a theory on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of risk, emphasizing that risk is always an inherent part of higher reward [9]. [PERSON] further introduced the industry to the capital asset pricing model (CAPM), which in the simplest forms, was a technique to combine the market portfolio with a risk-free asset to further improve the set of risk-return above the efficient frontier [10]. Modern portfolio theory and capital market theory provide a framework to specify and measure investment risk and to develop relationships between expected return and risk. These relationships are called asset pricing models. The arbitrage pricing theory (APT) as an alternative to the CAPM was developed in the work of [11]. Unlike in the CAPM, markets were assumed as perfectly efficient; APT was a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset's expected return and a number of macroeconomic variables that capture systematic risk. The Fama French three-factor model was an asset pricing model that expands on the CAPM by adding size risk and value risk factors to the market risk factors. This model considers the fact that value and small-cap stocks outperform markets on a regular basis. By including these two additional factors, the model adjusts for this outperforming tendency, which is thought to make it a better tool for evaluating manager performance [12]. The Black-Litterman model was essentially a combination of two main portfolio theories: the CAPM and the modern portfolio theory [13]. The main benefit of the Black-Litterman model is that it allows the portfolio manager to use it as a tool for producing a set of expected returns within the mean-variance optimization framework. In addition to developing portfolio theories as the principle of portfolio management, multiple optimization techniques have been proposed to extend the impact of modern portfolio theory. A 60-year review of different approaches developed to address the challenges encountered when using portfolio optimization in practice, such as the transaction costs, portfolio constraints, and estimates errors was provided in [14]. Mathematical optimization has also attracted widespread interest in multi-objective optimization. There exist a whole series of optimization algorithms such as convex programming, integer programming, linear programming, and stochastic programming developed to solve optimization problems not only for linear constraints but also for random constraints [15; 16]. Metaheuristic is a subfield of computational intelligence that represents an efficient way to deal with complex optimization problems and is applicable to both continuous and combinatorial optimization problems. Evolutionary algorithms such as genetic algorithms have shown an effective impact on complex objectives and constraint optimization tasks [17]. Much research in recent years has focused on uncertainty in financial investment. Probabilistic programming techniques have also been applied to handle the uncertainty of the financial markets to support portfolio selection. The fuzzy set theory has been widely used to solve many practical problems, including financial risk management. Using fuzzy approaches, quantitative analysis, qualitative analysis, experts' knowledge, and investors' subjective strategies can be better integrated into a portfolio selection model [18]. One significant difference between the discussed approaches and this work is that the input values such as expected returns and risk for optimization models, which either are calculated by a mathematical or statistical model, are based on historical data.