Constrained Optimization on Manifolds

Abstract

Optimization problems are present in many mathematical applications, and those that are particularly challenging arise when it comes to solving highly nonlinear problems. Hence, it is of benefit exploiting available nonlinear structure, and optimization on nonlinear manifolds provides a general and convenient framework for this task. Applications arise, for instance, in nonlinear problems for linear algebra, nonlinear mechanics, and many more. For the case of nonlinear mechanics, the aim is to preserve some specific structure, such as orientability, incompressibility or inextensibility, as in the case of elastic rods. Moreover, additional constraints can occur, as in the important case of optimal control problems. Therefore, for the solution of such problems, new geometrical tools and algorithms are needed. This thesis deals with the setting of constrained optimization problems on manifolds and with the construction of algorithms for their numerical solution. In the abstract formulation, we seek to minimize a real function on a manifold, where the constraint is given by a submersion that is a twice continuously differentiable map between two differentiable manifolds. Furthermore, for optimal control applications, we extend this formulation to vector bundles. Optimization algorithms on manifolds are available in the literature, mostly for the unconstrained case, and the usage of retraction maps is an indispensable tool for this purpose. Retractions, which play a fundamental role in the updating of the iterates in optimization algorithms, also allow us to pullback the involved maps to linear spaces, making possible the use of tools and results from the linear setting. In particular, KKT-theory and second-order optimality conditions are tractable thanks to such maps. In this work, we extend the concept of retraction to vector bundles, where first and second-order consistency conditions are also defined. On the other hand, at each point in the manifold, a 1-form Lagrange multiplier arises as a solution of a saddle point problem. We prove that the existence of a potential function for this Lagrange multiplier depends on the integrability, in the sense of Frobenius, of the horizontal distribution, i.e., the orthogonal complement of the linearized constraint map. For the algorithmic solution of constrained optimization problems on manifolds, we generalize the affine covariant composite step method to these spaces, and local superlinear convergence of the algorithm for first-order retractions is obtained. First, we test the algorithm in a constrained eigenvalue problem. Then, we consider numerical experiments on the mechanics of elastic inextensible rods. There, we compute the final configuration of an elastic inextensible rod under dead load. The case in which the rod enters in contact with one, or several planes, is considered. Hence, we exploit the Riemannian structure of the positive cone. In addition, we solve an optimal control problem of an elastic inextensible rod, as an application to constrained…