求解具有等式约束的随机优化问题的二阶微分方程方法

Second Order Differential Equation Algorithm for Solving Stochastic Optimization Problems with Equality Constraints

针对具有等式约束的随机优化问题,本文首先考虑基于SAA方法获取原始问题目标函数的SAA问题,并利用惩罚函数法将其转化为一般无约束凸优化问题。接下来建立二阶微分方程算法对所得到的问题进行求解。本文还证明了当算法的参数满足一定条件时,二阶微分方程算法具有良好的收敛性,其收敛率为,最后给出具体的数值算例验证算法的可行性。

For the stochastic optimization problem with equality constraint, this paper first considers the SAA problem of obtaining the objective function of the original problem based on the SAA method, and uses the penalty function method to transform it into a general unconstrained convex optimization problem. Next, a second-order differential equation algorithm is established to solve the obtained problem. This paper also proves that when the parameters of the algorithm meet certain conditions, the second-order differential equation algorithm has good convergence, and its convergence rate is , and finally a specific numerical example is given to verify the feasibility of the algorithm.