Algorithms For Optimization Problems With Second-Order Stochastic Dominance Constraints

In this thesis,we study algorithms for solving problems with second-order stochastic dominance constraints.Stochastic dominance is often used to compare the distribution of two random variables and is an important tool in decision theory and economics for theoretical analysis.Second-order stochastic dominance was first introduced as a constraint to optimization problems by Dentcheva and Ruszczy (?)ski in 2003,and as a risk-averse decision model,it has been widely used in the fields of economics,finance,insurance and investment.However,the nonsmoothness and semi-infinity nature of the constraint bring di culties in solving the problem theoretically and computationally.In this paper,we further explore the problem from the perspective of numerical optimization algorithms and propose to apply the alternating direction multiplier method(ADMM)and the adaptive primal-dual stochastic gradient method to optimization problems with second-order stochastic dominance constraints.For the case of univariate linear stochastic dominance constrains,we assume that the variable has finite realizations.By introducing auxiliary variables,we equivalently reformulate the dominance constraints as an equality constraint applicable to ADMM and an inequality constraint that is easy to project,and then ADMM is applied to solve the problem.For the case of multivariate stochastic dominance constrains,we deal with the problem under discrete distribution and nonlinear functions.We apply the adaptive primal-dual stochastic gradient method to the problem,and two methods for generating unbiased stochastic subgradient of Lagrange function are presented.Numerical experiments demonstrate the effectiveness of the above methods.