## 2 The MCVSK and MVSK Portfolio Optimization Model
### The Second-Order Stochastic Dominance Constraint  
The theory of stochastic dominance originates from the theory of discrete stochastic variable optimization, and later develops into the theory of generalized stochastic variable optimization, which is widely used in economic and financial fields nowadays. In the stochastic dominance theory, the comparison between random variables is under their k-order distribution function \(F^{\left(k\right)}\). Assuming that there are portfolios \(X\) and \(Y\), and the utility function of all investors is monotonically increasing. If all the investors prefer portfolio \(X\) to portfolio \(Y\), or believe that there are only part of them is no difference, then we say portfolio \(X\) stochastically dominates portfolio \(Y\) in the first order [31].  
let \(x\) and \(y\) be the decision vectors and \(\xi\) be the random variable, it's said that \(g\left(x,\xi\right)\) stochastically dominates \(g\left(y,\xi\right)\) in the first order, denoted by \(g\left(x,\xi\right)\succeq_{1}g\left(y,\xi\right)\), if:  
\[F\left(g\left(x,\xi\right);\eta\right)\leq F\left(g\left(y,\xi\right);\eta \right),\forall\eta\in\mathbb{R} \tag{5}\]  
where \(g\left(x,\xi\right)\) is the concave continuous function both in \(x\) and \(\xi\), \(F\left(g\left(x,\xi\right);\eta\right)\) is the cumulative distribution function of \(g\left(x,\xi\right)\). For a random variable \(X\), the first order distribution function of \(X\) is its right-continuous cumulative distribution function:  
\[F_{x}^{\left(1\right)}\left(\eta\right)=\int_{-\infty}^{\eta}P_{x}\left(d \xi\right)=P\left\{x\leq\eta\right\},\forall\eta\in\mathbb{R} \tag{6}\]  
In portfolio optimization problem, SSD is of interest because, for any decision maker with risk-averse and non-satiable preferences, a portfolio which dominates a benchmark portfolio is preferred to the benchmark [32]. Based on the definition of first-order stochastic dominance, we denote \(g\left(x,\xi\right)\) stochastically dominates \(g\left(y,\xi\right)\) in the second order by \(g\left(x,\xi\right)\succeq_{2}g\left(y,\xi\right)\), if:  
\[\int_{-\infty}^{\eta}F\left(g\left(x,\xi\right);\alpha\right)d\alpha\leq\int_ {-\infty}^{\eta}F\left(g\left(y,\xi\right);\alpha\right)d\alpha,\forall\eta \in\mathbb{R} \tag{7}\]  
Thereinto, for two different profolios \(X\) and \(Y\), the strict dominance relation \(\succ_{\left(k\right)}\) is defined as follows:  
\[X\succ_{\left(k\right)}Y\Leftrightarrow X\succeq_{\left(k\right)}Y\text{ and }Y\
subseteq_{\left(k\right)}X \tag{8}\]  
[PERSON] et al. [33] and [PERSON] et al. [34] prove that SSD is equivalent to the following two inequalities:  
\[E\left[\left(\eta-R_{x}\right)_{+}\right]\leq E\left[\left(\eta-R_{y}\right)_ {+}\right],\forall\eta\in\mathbb{R} \tag{9}\]  
where \(E\left(\cdot\right)\) denotes the expected value with respect to the probability distribution of \(\xi\). Moreover, for any increasing and concave utility function \(u\), the following inequality exists:  
\[E\left[u\left(R_{x}\right)\right]\geq E\left[u\left(R_{y}\right)\right] \tag{10}\]  
where \(R_{x}\) and \(R_{y}\) are two random variables, and they may represent the returns of two portfolios \(x\) and \(y\)[33]. Besides, let \(X\) and \(Y\) denote two different profolios, [PERSON] and [PERSON] [35] point out that the SSD constraint is equivalent to the continuum of CVaR constraints for all confidence levels \(\alpha\in\left(0,1\right]\):  
\[X\succeq_{\left(2\right)}Y\Leftrightarrow CVaR_{\alpha}\left(R_{x}\right) \geq CVaR_{\alpha}\left(R_{y}\right),\forall\alpha\in\left(0,1\right] \tag{11}\]