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A new problem qualification based on approximate KKT conditions for Lipschitzian optimization with application to bilevel programming

Isabella Käming, Andreas Fischer, Alain B. Zemkoho

TL;DR

The work tackles the challenge that many Lipschitzian optimization problems lack reliable constraint qualifications at local minima. It introduces Subset Mangasarian-Fromovitz Condition (subMFC), a problem qualification tied to approximate KKT (AKKT) conditions, ensuring that an AKKT point is a KKT point when a subset of active constraint subgradients is positively independent. subMFC is deliberately weaker than several classic CQs (e.g., CCP) and can hold even when error bounds or calmness fail, broadening the tractable problem classes. The authors further adapt subMFC to bilevel optimization via the lower-level value function reformulation, yielding LLVFR-subMFC, which can hold without partial calmness and under mild Lipschitz assumptions, enabling robust KKT characterizations for bilevel problems. Overall, subMFC provides a practical, analysis-friendly framework for verifying optimality without strong CQs and offers a promising avenue for bilevel and related nonsmooth problem classes.

Abstract

When dealing with general Lipschitzian optimization problems, there are many problem classes where even weak constraint qualifications fail at local minimizers. In contrast to a constraint qualification, a problem qualification does not only rely on the constraints but also on the objective function to guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For example, calmness in the sense of Clarke is a problem qualification. In this article, we introduce the Subset Mangasarian-Fromovitz Condition (subMFC). This new problem qualification is derived by means of a nonsmooth version of the approximate KKT conditions, which hold at every local minimizer without further assumptions. A comparison with existing constraint and problem qualifications reveals that subMFC is strictly weaker than quasinormality and can hold even if the local error bound condition, the cone-continuity property, the Guignard constraint qualification and calmness are violated. Furthermore, we emphasize the power of the new problem qualification within the context of bilevel optimization. More precisely, under mild assumptions on the problem data, we suggest a version of subMFC that is tailored to the lower-level value function reformulation. It turns out that this new condition can be satisfied even if the widely used partial calmness condition does not hold.

A new problem qualification based on approximate KKT conditions for Lipschitzian optimization with application to bilevel programming

TL;DR

The work tackles the challenge that many Lipschitzian optimization problems lack reliable constraint qualifications at local minima. It introduces Subset Mangasarian-Fromovitz Condition (subMFC), a problem qualification tied to approximate KKT (AKKT) conditions, ensuring that an AKKT point is a KKT point when a subset of active constraint subgradients is positively independent. subMFC is deliberately weaker than several classic CQs (e.g., CCP) and can hold even when error bounds or calmness fail, broadening the tractable problem classes. The authors further adapt subMFC to bilevel optimization via the lower-level value function reformulation, yielding LLVFR-subMFC, which can hold without partial calmness and under mild Lipschitz assumptions, enabling robust KKT characterizations for bilevel problems. Overall, subMFC provides a practical, analysis-friendly framework for verifying optimality without strong CQs and offers a promising avenue for bilevel and related nonsmooth problem classes.

Abstract

When dealing with general Lipschitzian optimization problems, there are many problem classes where even weak constraint qualifications fail at local minimizers. In contrast to a constraint qualification, a problem qualification does not only rely on the constraints but also on the objective function to guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For example, calmness in the sense of Clarke is a problem qualification. In this article, we introduce the Subset Mangasarian-Fromovitz Condition (subMFC). This new problem qualification is derived by means of a nonsmooth version of the approximate KKT conditions, which hold at every local minimizer without further assumptions. A comparison with existing constraint and problem qualifications reveals that subMFC is strictly weaker than quasinormality and can hold even if the local error bound condition, the cone-continuity property, the Guignard constraint qualification and calmness are violated. Furthermore, we emphasize the power of the new problem qualification within the context of bilevel optimization. More precisely, under mild assumptions on the problem data, we suggest a version of subMFC that is tailored to the lower-level value function reformulation. It turns out that this new condition can be satisfied even if the widely used partial calmness condition does not hold.
Paper Structure (21 sections, 18 theorems, 69 equations, 2 figures)

This paper contains 21 sections, 18 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Subset Mangasarian-Fromovitz Condition
  4. Comparison to constraint qualifications and calmness
  5. Mangasarian-Fromovitz constraint qualification
  6. Constant positive linear dependence constraint qualification
  7. Pseudonormality and quasinormality
  8. Local error bound condition
  9. Cone-continuity property
  10. Abadie constraint qualification and Guignard constraint qualification
  11. Calmness in the sense of Clarke
  12. Summary of the comparisons
  13. Application to bilevel optimization problems
  14. Comparison to partial calmness
  15. Final remarks
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let $\phi:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz continuous and $\bar{x}\in\mathbb{R}^n$.

Figures (2)

  • Figure 1: Strict implications between different conditions considered at a local minimizer of the inequality constrained problem \ref{['P']} are represented by solid arrows. Additional strict implications and one equivalence in case of differentiability of the constraint functions are visualized by dotted arrows. A crossed out arrow symbolizes that the condition at the start of the arrow does not imply the one at its end.
  • Figure A1: The feasible set $D$ of the optimization problem in Example \ref{['ex:mehrelementiges_subdiff_error_bound']} is depicted in blue.

Theorems & Definitions (49)

  • Proposition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 1
  • Remark 2
  • Definition 3.5
  • Theorem 3.6
  • ...and 39 more