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Relative Well-Posedness of Truncated Constrained Systems Accompanied by Variational Calculus

Boris S. Mordukhovich, Pengcheng Wu, Xiaoqi Yang

TL;DR

The paper addresses sensitivity and stability for truncated constrained systems modeled by set-valued mappings between Banach spaces, introducing relative well-posedness notions such as Lipschitz-like stability, metric regularity, and linear openness relative to constraint sets $\Omega$ and $\Theta$. It develops a novel framework of relative contingent coderivatives $\widehat{D}_\varepsilon^*S_\Omega^\Theta$ and its normal/mixed/mirror variants, together with the relative PSNC condition, to characterize these properties in both infinite and finite dimensions. Central results prove complete coderivative-based characterizations and establish pointwise calculus rules (chain, sum) for these relative notions, enabling verification and computation in structured truncated problems. The framework yields equivalences among stability properties, exact Lipschitz bounds, and PSNC criteria, offering a robust toolset for stability analysis in constrained optimization, variational inequalities, and related control and economic models.

Abstract

The paper concerns foundations of sensitivity and stability analysis in optimization and related areas, being primarily addressed truncated constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative contingent coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of such properties and developing basic rules of variational calculus interrelated with the obtained characterizations of well-posedness. Most of the our results valid in general infinite-dimensional settings are also new in finite dimensions.

Relative Well-Posedness of Truncated Constrained Systems Accompanied by Variational Calculus

TL;DR

The paper addresses sensitivity and stability for truncated constrained systems modeled by set-valued mappings between Banach spaces, introducing relative well-posedness notions such as Lipschitz-like stability, metric regularity, and linear openness relative to constraint sets and . It develops a novel framework of relative contingent coderivatives and its normal/mixed/mirror variants, together with the relative PSNC condition, to characterize these properties in both infinite and finite dimensions. Central results prove complete coderivative-based characterizations and establish pointwise calculus rules (chain, sum) for these relative notions, enabling verification and computation in structured truncated problems. The framework yields equivalences among stability properties, exact Lipschitz bounds, and PSNC criteria, offering a robust toolset for stability analysis in constrained optimization, variational inequalities, and related control and economic models.

Abstract

The paper concerns foundations of sensitivity and stability analysis in optimization and related areas, being primarily addressed truncated constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative contingent coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of such properties and developing basic rules of variational calculus interrelated with the obtained characterizations of well-posedness. Most of the our results valid in general infinite-dimensional settings are also new in finite dimensions.
Paper Structure (5 sections, 22 theorems, 153 equations)

This paper contains 5 sections, 22 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. New Tools of Generalized Differentiation
  3. Relative Well-Posedness Properties and Coderivative Characterizations
  4. Variational Calculus
  5. Concluding Remarks

Key Result

Lemma 2.3

Let $\Omega\subset X$, $x\in\Omega$, and $\varepsilon\geq0$. Suppose that $x^*\in\widehat{N}_\varepsilon(x;\Omega)\cap J_X(T(x;\Omega))$. Then we have $\|x^*\|_{X^*}\leq\varepsilon$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Lemma 3.2
  • ...and 36 more