Optimizing the sum of quadratic forms linked by orthogonality constraints

Question

I am looking to find a qualitative solution to the optimization problem:

$$\text{min}_{\{\mathbf{u}_i\}_i}\quad\sum_i \mathbf{u}_i^T\mathbf{M}_i\mathbf{u}_i \\ \text{s.t.}\quad \mathbf{u}_i^T\mathbf{u}_j = \delta_{ij}, \quad \forall i,j$$

where $\delta_{ij}$ is the Kronecker delta (so $\mathbf{u}_i \in \mathbb{R}^d$ are orthonormal) and all $\mathbf{M}_i \in \mathbb{R}^{d\times d}$ can be considered positive definite and symmetric.

My trouble here is that there a different matrices in each summand. If they all would be the same, the solutions should just be the eigenvectors corresponding to the smallest eigenvalues. Otherwise, if I wouldn't require the vectors to be orthogonal, the solutions would correspond to each smallest eigenvectors of each individual matrix $\mathbf{M}_i$.

I 'd be very grateful for any pointer -- for example, is there a name for these kind of problems? -- and sincerely hope that this is not too stupid of a problem.

Answer 1

I have some references if you want:

• Extrema of sums of heterogeneous quadratic forms
• On Minimization of Sums of Heterogeneous Quadratic Functions on Stiefel Manifolds