Critical-cone characterization of directional solution derivatives in Asplund spaces

Establish, in the infinite-dimensional Asplund-space setting of Theorem 3.1, that the direction \(\bar v(\bar y)\) associated with directional \(V\)-flatness belongs to the critical cone \(C(\bar x,\bar y;\bar u)\), rather than only to the linearization cone \(\mathbb L(\bar x,\bar y;\bar u)\).

Background

Theorem \ref{thm: directional subdifferential of V} derives upper estimates for the directional limiting and singular subdifferentials of the value function in Asplund spaces under directional VV-flatness and directional restricted inf-compactness. For each directional solution yˉ\bar y, the theorem produces a direction vˉ(yˉ)\bar v(\bar y) that is known to belong to the linearization cone L(xˉ,yˉ;uˉ)\mathbb L(\bar x,\bar y;\bar u).

In the corresponding finite-dimensional result, the direction is shown to belong to the smaller critical cone C(xˉ,yˉ;uˉ)C(\bar x,\bar y;\bar u), which additionally requires inequalities relating the directional derivatives of the objective and value functions. The paper explains that the lack of strong compactness in Asplund spaces prevents guaranteeing a sequence of exact minimizers with the required convergent difference quotients, so the critical-cone restriction remains unestablished in this setting.

References

In the finite-dimensional framework established in Theorem 3.1, the direction \bar{v}(\bar{y}) is shown to belong to the critical cone C(\bar{x}, \bar{y}; \bar{u}). However, in Theorem~\ref{thm: directional subdifferential of V}, we can only guarantee that \bar{v}(\bar{y}) \in \mathbb{L}(\bar{x}, \bar{y}; \bar{u}). This discrepancy arises because, in the infinite-dimensional setting, we cannot guarantee the existence of a sequence y_k \in S(x_k) satisfying \frac{y_k - \bar{y}{t_k} \to \bar{v}(\bar{y}). Consequently, we are unable to establish the following sequence of estimates:

— Directional Subdifferentials of the Value Function in Asplund Spaces  (2608.20241 - Mao et al., 20 Aug 2026) in Remark 3.7 (Remark \ref{rem: critical_cone}), Section 3, following Theorem \ref{thm: directional subdifferential of V}