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Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities

Juan Carlos De Los Reyes

TL;DR

This paper investigates the differentiability properties of the solution map for variational inequalities of the second kind involving discrete total variation ($TV$). It establishes Bouligand differentiability of the solution operator via a direct quotient analysis applied to a primal-dual reformulation and characterizes the Bouligand subdifferential by exploiting the directional derivative structure and a tailored subspace. It then derives optimality conditions for Bouligand-stationary and strongly-stationary points in discrete VI-constrained optimal control problems, and proposes a trust-region algorithm based on these characterizations, accompanied by a numerical example illustrating the solution behavior and algorithm performance. The work provides a rigorous framework for discrete $TV$-regularized VI control problems, enabling precise sensitivity analysis and robust computational methods with practical implications for imaging and related applications where total variation regularization is essential.

Abstract

We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping. We then derive optimality conditions characterizing Bouligand-stationary and strongly-stationary points for discrete VI-constrained optimal control problems. A trust-region algorithm for solving these control problems is proposed based on the obtained characterizations, and a numerical experiment is presented to illustrate the main properties of both the solution and the proposed algorithm.

Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities

TL;DR

This paper investigates the differentiability properties of the solution map for variational inequalities of the second kind involving discrete total variation (). It establishes Bouligand differentiability of the solution operator via a direct quotient analysis applied to a primal-dual reformulation and characterizes the Bouligand subdifferential by exploiting the directional derivative structure and a tailored subspace. It then derives optimality conditions for Bouligand-stationary and strongly-stationary points in discrete VI-constrained optimal control problems, and proposes a trust-region algorithm based on these characterizations, accompanied by a numerical example illustrating the solution behavior and algorithm performance. The work provides a rigorous framework for discrete -regularized VI control problems, enabling precise sensitivity analysis and robust computational methods with practical implications for imaging and related applications where total variation regularization is essential.

Abstract

We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping. We then derive optimality conditions characterizing Bouligand-stationary and strongly-stationary points for discrete VI-constrained optimal control problems. A trust-region algorithm for solving these control problems is proposed based on the obtained characterizations, and a numerical experiment is presented to illustrate the main properties of both the solution and the proposed algorithm.

Paper Structure

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem