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Parabolic optimal control problems with combinatorial switching constraints -- Part III: Branch-and-bound algorithm

Christoph Buchheim, Alexandra Grütering, Christian Meyer

Abstract

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance $\varepsilon>0$, we show how to compute in finite time an $\varepsilon$-optimal solution in function space, independently of any prior discretization. The main ingredients in our approach are an appropriate branching strategy in infinite dimension, an a posteriori error estimation in order to obtain safe dual bounds, and an adaptive refinement strategy in order to allow arbitrary switching points in the limit. The performance of our approach is demonstrated by extensive experimental results.

Parabolic optimal control problems with combinatorial switching constraints -- Part III: Branch-and-bound algorithm

Abstract

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance , we show how to compute in finite time an -optimal solution in function space, independently of any prior discretization. The main ingredients in our approach are an appropriate branching strategy in infinite dimension, an a posteriori error estimation in order to obtain safe dual bounds, and an adaptive refinement strategy in order to allow arbitrary switching points in the limit. The performance of our approach is demonstrated by extensive experimental results.
Paper Structure (25 sections, 16 theorems, 114 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 25 sections, 16 theorems, 114 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solution mapping
  4. Functions of bounded variation
  5. Examples of switching constraints
  6. Branch-and-bound algorithm
  7. Convex hull under fixings
  8. Restricted total variation
  9. Switching point constraints
  10. Computation of primal and dual bounds
  11. Dual bounds
  12. Primal bounds
  13. Discretization error and adaptive refinement
  14. Finite element discretization
  15. A posteriori discretization error of dual bounds
  16. ...and 10 more sections

Key Result

Theorem 3.1

For $N\in\mathbb{N}$, let $0\le\tau_1^{N}<\dots<\tau_N^N< T$ and $c_1^{N},\dots,c_N^N\in\{0,1\}$. Define where $\tau_0^N:=0$ and $\tau_{N+1}^N:=T$. If $\Delta \tau^{N} \to 0$ for $N\to\infty$, then

Figures (2)

  • Figure 1: Illustration of the second part of the proof of Lemma \ref{['lem:Cmax']}.
  • Figure 2: Complete branch-and-bound tree of an instance generated with $\theta=3$ jump points and with $\sigma=1$ allowed switchings. The path of the optimal solution is marked in bold and the branching decisions along the optimal path are listed. In the case of a single child node, the temporal discretization of the subproblem has been refined.

Theorems & Definitions (35)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 25 more