How to solve the following optimzation by lagrange multiplier method?
$$\begin{array}{ll} \text{minimize(P)}& \| \mathbf{B} - \mathbf{P}\mathbf{A} \|_F^2\\ \text{subject to} & {\mathbf{P}}^T\mathbb{1} = \mathbb{1}\\ & \mathbf{P} \geq 0\end{array}$$
where $A$ and $B$ are given and $\mathbb{1}$ is a vector containing only $1$'s, can we solve ot by lagrange multiplier method ? Thanks so much.
Let says if $\mathbf{P} \geq 0$ are ignored now. Can we solve it by largrange multiple method?
design a largrange function: $L(P,\lambda) = \| \mathbf{B} - \mathbf{P}\mathbf{A} \|_F^2 + \lambda({\mathbf{P}}^T\mathbb{1} - \mathbb{1})$ And then use KKT complementary condition to solve it. Can we do that? I am total new about optimization. thanks.