Project Description
Optimization on manifolds is a rich field of growing interest and with numerous applications. In contrast to classical optimization, where an objective function is minimized over a linear space, the objective’s domain of definition is a (nonlinear) smooth manifold. This setting allows for the consideration of new classes of problems and the natural exploitation of problem structure. This additional structure would be lost if instead the manifold were described via constraints in a linear embedding space.
Most of the literature to date deals with unconstrained optimization on manifolds. Some theory and algorithms exist that consider constrained problems, but these efforts are almost exclusively confined to finite dimensions and to constraints that map into a fixed linear space. This setting, however, has its shortcomings and excludes many interesting applications, including differential equations and variational problems on manifolds.
The goal of the project therefore is to develop theory and algorithms for constrained optimization problems defined on manifolds and with manifold-valued constraints. We will exploit our prior work on manifolds with corners as an adequate way to generalize both equality and inequality constraints to manifolds. In addition, we have recently identified vector bundles as a natural setting for Newton’s method and thus also as a suitable codomain for equality constraints on manifolds. Specifically, the cotangent bundle occurs frequently in the optimization of variational problems on manifolds, such as elastic geodesics under an external force field.
We will significantly extend these results, with the overall motivation to be able to deal with infinite-dimensional variational problems and constraints of equality and inequality type. To this end, we formulate three specific goals. (1) We will derive optimality conditions for equality- and inequality-constrained optimization problems, modeled in terms of manifolds with corners, where both the domain and codomain may be infinite-dimensional. (2) Building on this theory, we will develop and analyze an intrinsic sequential quadratic programming (SQP) algorithm for manifold-valued constrained optimization. (3) We will demonstrate the applicability of our theory and algorithm by studying a particular class of applications, namely the optimal control of variational problems on manifolds with and without inequality constraints. Specifically, we will consider the optimal control of elastic geodesics with state and control constraints, as well as the optimal control of the Landau-Lifshitz-Gilbert equation that describes the evolution of magnetization in a ferromagnetic material.
