An Algorithm For Finding the Optimal Embedding of a Symmetric Matrix into the Set of Diagonal Matrices

We investigate two two-sided optimization problems that have their application in atomic chemistry and whose matrix of unknowns $Y\in\R^{n\times p}$ ($n\ge p$) lies in the Stiefel manifold. We propose an analytic optimal solution of the first problem, and show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and $p$ unknowns. We prove that the latter problem can be solved by the active-set method in at most $2p$ iterations. Subsequently, we analyze the set of the optimal solutions of both problems, which is of the form of $\mathcal{C}=\{Y\in\R^{n\times p}:Y^TY=I_p, Y^T\Lambda Y=\Delta\}$ for $\Lambda$ and $\Delta$ diagonal and we address the problem how an arbitrary smooth function over $\mathcal{C}$ can be minimized. We find that a slight modification of $\mathcal{C}$ is a Riemannian manifold for which geometric objects can be derived that are required to make an optimization over this manifold possible. By using these geometric tools we propose then an augmented Lagrangian-based algorithm that minimizes an arbitrary smooth function over $\mathcal{C}$ and guarantees global convergence to a stationary point. Latter is shown by investigating when the LICQ (Linear Independence Constraint Qualification) is satisfied. The algorithm can be used to select a particular solution out of the set $\mathcal{C}$ by posing a new optimization problem. Finally we compare this algorithm numerically with a similar algorithm that, however, does not apply these geometric tools and that is to our knowledge not guaranteed to converge. Our results show that our algorithm yields a significantly better performance