Optimization on manifold with additional linear constraints

Question

I am looking for references about manifold optimization when additional constraints on the variable are present.

Specifically, the problem I'm interested in is something along this line

\begin{align} \min_{Q \in SO(n)} \quad & f(Q) \\ \text{s.t.} \quad & \Lambda Q \geq 0 \\ \text{s.t} \quad & M Q^T \geq 0 \end{align}

Answer 1

The field of unconstrained Riemannian optimization has a significant literature. But as far as I know, there had been no publications in constrained Riemannian optimization until the 2019 publication cited below.

Paywalled version: Simple Algorithms for Optimization on Riemannian Manifolds with Constraints, by Changshuo Liu & Nicolas Boumal

Freely accessible pre-publication version: Simple Algorithms for Optimization on Riemannian Manifolds with Constraints, by Changshuo Liu & Nicolas Boumal

Abstract: We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions.

Answer 2

As a supplement of @Mark L. Stone's answer, Karush-Kuhn-Tucker condition of constrained manifold optimization has recently formulated in the following publication.

Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds

Abstract: Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrained optimization problems on smooth manifolds are formulated. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. The linear independence, Mangasarian{Fromovitz, and Abadie constraint qualifications are also formulated, and the chain LICQ implies MFCQ implies ACQ implies GCQ" is proved. Moreover, classical connections between these constraint qualifications and the set of Lagrange multipliers are established, which parallel the results in Euclidean space. The constrained Riemannian center of mass on the sphere serves as an illustrating numerical example.