Strong Bi-Metric Regularity

• Strong bi-metric regularity is a property of set-valued mappings that guarantees unique local solutions and Lipschitz stability under perturbations measured by different metrics.
• It decouples the topological requirements for existence from those needed for sensitivity analysis, supporting rigorous treatment in infinite-dimensional optimization.
• Its application in affine optimal control, including bang-bang solutions, yields explicit error bounds and convergence guarantees in discretized numerical schemes.

Strong bi-metric regularity (Sbi-MR) is a property of set-valued mappings central to the analysis of variational and optimality systems, particularly in infinite-dimensional optimization and optimal control. The Sbi-MR property governs the local well-posedness, stability, and robustness of solution mappings subject to perturbations, especially when different metrics or norms are necessary to quantify “locality” versus “sensitivity” in domain and image spaces. This concept is crucial in settings like affine optimal control, where solutions (such as bang-bang controls) may display discontinuities or lack convexity, and classical regularity notions based on a single metric lose effectiveness (Corella et al., 18 Nov 2025, Jork et al., 28 Sep 2024).

1. Definition and Formal Structure

Let $(Y, d_Y)$, $(Z, d_Z)$, and $(\tilde{Z}, d_{\tilde{Z}})$ be metric spaces with $\tilde{Z} \subset Z$, and $d_Z(z, z') \leq d_{\tilde{Z}}(z, z')$ on $\tilde{Z}$. For a set-valued mapping $\Phi: Y \rightrightarrows Z$, define neighborhoods

$${}_Y(\hat{y}; a) := \{ y \in Y \mid d_Y(y, \hat{y}) \leq a \}, \quad {}_{\tilde{Z}}(\hat{z}; b) := \{ z \in \tilde{Z} \mid d_{\tilde{Z}}(z, \hat{z}) \leq b \}.$$

Sbi-MR is specified at $\hat{y}$ for $\hat{z} \in \tilde{Z}$ with constants $\kappa \geq 0$, $a, b > 0$ by:

• For every $z \in {}_{\tilde{Z}}(\hat{z}; b)$, $\Phi^{-1}(z) \cap {}_Y(\hat{y}; a)$ is a singleton,
• For all $z_1, z_2 \in {}_{\tilde{Z}}(\hat{z}; b)$,

$$d_Y\left( \Phi^{-1}(z_1) \cap {}_Y(\hat{y}; a), \; \Phi^{-1}(z_2) \cap {}_Y(\hat{y}; a) \right) \leq \kappa \, d_Z(z_1, z_2).$$

This bi-metric form decouples the metric structuring local neighborhoods (weak/strong topology for existence) from the metric quantifying solution sensitivity (e.g., $L^1$ vs $L^\infty$ norms). The property is generalized further to cases where two pairs of metrics appear in the domain and the codomain (Jork et al., 28 Sep 2024).

2. Motivation and Contrast with Single-Metric Regularity

Classical strong metric regularity (SMR) and strong metric subregularity (SMsR) employ a single norm in both argument and residual measurement: $$d_X(x, \bar{x}) \leq K\,d_Y(\bar{y}, F(x)) \quad \text{(SMsR)},$$

$$F^{-1}(y) \cap U \text{ singleton for each } y \in V,\quad d_X(F^{-1}(y), F^{-1}(y')) \leq K\,d_Y(y, y').$$

Bi-metric regularity reflects the necessities of infinite-dimensional variational settings, especially optimal control, where weak topologies (e.g., $L^\infty$) are natural for perturbations/data input while strong norms ($L^1, W^{1,1}$) are required for error bounds and differentiability (Corella et al., 18 Nov 2025, Jork et al., 28 Sep 2024). The flexibility of Sbi-MR enables sharp, simultaneously robust and fine-grained stability results, overcoming limitations where, e.g., single-metric regularity is vacuous due to discontinuity or lack of convexity.

3. Bi-Metric Regularity in Affine Optimal Control

Consider the affine-in-control ODE system: $$\min_{u(\cdot)} J(u) = \int_0^T [w(t, x(t)) + s(t, x(t)), u(t)]\, dt$$ subject to

$$\dot{x}(t) = a(t, x(t)) + B(t, x(t))u(t), \quad x(0) = x^0, \quad u(t) \in U,$$

where $U \subset \mathbb{R}^m$ is convex, compact, and $u \in L^\infty$. The associated Pontryagin optimality conditions can be recast as a generalized equation

$0 \in F(y), \quad y = (x, p, u) \in Y,$

where $Y$ and $Z$ are Banach spaces,

$$Y = \{ (x, p, u) \mid x, p \in W^{1,1},\; u \in L^1,\, x(0) = x^0,\; p(T) = 0 \} \cap (W^{1,\infty} \times W^{1,\infty} \times L^\infty),$$

$$Z = L^\infty \times L^\infty \times L^\infty, \quad \tilde{Z} = L^\infty \times L^\infty \times W^{1, \infty}.$$

Sbi-MR of the optimality mapping $F: Y \rightrightarrows Z$ is established at reference solutions under smoothness and second-order positivity conditions without requiring convexity of the Lagrangian (Corella et al., 18 Nov 2025). Notably, even directionally nonconvex functionals with bang-bang optimal solutions can exhibit Sbi-MR.

4. Sufficient Conditions, Key Theorems, and Proof Technique

The central result (Corella et al., 18 Nov 2025) provides sufficient conditions for Sbi-MR:

• (A1) Smoothness: coefficients and data ($a, B, w, s$) $C^2$ in $(t, x)$ with uniform Lipschitz constants.
• (A2) Second-order positive definiteness on critical cones: $$\int_0^T \langle \sigma(t), \delta u(t) \rangle\, dt + \Gamma(\delta u) \geq c_0 \|\delta u\|_1^2,$$ for all small $\|\delta u\|_1$ and suitably close $\sigma$.

The proof involves:

1. Partial linearization of $F$ (as in [QSV-18]), reducing Sbi-MR to that of a linear-normal inclusion.
2. Reduction to a control-only mapping on $L^1$, using Fréchet differentiability.
3. Establishment of strong monotonicity via second-order variations.
4. Application of variational inequality characterizations and splitting techniques to obtain Lipschitz estimates.
5. Verification of Lipschitz dependence of the solution mapping with respect to disturbances measured in the strong metric.

This analytic structure supports uniform distance estimates and stability under perturbations, crucial for qualitative and numerical analysis (Corella et al., 18 Nov 2025, Jork et al., 28 Sep 2024).

5. Stability, Perturbation, and Discretization Results

Sbi-MR provides the foundation for perturbation-stability and discretization convergence in optimal control mappings. If $F$ is Sbi-MR and $P$ is a sufficiently small, single-valued perturbation, then $F+P$ retains the Sbi-MR property, possibly with adjusted constants (Jork et al., 28 Sep 2024): $K' = \frac{K}{1 - \mu K},$ where $\mu$ is a Lipschitz constant for $P$ in the disturbance metric.

An application is to the Euler discretization of control problems. For a family of perturbations of the system data, the discrete-time optimality system yields residuals proportional to the discretization step in the strong norm, and Sbi-MR yields global error bounds: $$\|x_h - x^*\|_{W^{1,1}} + \|p_h - p^*\|_{W^{1,1}} + \|u_h - u^*\|_{L^1} \leq C h,$$ uniformly for all perturbations in a neighborhood (Corella et al., 18 Nov 2025). This establishes numerical convergence with explicit constants and robustness.

6. Relationship to Broader Regularity Theory

Sbi-MR generalizes and refines established notions of strong metric subregularity and regularity, widely studied in convex-composite optimization, variational inequalities, and KKT mappings (Burke et al., 2018). While strong metric regularity ensures local single-valuedness and Lipschitz stability in a single norm, Sbi-MR addresses cases where, due to structural or analytic constraints, different topologies are required on domain and codomain for the theory to be both nontrivial and quantitatively precise.

The analytical tools include coderivative criteria, implicit mapping theorems for set-valued maps, and second-order expansion techniques, as detailed in (Jork et al., 28 Sep 2024) and (Corella et al., 18 Nov 2025). In convex-composite and nonlinear programming, Sbi-MR concepts naturally emerge as both necessary and sufficient for local isolability and quadratic convergence properties under Newton-type algorithms (Burke et al., 2018).

7. Concrete Examples and Implications

Explicit examples substantiating Sbi-MR without convexity include control problems with nonconvex integrands yielding bang-bang optimal controls: $$\min \int_0^1 \left( -\frac{\alpha}{2} x^2(t) - \beta x(t) + u(t) \right) dt, \quad \dot{x} = u,\, x(0)=0,\, u\in[0,1],$$ with certain ranges for $\beta$ and $\alpha$. Despite the directional nonconvexity, the optimality system satisfies Sbi-MR (Corella et al., 18 Nov 2025).

A plausible implication is that whenever second-order positivity and a mild symmetry hold, Sbi-MR can be leveraged for robust stability and convergence results in both theory and computational practice, irrespective of convexity.

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References:

(Corella et al., 18 Nov 2025) Domínguez Corella, Quincampoix, Veliov, "Strong bi-metric regularity in affine optimal control problems" (Jork et al., 28 Sep 2024) "Strong metric (sub)regularity in optimal control" (Burke et al., 2018) Burke & Engle, "Strong Metric (Sub)regularity of KKT Mappings for Piecewise Linear-Quadratic Convex-Composite Optimization"