I'm using MANOPT's (MATLAB toolbox for Riemannian optimization) implementation of the Riemannian trust region method from
Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization algorithms on matrix manifolds., Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008). ZBL1147.65043.
to minimize a smooth, partially separable cost function $$f(\mathbf{x}_1,\ldots,\mathbf{x}_n)=\sum_{i,j} f_{i,j}(\mathbf{x}_i,\mathbf{x}_j)$$ on the product manifold $\mathcal{N}=\mathcal{M}^n$ where $\mathcal{M}\subset\mathbb{R}^m$ is a regular submanifold. In other words: each $\mathbf{x}_i$ is an element of $\mathcal{M}$. In general, the matrix representation (in the ambient space) of the Riemannian Hessian of the cost must be assumed to be indefinite. Due to the partially separable structure it is a sparse block matrix.
Unfortunately the performance of the algorithm in terms of execution time is very unsatisfactory. It performs lots of inner iterations (Steihaug-Toint CG), often exceeding the maximum number of inner iterations allowed. Checking the eigenvalues of the Hessian shows that they are scattered over a wide range and some are quite close to zero, which would suggest using a preconditioner. The MANOPT tutorial states, that the preconditioner should be a symmetric positive definite operator on the tangent space at the current (outer) iterate.
I've implemented a preconditioner $P$ based on an incomplete Cholesky factorization of the Hessian $$ H+\alpha I = LL^\top+R$$ with a diagonal shift $\alpha$ s.t. $H+\alpha I>0$ (i.e. Cholesky works). Currently $\alpha$ is selected by trial and error, starting with an estimate of the absolute value of the most negative eigenvalue and gradually increasing the parameter until the decomposition succeeds. The preconditioner itself is given by: $$P=P_kLL^\top P_k$$ where $P_k$ is the orthogonal projection onto the tangent space at the k-th (outer) iterate. With this preconditioner I'm indeed observing a reduced inner iteration count. However, given the same initial iterate, the optimization is now converging to a different solution.
Now my question is, whether this is normal/can be expected to happen and if yes, why? Could this be an issue of several local minima (the cost definitely has several local minima)? Moreover I'm wondering whether a different approach to the preconditioning issue might give better results. I've found few publications dealing with general-purpose preconditioners for the Riemannian trust region method and none for the indefinite Hessian case. Any input would be appreciated!