A portfolio optimization model with relaxed second order stochastic dominance (SSD) constraints is presented. The proposed model uses Conditional Value at Risk (CVaR) constraints at probability level $ \beta\in(0,1) $ to relax SSD constraints. The relaxation is justified by theoretical convergence results based on sample average approximation (SAA) method when sample size $ N\to\infty $ and CVaR probability level $ \beta $ tends to 1. SAA method is used to reduce infinite number of inequalities of SSD constraints to finite ones and also to calculate the expectation value. The proposed relaxation on the SSD constraints in portfolio optimization problem is achieved when the probability level $ \beta $ of CVaR takes value less than but close to 1, and the model can then be solved by cutting plane method. The performance and characteristics of the portfolios constructed by solving the proposed model are tested empirically on three sets of market data, and the experimental results are analyzed and discussed. Furthermore, it is shown that with appropriate choices of CVaR probability level $ \beta $, the constructed portfolios are sparse and outperform the portfolios constructed by solving portfolio optimization problems with SSD constraints, with either index portfolios or mean-variance (MV) portfolios as benchmarks.

中文翻译：

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通过风险条件值放宽随机二阶优势约束的投资组合优化

提出了具有松弛二阶随机优势（SSD）约束的投资组合优化模型。所提出的模型使用概率水平为\β\ in（0,1）$的条件风险值（CVaR）约束来放宽SSD约束。当样本量$ N \ to \ infty $和CVaR概率水平$ \ beta $趋于1时，基于样本平均逼近（SAA）方法的理论收敛结果可以证明这种松弛是正确的。SAA方法用于减少无穷大的不等式。 SSD约束为有限约束，还可以计算期望值。当CVaR的概率水平$ \ beta $的值小于但接近1时，就可以实现对投资组合优化问题中SSD约束的拟议放松，然后可以通过切平面法求解该模型。在三组市场数据上对通过求解该模型而构建的投资组合的性能和特征进行了实证测试，并对实验结果进行了分析和讨论。此外，它表明险值的概率水平$ \测试$的适当的选择，所构造的组合是稀疏和优于通过求解与SSD约束组合优化问题，与任一索引组合或平均方差（MV）组合作为构成的组合基准。