Limiting BCQ: Dual Conditions in Asplund Spaces

• Limiting BCQ is a dual inclusion condition for closed multifunctions using limiting normal cones and coderivatives, essential for metric subregularity in Asplund spaces.
• It generalizes convex BCQ by removing unit-ball restrictions and quantifying dual relationships in nonconvex settings.
• The framework employs both exact and fuzzy Fréchet inclusions, providing precise criteria to verify stability and error-bound conditions in variational analysis.

Limiting Basic Constraint Qualification (BCQ) is a dual inclusion for closed multifunctions $F:X\rightrightarrows Y$ between Banach spaces, introduced in the study of metric subregularity in Asplund spaces. For $(\bar x,\bar y)\in \operatorname{gph}(F)$, it is defined by

$$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$

where $N$ is the Mordukhovich normal cone and $D^*F$ is the Mordukhovich coderivative. In the formulation developed for multifunctions with closed graph, the inclusion is presented as a necessary dual condition implied by metric subregularity, and its validity for all such multifunctions characterizes the Asplund property of the underlying space (Wei et al., 13 Sep 2025).

1. Definition and analytic framework

The ambient setting is the class $\Gamma(X,Y)$ of multifunctions with closed graph, that is, $\operatorname{gph}(F)\subset X\times Y$ is closed. Metric subregularity at $(\bar x,\bar y)\in \operatorname{gph}(F)$ means that there exist $\tau,r\in(0,+\infty)$ such that

$$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$

This is the basic primal regularity property to which limiting BCQ is tied (Wei et al., 13 Sep 2025).

The construction of limiting BCQ uses two levels of normal-cone and coderivative calculus. For a closed set $(\bar x,\bar y)\in \operatorname{gph}(F)$0 and $(\bar x,\bar y)\in \operatorname{gph}(F)$1, the Fréchet normal cone is

$(\bar x,\bar y)\in \operatorname{gph}(F)$2

and the Mordukhovich normal cone is

$(\bar x,\bar y)\in \operatorname{gph}(F)$3

with

$(\bar x,\bar y)\in \operatorname{gph}(F)$4

The associated coderivatives at $(\bar x,\bar y)\in \operatorname{gph}(F)$5 are

$(\bar x,\bar y)\in \operatorname{gph}(F)$6

and

$(\bar x,\bar y)\in \operatorname{gph}(F)$7

In convex settings, Fréchet and limiting normal cones coincide with the convex-analysis normal cone (Wei et al., 13 Sep 2025).

Within this framework, the limiting BCQ is qualitative rather than quantitative: it does not involve a scalar modulus, only the inclusion of the solution-set normal cone into the coderivative image of $(\bar x,\bar y)\in \operatorname{gph}(F)$8. This distinguishes it from strong BCQ formulations based on unit-ball truncations and multiplicative constants.

2. Relation to convex BCQ and other nonconvex BCQ formalisms

For convex multifunctions, the relevant benchmark is the BCQ of Zheng and Ng. If $(\bar x,\bar y)\in \operatorname{gph}(F)$9 is convex, then BCQ at $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$0 is

$$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$1

Because the inclusion

$$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$2

always holds in convex settings, this equality is equivalent to

$$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$3

The limiting BCQ for general closed multifunctions keeps precisely this reverse inclusion and extends it beyond convex graphs (Wei et al., 13 Sep 2025).

A distinct nonconvex line of work studies BCQ and strong BCQ for generalized equations $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$4 subject to $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$5. In that setting, Huang–He–Wei define Clarke-, Fréchet-, and Mordukhovich-type strong BCQ conditions, including the limiting strong BCQ

$$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$6

and show that metric subregularity yields localized necessary conditions of this kind near the reference solution (Huang et al., 2015). Relative to those strong BCQ statements, the limiting BCQ of the multifunction setting is an exact inclusion without unit-ball restriction or scalar $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$7.

For scalar nonconvex inequalities $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$8, a separate formulation is given through Clarke and Fréchet constructions: $$N(F^{-1}(\bar y),\bar x)\subseteq D^*F(\bar x,\bar y)(Y^*),$$9 That development explicitly states that a limiting/Mordukhovich version is not treated there; the paper focuses instead on Clarke and Fréchet objects and notes that a natural limiting analog would require replacing them by $N$0 and $N$1, but such statements are not provided or proved (Wei et al., 2017).

This comparison suggests that the limiting BCQ of multifunctions occupies a specific position in the CQ landscape: it is not the classical convex equality, not the quantified strong BCQ of generalized equations, and not the Clarke- or Fréchet-based nonconvex BCQ of scalar inequalities. Its defining feature is the use of limiting normals and coderivatives as a necessary condition aligned with metric subregularity.

