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Dual Regularization and Outer Approximation of Optimal Control Problems in BV

Christian Meyer, Annika Schiemann

TL;DR

This work tackles PDE-constrained optimization with a total-variation constraint on BV controls by introducing a dual-regularized TV $TV_\varepsilon$ and solving the resulting regularized problems via an outer-approximation algorithm in function space. It establishes existence of solutions, proves that $TV_\varepsilon(u) \to TV(u)$ as $\varepsilon \to 0$, and proves convergence of the regularized minimizers $u_\varepsilon$ to the true minimizer $u^*$, including a rate $0 \le J(u^*)-J(u_\varepsilon) \le C\varepsilon$ and $\|u^*-u_\varepsilon\|_{L^2} \le C\sqrt{\varepsilon}$ under additional smoothness assumptions. An outer-approximation framework is developed to handle the infinite family of TV constraints, with provable convergence to the regularized minimizer $u_\varepsilon$ and feasibility preservation. The paper also provides an exact BV-solution construction for testing, and demonstrates numerical viability on two 2D examples, using a finite-element discretization and a semismooth Newton solver, including a path-following strategy in $\varepsilon$ for robustness. Overall, the approach delivers convergence-guaranteed, outside-in approximations of BV-optimal controls for elliptic PDEs and offers practical algorithms for BV-constrained optimal control in function space.

Abstract

This paper is concerned with an elliptic optimal control problem with total variation (TV) restriction on the control in the constraints. We introduce a regularized optimal control problem by applying a quadratic regularization of the dual representation of the TV-seminorm. The regularized optimal control problem can be solved by means of an outer approximation algorithm. Convergence of the regularization for vanishing regularization parameter as well as convergence of the outer approximation algorithm is proven. Moreover, we derive necessary and sufficient optimality conditions for the original unregularized optimal control problem and use these to construct an exact solution that we use in our numerical experiments to confirm our theoretical results.

Dual Regularization and Outer Approximation of Optimal Control Problems in BV

TL;DR

This work tackles PDE-constrained optimization with a total-variation constraint on BV controls by introducing a dual-regularized TV and solving the resulting regularized problems via an outer-approximation algorithm in function space. It establishes existence of solutions, proves that as , and proves convergence of the regularized minimizers to the true minimizer , including a rate and under additional smoothness assumptions. An outer-approximation framework is developed to handle the infinite family of TV constraints, with provable convergence to the regularized minimizer and feasibility preservation. The paper also provides an exact BV-solution construction for testing, and demonstrates numerical viability on two 2D examples, using a finite-element discretization and a semismooth Newton solver, including a path-following strategy in for robustness. Overall, the approach delivers convergence-guaranteed, outside-in approximations of BV-optimal controls for elliptic PDEs and offers practical algorithms for BV-constrained optimal control in function space.

Abstract

This paper is concerned with an elliptic optimal control problem with total variation (TV) restriction on the control in the constraints. We introduce a regularized optimal control problem by applying a quadratic regularization of the dual representation of the TV-seminorm. The regularized optimal control problem can be solved by means of an outer approximation algorithm. Convergence of the regularization for vanishing regularization parameter as well as convergence of the outer approximation algorithm is proven. Moreover, we derive necessary and sufficient optimality conditions for the original unregularized optimal control problem and use these to construct an exact solution that we use in our numerical experiments to confirm our theoretical results.
Paper Structure (12 sections, 17 theorems, 112 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 17 theorems, 112 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dual regularization of the total variation
  3. Existence of optimal controls
  4. Convergence of the dual regularization
  5. Convergence rate under additional assumptions
  6. Outer approximation
  7. Optimality conditions for \ref{['eq:P']}
  8. Construction of an exact optimal solution
  9. Numerical examples
  10. Numerical realization
  11. Numerical example with known exact optimal solution
  12. Numerical example without known exact optimal solution

Key Result

Lemma 2.3

Let Assumption assu:density be fulfilled. Then, for every $u\in L^2(\Omega)$, it holds that $\mathrm{TV}_\varepsilon(u) < \infty$ and where the maximum is attained by a unique maximizer $\varphi_\varepsilon \in \mathcal{H}$.

Figures (2)

  • Figure 1: Plots of the exact solution $\bar{u}$ to \ref{['eq:P']} corresponding to the instance from \ref{['sec:numeric_exact_solution']} and the associated solution $u_k$ to \ref{['eq:Peps']} with $\varepsilon = 7.8e-08$ obtained by applying \ref{['alg:Reps']}.
  • Figure 2: Plots of the desired control $u_d$ from \ref{['eq:P']} corresponding to the instance from \ref{['sec:numeric_without_exact_solution']} and the associated solution $u_k$ to \ref{['eq:Peps']} with $\varepsilon = 1.6e-07$ obtained by applying \ref{['alg:Reps']}.

Theorems & Definitions (41)

  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 31 more