Smoothed Moreau-Yosida Tensor Train Approximation of State-constrained Optimization Problems under Uncertainty

We propose an algorithm to solve optimization problems constrained by ordinary or partial differential equations under uncertainty, with additional almost sure inequality constraints on the state variable. To alleviate the computational burden of high-dimensional random variables, we approximate all random fields by the tensor-train (TT) decomposition. To enable efficient TT 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. We propose a practical recipe for selecting the smoothing parameter as a function of the penalty parameter, and 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.