3. Metric subregularity and characterization of Asplund spaces

The central theorem states that the limiting BCQ is equivalent, at the level of universal implication from metric subregularity, to the Asplund property of the domain space. Let $N$2 be a Banach space. The following are equivalent: $N$3 is Asplund; for every Asplund space $N$4 and every $N$5 metrically subregular at $N$6, there exists $N$7 such that

$N$8

and, equivalently, the same inclusion holds at the reference point: $N$9 Thus, in Asplund spaces, metric subregularity implies the limiting BCQ both locally along the solution set and at the base point; conversely, if the implication “metric subregularity $D^*F$0 limiting BCQ” holds for all closed multifunctions with Asplund target, then $D^*F$1 must be Asplund (Wei et al., 13 Sep 2025).

The proof uses a norm on $D^*F$2 defined by

$D^*F$3

together with a localization inequality. These devices are quantitative, but the limiting BCQ conclusion itself remains qualitative.

The necessity of the Asplund assumption is exhibited by a non-Asplund construction. Let $D^*F$4 with $D^*F$5 non-Asplund, let

$D^*F$6

define

$D^*F$7

and set $D^*F$8 with the $D^*F$9 norm. For $\Gamma(X,Y)$0, the multifunction satisfies

$\Gamma(X,Y)$1

so metric subregularity holds. However, limiting BCQ fails because

$\Gamma(X,Y)$2

This counterexample shows that the Asplund property is essential for the exact dual implication (Wei et al., 13 Sep 2025).

4. Fréchet fuzzy inclusions and exact limiting statements

Alongside the limiting BCQ, the same analysis derives necessary conditions via Fréchet normal cones and Fréchet coderivatives in a fuzzy form. If $\Gamma(X,Y)$3 is a Banach space, then the following are equivalent: $\Gamma(X,Y)$4 is Asplund; for every Asplund space $\Gamma(X,Y)$5 and every $\Gamma(X,Y)$6 metrically subregular at $\Gamma(X,Y)$7, there exists $\Gamma(X,Y)$8 such that for all $\Gamma(X,Y)$9 and all $\operatorname{gph}(F)\subset X\times Y$0,

$\operatorname{gph}(F)\subset X\times Y$1

and, equivalently, the analogous inclusion holds at $\operatorname{gph}(F)\subset X\times Y$2 (Wei et al., 13 Sep 2025).

The term “fuzzy” means that the inclusion is not exact at $\operatorname{gph}(F)\subset X\times Y$3 alone: it uses nearby graph points $\operatorname{gph}(F)\subset X\times Y$4 and an additive $\operatorname{gph}(F)\subset X\times Y$5-ball in $\operatorname{gph}(F)\subset X\times Y$6. These inclusions are therefore necessary in Asplund spaces, but only approximate.

A key technical input is the fuzzy sum rule for Fréchet subdifferentials in Asplund spaces. That rule allows the subdifferential of a sum to be approximated by sums of subdifferentials at nearby points, plus an $\operatorname{gph}(F)\subset X\times Y$7-ball, and underpins the fuzzy coderivative inclusions. In finite-dimensional $\operatorname{gph}(F)\subset X\times Y$8, one can often pass from fuzzy Fréchet inclusions to exact limiting inclusions, whereas in general Banach spaces the exact limiting statements are tied to the Asplund characterization above (Wei et al., 13 Sep 2025).

This division between fuzzy Fréchet conditions and exact limiting conditions is structurally important. It shows that approximate regular-normal calculus is broadly available, but exact normal-coderivative necessary conditions of BCQ type depend decisively on the ambient geometry of Asplund spaces.

5. Conic inequalities and recovery of scalar error-bound results

The paper applies the limiting BCQ framework to conic inequalities

$\operatorname{gph}(F)\subset X\times Y$9

where $(\bar x,\bar y)\in \operatorname{gph}(F)$0 is a closed cone with nontrivial recession cone $(\bar x,\bar y)\in \operatorname{gph}(F)$1, and $(\bar x,\bar y)\in \operatorname{gph}(F)$2 is a proper mapping with closed $(\bar x,\bar y)\in \operatorname{gph}(F)$3-epigraph. Metric subregularity at $(\bar x,\bar y)\in \operatorname{gph}(F)$4 means that

$(\bar x,\bar y)\in \operatorname{gph}(F)$5

The associated $(\bar x,\bar y)\in \operatorname{gph}(F)$6-relative subdifferentials are defined through normal cones to $(\bar x,\bar y)\in \operatorname{gph}(F)$7, using the dual cone

$(\bar x,\bar y)\in \operatorname{gph}(F)$8

namely

$(\bar x,\bar y)\in \operatorname{gph}(F)$9

together with the singular part

$\tau,r\in(0,+\infty)$0

Lemma 4.1 relates these objects to coderivatives through

$\tau,r\in(0,+\infty)$1

for $\tau,r\in(0,+\infty)$2 (Wei et al., 13 Sep 2025).

