I have the following problem:
Define $H$ and $R_k$ for $k=1\dots N$, to be $M\times M$ positive definite matrices.
The problem is to find optimal weights $p_k$that solves the following problem
\begin{equation*} \begin{aligned} & \underset{p}{\text{minimize}} & & tr\left(\sum_{k=1}^N p_k^2 H^{-1}R_k\right) \\ & \text{subject to} & & \sum_{k=1}^Np_k = 1, \;\;\; p_k>0 \;\;\;\forall \; k \end{aligned} \end{equation*}
I know that the solution is given by the following
\begin{equation*} p^0_k = \frac{1}{tr\left(H^{-1}R_k\right)} \left( \sum_{l=1}^N\frac{1}{tr\left(H^{-1}R_l\right)}\right)^{-1}. \end{equation*}
It is obvious that $\sum_{k=1}^Np^0_k = 1$ and that $p^0_k>0 \;\;\;\forall \; k$, so the given solution satisfies the constraint. But how to show that it minimizes the trace?
A constrained optimisation problem can be tackled by the use of Lagrangian multipliers.
The Lagrangian of the problem is given as
$$\Lambda=tr(\sum_{k=1}^Np_k^2H^{-1}R_k)+\lambda(1-\sum_{k=1}^Np_k)$$
where $\lambda$ is known as the Lagrange multiplier.
In order to find the values of $p_k$ that minimises $tr(\sum_{k=1}^Np_k^2H^{-1}R_k)$ we need to solve for the following sets of equations
$$\frac{\partial \Lambda}{\partial p_m}=2(tr(H^{-1}R_m))p_m-\lambda=0\text{ for } m\in\{1,2,..,N\}$$
$$\frac{\partial \Lambda}{\partial \lambda}=1-\sum_{k=1}^Np_k=0$$
We have
$\large p_m^0=\frac{\lambda}{2(tr(H^{-1}R_m))}$ for $m\in\{1,2,..,N\}$
where $p_m^0$ is the optimal value of $p_m$ that minimises the cost function.
Summing across the $p_k$
$\large \sum_{k=1}^Np_k^0=\lambda(\sum_{k=1}^N\frac{1}{2(tr(H^{-1}R_k)})=1$
leading to
$\large \lambda=\frac{1}{(\sum_{k=1}^N\frac{1}{2(tr(H^{-1}R_k)})}$
We can then substitute the expression for $\lambda$ back into the optimal values of $p_k$ resulting in
$\large p_k^0=\frac{\lambda}{2(tr(H^{-1}R_k))}=\frac{1}{(\sum_{k=1}^N\frac{1}{2(tr(H^{-1}R_k)})}\frac{1}{2(tr(H^{-1}R_k))}\\\large=\frac{1}{2(tr(H^{-1}R_k))}(\sum_{k=1}^N\frac{1}{2(tr(H^{-1}R_k)})^{-1}$