Optimization on Manifolds for the Numerical Solution of Equality Constrained Variational Problems

• Applicant: Professor Dr. Anton Schiela
• Subject Area: Mathematics
• Term: from 2016 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 318513007

• Final Report Year: 2021

Final Report Abstract

In this project we developed new algorithms for equality constrained optimization on smooth manifolds. In view of variational problem, coming from mathematical physics, our research was done in the infinite dimensional context of Hilbert manifolds. We were able to establish some theoretical results for our new algorithm, such as transition to fast local convergence. Moreover we successfully applied our algorithm to the simulation and optimal control of inextensible rods, examples for constrained variational problems on manifolds. Our approach makes it possible to exploit nonlinear structure of the given problem and enable a more efficient solution. A particularly surprising insight of the project was the observation that constraint mappings with nonlinear codomains exhibit a much richer structure than their classical counterparts, which map into linear spaces. In this direction there are still many things to explore, theoretically and algorithmically. Our project also opened the door to new applications of optimization on manifolds, such as optimal control of variational problems on manifolds or shape optimization.

Publications

• “Constrained Optimization on Manifolds”. PhD thesis. Bayreuth, Dec. 2020
  J. Ortiz
  
• “An SQP Method for Equality Constrained Optimization on Hilbert Manifolds”. In: SIAM Journal on Optimization 31.3 (2021), pp. 2255– 2284
  A. Schiela and J. Ortiz
  
• “Second order directional shape derivatives of integrals on submanifolds”. In: Mathematical Control & Related Fields 11.3 (2021), pp. 658– 679
  A. Schiela and J. Ortiz