I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis. Specifically, I am concerned with the following convex optimization problem \begin{align*} &\min_{K_k, \Sigma_{k}} \ &&J_{\Sigma} = \sum_{k=0}^{N-1} \mathrm{tr}\big((Q_k + K_k^\intercal R_k K_k)\Sigma_{k}\big), \\ &\ \mathrm{s.t.} && \Sigma_{k+1} = (A_k + B_k K_k)\Sigma_{k} (A_k + B_k K_k)^\intercal, \\ &&&\Sigma_{N} = \Sigma_{f}, \quad \Sigma_{0} = \Sigma_{i}, \end{align*} where $\{A_k, B_k, Q_k, R_k\}_{k=0}^{N-1}$ and $\Sigma_{f}$ are problem parameters, and $K_k\in\mathbb{R}^{m\times n}, \Sigma_{k} \in\mathbb{R}^{n} \succ 0$. On the surface, this might seem like a nonlinear, non-convex program but it can be shown through change of variables and lossless relaxations that this is indeed a convex program and is semi-definite (SDP) representable.
However, I want to take a look at this problem from a more group-theoretic perspective, if possible. The constraints in this program involve covariance matrices, which are assumed to be positive definite symmetric. Thus, we aim for solutions on this "manifold" of positive definite matrices, I believe. I am not familiar with the properties of this manifold or if it possible to do optimization on it, but this is just my preliminary thoughts.
Also, for the covariance matrix propagation, we have constraints relating the covariance at time step $k+1$ to the covariance at time step $k$ through a "conjugation" of sorts on this manifold. After some preliminary reading, I believe it is possible to view the one-step covariance propagation of $\Sigma_{k}$ via $A_k + B_k K_k$ as an action of the Lie group $GL(n)$ (i.e., the group of invertible $n\times n$ matrices) on the manifold of symmetric positive definite matrices $\mathbb{S}_{++}^{n}$ through conjugation. To encode this into the optimization problem, we then need to recognize that since $\Sigma_{k}\in\mathbb{S}_{++}^{n}$ the optimization variables $K_k$ must be chosen such that the propagated $\Sigma_{k+1}$ remains in $\mathbb{S}_{++}^{n}$. That is, the group action through conjugation $G_k \cdot \Sigma_{k} = G_k \Sigma_k G_k^\intercal$, where $G_k \triangleq A_k + B_k K_k$ must preserve symmetric positive definiteness of $\Sigma_{k}$. The way to encode these constraints I'm not sure about, though.
TL-DR: Is it possible to formulate my (convex) optimization problem in a group-theoretic setting or as an optimization over the manifold of symmetric positive definite matrices? If so, is it tractable to solve and are there methods to solve these types of problems efficiently with constraints?