Determine whether robust noncriticality guarantees neighboring-multiplier noncriticality

Determine whether defining noncriticality through a robust generalized derivative, such as the limiting graphical derivative or the semi-strict graphical derivative, automatically guarantees noncriticality for all multipliers sufficiently close to the reference multiplier.

Background

The paper defines noncriticality using the ordinary graphical derivative and emphasizes that this derivative is not robust under perturbations of the reference point. The authors therefore consider stronger notions based on robust derivatives, including the limiting graphical derivative and the semi-strict graphical derivative. An unresolved issue is whether such a robust notion would ensure that noncriticality persists for neighboring multipliers, which would address the instability illustrated by the paper’s example of a noncritical multiplier approached by critical multipliers.

References

Such a robust notion of noncriticality might automatically ensure noncriticality of all neighboring multipliers as well, a feature closely related to the semi-strict graphical derivative, a very interesting concept from , and we leave this direction open for future research.

— On error bounds and noncritical Lagrange multipliers  (2610.11491 - Benko et al., 8 Oct 2026) in Section “An exemplary setting from parametric optimization,” final paragraph before Section “Consequences in composite optimization”