Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds

TL;DR

This work introduces a stochastic augmented Lagrangian framework for constrained optimization on Riemannian shape manifolds, enabling stable handling of stochastic objectives with deterministic geometry constraints. By combining a random‑stopped inner stochastic gradient loop with a safeguarded outer AL loop and shaping updates through a multi‑exponential (approximated by a multi‑retraction), the method achieves provable convergence to KKT points (or AKKT in weaker settings) and provides explicit complexity and rate results in expectation. The approach is demonstrated on a two‑dimensional Stokes flow problem with uncertain inflow, optimized over a multi‑shape product manifold with volume and perimeter constraints, and numerical experiments illustrate objective reduction, mesh deformation control, and feasibility improvement. The work highlights the feasibility and potential of Bayesian‑style stochastic optimization in geometric PDE‑constrained shape design, while also noting open questions about manifold assumptions and constraint qualifications in infinite‑dimensional settings.

Abstract

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints.We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically.

Figures (3)

• Figure 1: Shapes $u_1$ (left, green), $u_2$ (top right, orange) and $u_3$ (bottom right, blue) at the start of the stochastic optimization (left) and the magnitude of the target fluid velocity $\| \bar{\boldsymbol{q}} \|_2$ together with the shapes used to obtain the target velocity in white (right).
• Figure 2: Shapes $u_1$ (left, green), $u_2$ (top right, orange) and $u_3$ (bottom right, blue) after $k=11$ iterations of stochastic optimization with different seeds and shapes used to obtain the target velocity (black).
• Figure 3: Objective functional (left) and $H^1$ norm of the mesh deformation (right) as a function of cumulative stochastic gradient steps using seed $421507$, $c_1=4$ and $c_2=\frac{1}{2}$. The changes of augmented Lagrange parameters are indicated with a red, dashed, vertical line.

Theorems & Definitions (15)

• Definition 2.1
• Remark
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark
• Theorem 2.1
• proof
• Theorem 2.2