I'm working with the following optimization problem below. $$\min_{\Pi} \left[ \frac{1}{4 \lambda }\left((\Pi\vec{1}-s)^T K(\Pi\vec{1}-s) + \left(\Pi^T \vec{1}-t\right)^T K \left(\Pi^T \vec{1}-t\right)\right)-\text{Tr}[\Pi K] \right]$$
Here, $\lambda$ is a scalar, $\Pi$ and $K$ are $n\times n$ matrices, and $s,t$ are $n\times 1$ probability vectors (i.e. the components sum up to $1$). Also $K$ is PSD and $\Pi$ has elements summing up to $1$.
I was able to empirically show that the diagonal elements of $\Pi^{*}$, the optimal $\Pi$, correspond to $\frac{s+t}{2}$ regardless of the choice of $\lambda$, but I am having difficulty showing this to be the case theoretically. Does anyone have any advice? I tried for the case when $n=2$ using Lagrange multipliers and even then it gets very messy.
I set up a Lagrange multipier where the matrix constraint $1^T \Pi 1=1$, and I end up getting as my first order conditions
$$2K [(\Pi+\Pi^T) \vec{1}-(s+t)] \vec{1}^T - K - \lambda \vec{1}\vec{1}^T = 0$$ and $$1-\vec{1}^T \Pi \vec{1}=0.$$
I just don't see where to go from here. I'd like to use these to show $\vec{diag(\Pi^*)} = \frac{1}{2}(\vec{s+t})$. Also, how do I come up with KKT duals when every element of $\Pi$ must be nonnegative?
Anyone have any ideas?