The corresponding exact inclusion is the conic analogue of limiting BCQ. If $\tau,r\in(0,+\infty)$3 is a Banach space, then the following are equivalent: $\tau,r\in(0,+\infty)$4 is Asplund; for any Asplund space $\tau,r\in(0,+\infty)$5, any closed cone $\tau,r\in(0,+\infty)$6 with $\tau,r\in(0,+\infty)$7, and any $\tau,r\in(0,+\infty)$8 metrically subregular at $\tau,r\in(0,+\infty)$9 with $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$0, there exists $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$1 such that for all $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$2 with $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$3,

$$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$4

and, equivalently, the same inclusion holds at $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$5 (Wei et al., 13 Sep 2025).

There is also a fuzzy Fréchet counterpart: when $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$6 is Asplund, $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$7 is Asplund, $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$8 is closed convex, and $$\operatorname{dist}(x,F^{-1}(\bar y))\le \tau\,\operatorname{dist}(\bar y,F(x)), \qquad \forall x\in \mathbf B(\bar x,r).$$9 is metrically subregular, then for all $(\bar x,\bar y)\in \operatorname{gph}(F)$00 and all $(\bar x,\bar y)\in \operatorname{gph}(F)$01,

$(\bar x,\bar y)\in \operatorname{gph}(F)$02

In the scalar case $(\bar x,\bar y)\in \operatorname{gph}(F)$03 and $(\bar x,\bar y)\in \operatorname{gph}(F)$04, metric subregularity of $(\bar x,\bar y)\in \operatorname{gph}(F)$05 yields the local error bound

$(\bar x,\bar y)\in \operatorname{gph}(F)$06

and the conic inclusion reduces to

$(\bar x,\bar y)\in \operatorname{gph}(F)$07

The paper identifies this as precisely the Lewis–Pang BCQ implication for convex inequalities recovered as a special case (Wei et al., 13 Sep 2025).

6. Interpretation, verification, and limitations

The limiting BCQ is presented as a necessary dual inclusion tailored to metric subregularity rather than metric regularity. It is weaker than classical constraint qualifications such as Robinson’s CQ, MFCQ, Slater’s condition, and Abadie CQ, which are typically used to certify metric regularity, Lagrangian KKT conditions, or strong duality and often require interiority or linear-independence properties. By contrast, limiting BCQ does not require convexity, is formulated through limiting normal cones and coderivatives rather than tangent or constraint linearizations, and in convex settings collapses to the equality

$(\bar x,\bar y)\in \operatorname{gph}(F)$08

(Wei et al., 13 Sep 2025).

The paper also makes clear that limiting BCQ is not, in general, sufficient for metric subregularity. A one-dimensional example after Theorem 3.3 shows that the exact inclusion may hold while metric subregularity fails. A plausible implication is that limiting BCQ should be read as a sharp dual diagnostic of subregularity-compatible geometry, not as a stand-alone regularity criterion.

For verification, the proposed procedure is direct. One computes or estimates $(\bar x,\bar y)\in \operatorname{gph}(F)$09, computes $(\bar x,\bar y)\in \operatorname{gph}(F)$10 through the normal cone to $(\bar x,\bar y)\in \operatorname{gph}(F)$11, and then checks

$(\bar x,\bar y)\in \operatorname{gph}(F)$12

In set-based models $(\bar x,\bar y)\in \operatorname{gph}(F)$13, Lemma 3.1 provides explicit coderivative calculations. In composite-convex settings $(\bar x,\bar y)\in \operatorname{gph}(F)$14 with convex $(\bar x,\bar y)\in \operatorname{gph}(F)$15 and smooth $(\bar x,\bar y)\in \operatorname{gph}(F)$16, Lemma 3.2 reduces the normal-cone computation via the mapping $(\bar x,\bar y)\in \operatorname{gph}(F)$17. For conic inequalities, one compares $(\bar x,\bar y)\in \operatorname{gph}(F)$18 with $(\bar x,\bar y)\in \operatorname{gph}(F)$19 using the coderivative estimate of Lemma 4.1 (Wei et al., 13 Sep 2025).

Taken together, these results place limiting BCQ at the intersection of variational analysis, error-bound theory, and the geometry of Banach spaces. Its defining role is not merely as another CQ, but as an exact dual condition whose necessity under metric subregularity is equivalent to the Asplund property and whose failure in non-Asplund spaces is explicit rather than merely formal (Wei et al., 13 Sep 2025).