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A globalized inexact semismooth Newton method for strongly convex optimal control problems

Daniel Wachsmuth

Abstract

We investigate a globalized inexact semismooth Newton method applied to strongly convex optimization problems in Hilbert spaces. Here, the semismooth Newton method is appplied to the dual problem, which has a continuously differentiable objective. We prove global strong convergence of iterates as well as transition to local superlinear convergence. The latter needs a second-order Taylor expansion involving semismooth derivative concepts. The convergence of the globalized method is demonstrated in numerical examples, for which the local unglobalized method diverges.

A globalized inexact semismooth Newton method for strongly convex optimal control problems

Abstract

We investigate a globalized inexact semismooth Newton method applied to strongly convex optimization problems in Hilbert spaces. Here, the semismooth Newton method is appplied to the dual problem, which has a continuously differentiable objective. We prove global strong convergence of iterates as well as transition to local superlinear convergence. The latter needs a second-order Taylor expansion involving semismooth derivative concepts. The convergence of the globalized method is demonstrated in numerical examples, for which the local unglobalized method diverges.

Paper Structure

This paper contains 20 sections, 27 theorems, 109 equations, 2 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on the primal and dual problems
  3. Conjugate functions and proximal operators
  4. Optimality conditions for the dual problem
  5. Second-order semismoothness
  6. Conjugate Gradients
  7. Semismooth Newton
  8. Semismoothness of Nemyzki operators
  9. Piecewise smooth proximals
  10. Examples
  11. Box constraints.
  12. $L^1$-norm.
  13. Box constraints plus $L^1$-norm.
  14. $L^2$-ball constraint.
  15. Directional sparsity.
  16. ...and 5 more sections

Key Result

Lemma 2.1

The problem eq001 is uniquely solvable. An element $\bar{u} \in U$ is a solution of eq001 if and only if

Figures (2)

  • Figure 1: Example 1: optimal control (left), $\partial\operatorname{prox}_{g/\alpha}$ (right)
  • Figure 2: Example 2: optimal controls for $\alpha =10^{-8}$ (left), $\alpha =10^{-10}$ (right)

Theorems & Definitions (56)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3: BauschkeCombettes2017
  • Proposition 2.4: BauschkeCombettes2017
  • Proposition 2.5: BauschkeCombettes2017
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 46 more