This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.

本文提出了一种解决特定类别的结构优化问题的新颖方法：连续随机梯度法（CSG）。在最简单的情况下，我们假定目标函数是在连续参数集上作为所需属性的积分给出的。求积分近似的正交规则的应用会产生人为的和不希望的局部最小值。但是，CSG方法不依赖于积分的近似值，而是以最佳方式利用来自先前迭代的梯度近似值。尽管CSG方法不需要每次优化迭代解决一个状态问题（无限多个），有可能以数学上严格的方式证明，在优化过程中，函数值以及目标的整个梯度可以任意精度近似。此外，描述了具有无限多个工况的线性弹性问题的数值实验。对于所选示例，事实证明，相比经典的随机梯度法（SG）和随机平均梯度法（SAG），CSG方法明显优越。