First- and Second-Order Analysis for Optimization Problems with Manifold-Valued Constraints

TL;DR

This paper develops first- and second-order optimality conditions for optimization problems constrained by manifold-valued conditions, extending classical methods to a more general geometric setting involving smooth manifolds and convex cones.

Contribution

It introduces a framework for analyzing optimization problems with manifold-valued constraints, including invariance properties and conditions for optimality.

Findings

Established first- and second-order optimality conditions.

Proved invariance of optimality quantities under local problem representations.

Extended classical constraint analysis to manifold-valued settings.

Abstract

We consider optimization problems with manifold-valued constraints. These generalize classical equality and inequality constraints to a setting in which both the domain and the codomain of the constraint mapping are smooth manifolds. We model the feasible set as the preimage of a submanifold with corners of the codomain. The latter is a subset which corresponds to a convex cone locally in suitable charts. We study first- and second-order optimality conditions for this class of problems. We also show the invariance of the relevant quantities with respect to local representations of the problem.