We propose an algorithm to solve optimization problems constrained by partial (ordinary) differential equations under uncertainty, with almost sure constraints on the state variable. To alleviate the computational burden of high-dimensional random variables, we approximate all random fields by the tensor-train decomposition. To enable efficient tensor-train approximation of the state constraints, the latter are handled using the Moreau-Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by a softplus function. In a special case of a quadratic cost minimization constrained by linear elliptic partial differential equations, and some additional constraint qualification, we prove strong convergence of the regularized solution to the optimal control. This result also proposes a practical recipe for selecting the smoothing parameter as a function of the penalty parameter. We develop a second order Newton type method with a fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida problem. This algorithm is tested on benchmark elliptic problems with random coefficients, optimization problems constrained by random elliptic variational inequalities, and a real-world epidemiological model with 20 random variables. These examples demonstrate mild (at most polynomial) scaling with respect to the dimension and regularization parameters.

翻译：我们提出一种算法，用于求解在不确定性下受偏（常）微分方程约束的优化问题，其中状态变量需满足几乎必然约束。为了减轻高维随机变量带来的计算负担，我们采用张量列分解来近似所有随机场。为了实现状态约束的高效张量列近似，我们使用Moreau-Yosida罚函数处理这些约束，并通过softplus函数对正部（plus/ReLU）函数进行额外平滑。在线性椭圆型偏微分方程约束的二次成本最小化这一特殊情形下，并附加某些约束规范条件，我们证明了正则化解强收敛于最优控制。该结果还提出了一种实用的方案，用于根据罚参数选择平滑参数。我们开发了一种二阶牛顿型方法，利用近似Hessian矩阵的快速无矩阵作用来求解平滑后的Moreau-Yosida问题。该算法在具有随机系数的基准椭圆问题、受随机椭圆变分不等式约束的优化问题以及一个包含20个随机变量的真实世界流行病学模型上进行了测试。这些示例表明，算法在维度和正则化参数方面具有温和的（至多多项式）复杂度增长。