I am facing following minimization problem: \begin{equation} \begin{aligned} &W'=\underset{F}{\text{argmin}} \| W-F \|_F^2 \\ s.t. ~ &F\mathbf{1}=\mathbf{1},~F=F^T \end{aligned} \end{equation} $\mathbf{1}$ is being a vector containing ones only. It's also worth mentioning, that from the constrains it follows that the matrix $F$ is doubly-stochastic. From the paper I found this problem in, I know that the Lagrangian to this problem is having following form:
$$L(F,\mu_1,\mu_2)=trace(F^TF-2W^TF)-\mu_1^T(F\mathbf{1}-\mathbf{1})-\mu_2^T(F^T\mathbf{1}-\mathbf{1})$$
I am totally lost how to form this Lagrangian equation out of the formulation given above. Can somebody help me out here?
EDIT:
Expanding the Frobenius norm and after dropping the constant I get following: \begin{equation} \begin{aligned} \| W-F\|^2_F &= trace((W^T-F^T)(W-F)) \\ &= trace(W^TW - W^TF - F^TW + F^TF) \\ &= trace(F^TF - W^TF - F^TW) \end{aligned} \end{equation} Is $F^TW = W^TF$ in this case? If so, I don't understand why that should be the case.