## 2 The Model of Second Order Stochastic Dominance
### The Second Order Stochastic Dominance Model  
[PERSON] and [PERSON] [4] propose the following fundamental model of second order stochastic dominance:  
\[\begin{array}{ll}max&f(x)\\ s.t.&g(x,\xi)\succeq_{2}g(y,\xi)\\ &x\in X\end{array} \tag{7}\]  
Here \(f:X\rightarrow\mathbb{R}\) is a concave continuous function. In particular, we may use  
\[f(x)=E[f(x,\xi)] \tag{8}\]  
where \(g:\mathbb{R}^{n}\times\mathbb{R}^{k}\rightarrow\mathbb{R}\) are concave continuous functions both in \(x\) and \(\xi\), \(x\in X\) is a decision vector with \(X\) being a nonempty convex subset of \(\mathbb{R}^{n}\), and \(\xi\) is a random vector defined on probability space \((\Omega,\mathbb{F},\mathbb{P})\) with support \(\Xi\), \(y\in X\) is a predefined vector, \(E(\cdot)\) denotes the expected value w.r.t. the probability distribution of \(\xi\)[9].  
The constraints of model (7) are based on the definition of second order stochastic dominance, [PERSON] [7] and [PERSON] [22] prove that the constraint \(g(x,\xi)\succeq_{2}g(y,\xi)\) is equivalent to the following two inequalities:  
\[E[(\eta-X)_{+}]\leq E[(\eta-Y)_{+}],\forall\eta\in\mathbb{R},\text{where }[t-R_{x}]_{+}=max\{t-R_{x},0\} \tag{9}\]  
\[E(U(R_{x}))\geq E(U(R_{y})),\text{for any increasing and concave utility function U} \tag{10}\]  
Combined with the constraint (9), the model (7) can be formulated as a stochastic semi-infinite programming problem [3; 9] :  
\[\begin{array}{ll}min&-E[f(x,\xi)]\\ s.t.&G(x,\eta):=E[(\eta-g(x,\xi))_{+}]-E[(\eta-g(y,\xi))_{+}]\leq 0,\forall\eta \in\mathbb{R}\\ &x\in X\end{array} \tag{11}\]  
For the random variable \(\eta\in\mathbb{R}\), it is clear that there is no \(x\in X\) seriously satisfies the constraint condition when \(\eta\) takes a sufficiently small value. So the inequality does not hold and the model (11) does not satisfy the Slater condition. In the actual research, in order to overcome the serious technical difficulties and seek the optimal solution of algorithm, we get the following relaxed model:  
\[\begin{array}{ll}min&-E[f(x,\xi)]\\ s.t.&G(x,\eta)\leq 0,\forall\eta\in[a,b]\\ &x\in X\end{array} \tag{12}\]  
where \([a,b]\) is a closed interval in \(\mathbb{R}\). [PERSON] et al. [3] propose that model (12) is equivalent to model (11) for some appropriate interval \([a,b]\) if \(\xi\) has uniformly bounded distribution. As a relation of model (11), model (12) has a larger set of feasible solutions and subsequently its optimal value gives a lower bound for model. Besides, the relaxed model is more likely to satisfy the Slater condition which is closely related to numerical stability [9].  
Although the constraints are relaxed, the model (12) is a stochastic semi-infinite and non-smooth programming problem in nature. So the difficulty of solving is still large. [PERSON] and [PERSON] [PERSON] and [PERSON] (2013) change the model into a liner programming problem which solves the problem of non-smooth of SSD constraints. Besides, this method also applies to non-linear cases. However, facing the multi-dimensional problems, due to the increase in variables, the efficiency of the method will be greatly affected.  
In this paper, we mainly discuss the discrete condition, assuming the random variable \(\xi\) satisfies the discrete distribution, that is \(P(\xi=\xi^{i})=p_{i},i=1,\ldots,m\). [PERSON] (1993) proposes the following model:  
\[\begin{array}{ll}min&\{-\sum_{i=1}^{m}p_{i}f(x,\xi^{i})\}\\ s.t.&\sum_{i=1}^{m}p_{i}(\eta_{j}-g(x,\xi^{i}))_{+}\leq\sum_{i=1}^{m}p_{i}( \eta_{j}-g(y,\xi^{i}))_{+},\forall j=1,\ldots,m\\ &x\in X\end{array} \tag{13}\]  
where \(\eta_{j}=g(y,\xi^{j}),j=1,\ldots,m\). In particular, \(p_{i}=\frac{1}{m}\). In this case, model (13) is the sample average approximation of model (12). However, the model (13) is still non-smooth and does not satisfy the Slater condition.