Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control

TL;DR

This work analyzes the sample-average approximation (SAA) of risk-neutral composite optimal control problems with potentially nonsmooth regularizers, formulated as $\min_{u \in U} G(u) = F(Bu) + \psi(u)$ where $F(w)=\mathbb{E}[f(w,\boldsymbol{\xi})]$ and $B$ is a compact linear operator. It establishes both asymptotic consistency (for optimal values, solutions, and critical points) and nonasymptotic sample-size bounds on the gap functional, with rates reflecting Monte Carlo sampling and problem structure; convex cases yield stronger nonasymptotic results. The framework is then specialized to PDE-constrained controls, including affine-linear and bilinear elliptic equations, enabling bang-bang-type regularization via an $L^1$ term and demonstrating gradient representations and Lipschitz properties of the composite objective. Numerical experiments for affine-linear and bilinear bang-bang problems under uncertainty validate the theory and illustrate practical convergence behavior, including cases where observed rates outperform the conservative bounds. The results advance understanding of SAA in infinite-dimensional, nonsmooth stochastic optimization and suggest avenues for extension to risk-averse settings and variational-inequality formulations with practical PDE applications.

Abstract

Nonsmooth composite optimization problems under uncertainty are prevalent in various scientific and engineering applications. We consider risk-neutral composite optimal control problems, where the objective function is the sum of a potentially nonconvex expectation function and a nonsmooth convex function. To approximate the risk-neutral optimization problems, we use a Monte Carlo sample-based approach, study its asymptotic consistency, and derive nonasymptotic sample size estimates. Our analyses leverage problem structure commonly encountered in PDE-constrained optimization problems, including compact embeddings and growth conditions. We apply our findings to bang-bang-type optimal control problems and propose the use of a conditional gradient method to solve them effectively. We present numerical illustrations.

Figures (4)

• Figure 1: For the affine-linear control problem, nominal solution (left), and reference SAA solution $u^*$ with $N = N_{\text{ref}}$(right).
• Figure 2: For the affine-linear control problem, empirical estimate of $\E{G_{\text{ref}}(u_N^*)-G_{\text{ref}}(u^*)}$ as a function of the sample size $N$(left) and empirical estimate of $\E{\|u_{N}^*-u^*\|_{L^1(D)}}$ as a function of the sample size $N$(right).
• Figure 3: Empirical estimate of $\E{\Psi_{\text{ref}}(u_{N}^*)}$ as a function of the sample size $N$ for the affine-linear control problem (left) and bilinear control problem (right).
• Figure 4: For the bilinear control problem, nominal critical point (left) and reference SAA critical point $u^*$ with $N = N_{\text{ref}}$(right).

Theorems & Definitions (37)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof