Outer-Layer Optimization Approach

• Outer-layer optimization is a framework that indirectly approximates feasible sets via ambient group actions and Sobolev-type metrics.
• It leverages a Riemannian steepest descent algorithm with elliptic operator inversion to generate globally regularized, volume-filling descent directions.
• The method extends to applications such as fracture propagation, neural inverse problems, and convex programming, offering enhanced mesh quality and convergence.

The outer-layer optimization approach refers to optimization methodologies wherein the feasible set, shape, or solution region is not directly parameterized or manipulated, but is indirectly approximated, updated, or improved via systematic outer constructions, ambient group actions, or external regularization mechanisms. These strategies are essential in PDE-constrained optimization, shape optimization, convex programming, neural inverse problems, and physical modeling. Typical elements include group-theoretic embeddings, outer metrics, cutting planes/balls, and Riemannian manifold machinery, all aimed at robustifying and regularizing global descent directions and solution representations.

1. Sobolev-Type Outer Metrics on Diffeomorphism Groups

Shape optimization constrained by PDEs is often best formulated by identifying the admissible set of shapes with the orbit of a reference domain under the action of the compactly-supported diffeomorphism group $$Diff_c(\mathbb{R}^n) = \{ \phi:\mathbb{R}^n \to \mathbb{R}^n \mid \phi,\phi^{-1} \in C^\infty, supp(\phi-id) \; \text{compact} \}$$ endowed with a right-invariant Riemannian "outer" metric (Loayza-Romero et al., 28 Mar 2025).

The canonical construction employs a Sobolev-type inner product on the Lie algebra $X_c(\mathbb{R}^n)$:

$$\left\langle U, V \right\rangle_{H^s} = \int_{\mathbb{R}^n} \left\langle L U, V \right\rangle \, dx,$$

where $L = (Id - A \Delta)^s$, $A > 0$, $\Delta$ is the Laplacian, and $s$ denotes regularity. Expanding via derivatives:

$$\left\langle U, V \right\rangle_{H^s} = \sum_{|\alpha| \leq s} C_\alpha A^{|\alpha|} \int_{\mathbb{R}^n} \left\langle \partial^\alpha U, \partial^\alpha V \right\rangle dx,$$

with $C_\alpha$ combinatorial. For $s > n/2+1$, the metric is geodesically and metrically complete. This metric penalizes high-frequency deformation and naturally yields gradients that deform the full ambient domain, not merely boundary objects.

2. Push-Forward, Shape Derivative, and Riemannian Gradients

Given a classical shape functional $J(u)$ with Hadamard/Eulerian derivative $$DJ(u)[W] = \left.\frac{d}{dt}\right|_{t=0}J((Id + tW)(u))$$, one embeds the shape via a reference immersion $i:S^{n-1}\to \mathbb{R}^n$ and lifts the functional to the diffeomorphism group:

$$j(\phi) := J(\phi \circ i(S^{n-1})), \quad j: Diff_c \to \mathbb{R},$$

with differential:

$dj_\phi(\gamma) = DJ(\phi(u))[X \circ \phi^{-1}],$

where $X$ is the vector field generating the deformation. The Riemannian gradient $\nabla^G j(\phi)$ solves:

$$G_\phi(\nabla^G j(\phi), \gamma) = dj_\phi(\gamma), \quad \forall \gamma \in T_\phi Diff_c,$$

which reduces to inverting $L$:

$$L W = F, \quad \nabla^G j(\phi) = (L^{-1}F) \circ \phi.$$

This yields globally regularized volume-filling descent directions.

3. Riemannian Outer-Metric Steepest Descent Algorithm

The practical implementation is a Riemannian steepest descent scheme:

1. Compute the current shape $u^k = \phi^k(i(S^{n-1}))$.
2. Solve the state PDE and adjoint PDE to acquire classical shape derivative $F^k$.
3. Solve $(Id - A \Delta)^s W^k = F^k$ for compactly-supported $W^k$.
4. Form the outer-metric gradient: $\nabla^G j(\phi^k) = W^k \circ \phi^k$.
5. Line search $t_k$ via Armijo/backtracking.
6. Update via retraction: $\phi^{k+1} = \phi^k \circ (Id - t_k W^k)$.
7. Terminate on small $H^s$ norm or sufficient decrease.

Mesh-quality controls or remeshing signal excessive $t_k$ violating diffeomorphic constraints. This method handles PDE-constrained tomography and compliance minimization robustly (Loayza-Romero et al., 28 Mar 2025).

4. Comparative Analysis: Outer vs. Inner Metrics

Boundary-only (inner-metric) optimization endows the embedding manifold with metrics whose tangent space consists solely of surface vector fields. This restricts gradient descent to moving boundary points, which often degrades mesh quality, promotes tangential distortion, and limits deformation amplitude.

By contrast, the outer-metric approach via ambient diffeomorphism ensures that elasticity, smoothness, and global regularity are preserved:

• Mesh preservation: Interior nodes deform coherently with boundaries.
• Descent regularization: $H^s$ damping mitigates high-frequency boundary wiggles.
• Unified extension: Ad-hoc boundary-to-volume smoothing (e.g., harmonic extension) subsumed by $L^{-1}$.

Numerical tests confirm that for $s \geq 2$ outer-metric steepest descent achieves lower objective values, faster convergence, and superior mesh regularity compared to inner-metric or $L^2$-gradient approaches, especially under large shape evolutions (Loayza-Romero et al., 28 Mar 2025).

5. Applications and Variants of Outer-Layer Optimization in Related Fields

The outer-layer optimization paradigm generalizes to diverse problem classes:

• Fracture Propagation: Shape optimization over $Diff_{cn}(\mathbb{R}^2)$, using an outer Sobolev metric, enables direct modeling of brittle fracture and its propagation, exploiting spectral splitting for realistic constraints on tension-driven crack growth (Suchan et al., 25 Jul 2025).
• Weakly Convex Constraints: Outer approximation via quadratic “sphere” cuts solves weakly convex constrained problems, iteratively shrinking a QCQP-outer hull to reach global optimizers (Bednarczuk et al., 23 Sep 2024).
• Neural Inverse Problems: Region-wise convexification of ReLU networks allows outer-approximation phases exploiting tangent hyperplane cuts for rapid local descent, with subsequent global verification across activation regions (Cheon, 2020).
• Convex Vector Optimization: The Pascoletti-Serafini scalarization iteratively renders outer-approximation polyhedra, optimizing reference points/directions to cut off non-optimal vertices and converge to Hausdorff-accurate weak minimal sets (Keskin et al., 2021).

6. Theoretical and Practical Consequences

A coherent theme in outer-layer optimization is the lifting of optimization dynamics to an ambient or enclosing mathematical structure—diffeomorphism groups, outer hulls, QCQP relaxations, or manifold-valued parameterizations—where higher regularity, robustness, and globality of descent directions are ensured.

• Theoretical guarantees: For well-chosen metrics (Sobolev order $s > n/2+1$), the optimization problem becomes geodesically complete, possesses global minimizers, and regularizes mesh evolution.
• Algorithmic stability: Elliptic operator inversion automatically filters boundary-only derivative signals into well-behaved volume vector fields.
• Extensibility: This approach is extensible to multi-physics, weakly convex, variable-metric, and composite objectives, yielding tractable, unified frameworks for complex domain evolution and constrained minimization.

In summary, the outer-layer optimization approach enables PDE-constrained shape optimization and related problems to be formulated and solved in a Riemannian manifold context with Sobolev-type outer metrics. This leads to regularized, volume-preserving descent directions, enhanced mesh quality, natural solution regularity, and robust handling of large deformations and constraints (Loayza-Romero et al., 28 Mar 2025).