Stochastic augmented Lagrangian method in Riemannian shape manifolds
Motivated by applications in shape optimization, where an objective functional is supposed to be minimized with respect to a shape, or a subset of a finite dimensional space, this manuscript presents and analise a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. It is investigated 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 then applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. Finding a correct model to describe the set of shapes is one of the main challenges in shape optimization. From a theoretical and computational point of view, it is attractive to optimize in Riemannian manifolds because algorithmic ideas from [\textit{P. A. Absil} et al., Optimization algorithms on matrix manifolds. Princeton, NJ: Princeton University Press (2008; Zbl 1147.65043)] can be combined with approaches from differential geometry as outlined in [\textit{C. Geiersbach} et al., in: Handbook of mathematical models and algorithms in computer vision and imaging. Mathematical imaging and vision. Cham: Springer. 1585--1630 (2023; Zbl 1547.94038)].
scientific article; zbMATH DE number 7942668
scientific article