Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds

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.

References in corpus (13)

• Stochastic gradient descent on Riemannian manifolds
• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics
• Sobolev metrics on shape space of surfaces
• Riemannian stochastic variance reduced gradient algorithm with retraction and vector transport
• Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds
• An Augmented Lagrangian Method for Optimization Problems in Banach Spaces
• Algorithmic aspects of multigrid methods for optimization in shape spaces
• On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces
• Shape Sensitivity Analysis for a Microchannel Cooling System
• A shape optimization algorithm for cellular composites
• A product shape manifold approach for optimizing piecewise-smooth shapes
• Encoder-decoder neural network for solving the nonlinear Fokker-Planck-Landau collision operator in XGC