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

TL;DR

This work develops a tractable framework for PDE-constrained optimization under uncertainty with almost-sure state constraints by combining a Smoothed Moreau-Yosida (with softplus smoothing) formulation and tensor-train (TT) surrogates for high-dimensional random fields. A matrix-free Gauss-Newton method leverages a deterministic surrogate Hessian, while TT-Cross and related techniques enable efficient computation of expectations and derivatives, yielding polynomial scaling with dimension. Theoretical results establish strong convergence of the regularized solutions to the true optimum in the linear-quadratic setting under a precise relation $\varepsilon_{\gamma}=o(\gamma^{-1/2})$, and practical algorithms are validated on 1D and 2D elliptic problems, variational inequalities, and a 20-variable SEIR model. The approach provides a scalable, accurate tool for risk-neutral optimization under uncertainty with state constraints, with potential impact on digital twins, epidemiology, and engineering design.

Abstract

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.

Figures (6)

• Figure 1: Left: control signals $u_{\gamma_*}(x)$ for different $\gamma_*$. Right: mean (solid lines) and 95% confidence interval (shaded area, for $\gamma_*=3000$ only) of the state $y_{\gamma_*}(x,\xi)$.
• Figure 2: Left: probability of the constraint violation, $\mathbb{P}[y_{\gamma_*}(x,\xi)>0]$. Right: total final cost $j(u_{\gamma^*})$.
• Figure 3: Relative $L^2$-norm difference from $y$ and $u$ to the reference solutions with $\gamma_*=10^4$ with fixed $n_{\xi}=257$, $n_y=63$ (left), $n_{\xi}=129$ with fixed $\gamma_*=100$, $n_y=63$ (middle) and $n_y=511$ with fixed $\gamma_*=100$, $n_{\xi}=25$ (right).
• Figure 4: Left: control signal $u_{\gamma_*}(x)$. Middle: mean $\mathbb{E}_{\mathbb{P}}[y_{\gamma_*}(x,\xi)]$. Right: standard deviation $\sqrt{\mathbb{E}_{\mathbb{P}}[(y_{\gamma_*}(x,\xi) - \mathbb{E}_{\mathbb{P}}[y_{\gamma_*}(x,\xi)])^2]}$.
• Figure 5: Left: mean optimised state $\mathbb{E}_{\mathbb{P}}[-y]$ with $d=20,$$n_{\xi}=3$ and $\hbox{tol}=10^{-3}$. Middle: variance of the optimized state $\mathbb{E}_{\mathbb{P}}[(y - \mathbb{E}_{\mathbb{P}}[y])^2]$. Right: optimised control $u$.

Theorems & Definitions (21)

• Theorem 1.1: Informal statement of the main theoretical result, Thm. \ref{['thm:conv']}
• Proposition 1.2: Informal statement of the main practical results
• Remark 2.3
• Lemma 3.1
• Proof 1
• Theorem 3.4
• Lemma 3.5
• Proof 2
• Lemma 3.6
• Proof 3