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

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.

Figures (2)

• Figure 1.1: A geodesic polygon on the $2$-sphere. Unlike in $\mathbb{R}^2$, this set cannot be described as the intersection of half spaces. Notice that, for instance, at the tangent space at the light blue point in the middle of the horizontal geodesic, the image of the upper half space under the exponential map is the entire sphere.
• Figure 2.1: Illustration of a $k = 2$-dimensional manifold with corners $\mathcal{K}$ (teal) as a subset of the $n = 2$-dimensional sphere $\mathcal{N} = \mathcal{S}^{2}$. Due to $k = n$, the tangent space satisfies $\mathcal{T}_{q}\mathcal{K} = \mathcal{T}_{q}\mathcal{N}$ for every $q \in \mathcal{K}$. At the particular point $q$, which is a corner of index $\ell = 2$, the cone of inner tangent vectors $\mathcal{T}_{q}^i{\mathcal{K}}$ is shown in green.

Theorems & Definitions (45)

• Definition 2.1: Submanifold with corners (Michor:1980:1)
• Example 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Definition 3.1: Tangent cone
• Lemma 3.2
• proof
• Lemma 3.3
• proof