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The adjoint state method for parametric definable optimization without smoothness or uniqueness

Jérôme Bolte, Edouard Pauwels, Cheik Traoré

Abstract

Definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool which can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.

The adjoint state method for parametric definable optimization without smoothness or uniqueness

Abstract

Definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool which can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.

Paper Structure

This paper contains 14 sections, 9 theorems, 38 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. The adjoint state method: smooth case
  3. Preliminaries
  4. The adjoint method
  5. Comparison to alternative methods
  6. Elements of nonsmooth analysis and o-minimality
  7. Conservative set-valued fields
  8. Definition of o-minimality
  9. The adjoint state method: general conservative case
  10. Preliminaries
  11. Parametric subdifferentiation of the value function
  12. Conservative adjoint formula
  13. Failure of the method without the definability assumption
  14. Technical results

Key Result

Theorem 2.4

Let Assumptions ass:specialcase stand. Then, the adjoint state method alg:asm is a selection of a def:conservative for $f$, i.e., where $D_f$ is a conservative field for $f$.

Figures (1)

  • Figure 1: The fractal construction in pauwels2023conservative. They start with the closed unit square in black. It is divided into 16 squares, each with a side length equal to one fourth of the original square’s side length. Only four of them at specific positions are kept and the others dropped. This process is repeated recursively on each square ad infinitum. The additional red lines represent projection of these sets on rotated axes. Considering $C_{\ell}$, $\ell \in \mathbb{N}$, the set obtained after $\ell$ steps ($C_{0}$ is the original square). We have that $C_{\ell+1} \subset C_{\ell}$ for all $\ell$. Then, let $C = \cap_{\ell \in \mathbb{N}} C_{\ell}$, which is closed. The projection of $C$ on each axes are full segments. Furthermore, in the limit, both projections on rotated axes are Cantor sets of zero measure.

Theorems & Definitions (26)

  • Definition 2.1: Mangasarian-Fromovitz constraint qualification
  • Remark 2.2
  • Theorem 2.4
  • Remark 2.5: On conservative fields
  • Remark 2.6: On the semialgebraic assumption
  • Example 2.7
  • Remark 2.8
  • Definition 3.1: Clarke generalized gradients clarke1983nonsmooth
  • Definition 3.2: Conservative fields bolte2021conservative
  • Definition 3.3: o-minimal structures
  • ...and 16 more