For given real matrix $H,T\in \mathbb{R}^{n,n}$ ($H$ is full column rank, and thus $H^\top W H$ is well-defined), we intend to find a symmetric positive definite matrix $W\in \mathbb{R}^{n,n}$ such that the following optimization problem is solved:
$$\min_{W\in\mathbb{S}^{n}_+} \|(H^\top W H)^{-1}H^\top WT\|_2$$
where $\mathbb{S}^{n}_+$ is the set of symmetric positive-semidefinite matrices of order $n$. However, this problem is not easily solved by off-the-shelf solvers (I am not sure if it is convex). I intend to formulate it as an SDP problem. Firstly, it is equivalent to the following problem:
$$\min_{t>0,W\in\mathbb{S}^{n}_+} t \text{ such that } $$ $$T^\top W H (H^\top W H)^{-\top}(H^\top W H)^{-1}H^\top WT \preceq t I$$
By the result of the Shcur complement it is equivalent to：
$$\min_{t>0,W\in\mathbb{S}^{n}_+} t \text{ such that } $$ $$\begin{bmatrix}(H^\top W H)^2 & H^\top WT\\ T^\top W H& tI\end{bmatrix}\succeq 0$$
However, due to the quadratic term $(H^\top W H)^2$, this is not an SDP problem and can not be directly solved by convex problem solvers. Are there any ideas to reformulate it to convex opt problem or how to solve it?