Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

Question

The problem seems to be easy but I can't find a solution :(

Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$.

Thank you in advance for any help.

Update:

Since $1^Tx=1^Ty=1$ we have $$x^TAy+b^Tx+c^Ty = x^TAy+(x^Tb)(1^Ty)+(x^T1)(c^Ty) = x^T(A+b1^T+1c^T)y.$$ Thus if we denote $D=A+b1^T+1c^T$ then the problem is reduced to: \begin{align} \mbox{minimize}\quad & x^TDy \\ \mbox{subject to}\quad & 1^Tx=1^Ty=1, \\ & x\ge 0,y\ge 0. \end{align}

Answer 1

The minimum value is the smallest component $d_{ij}$ of $D$, attained when $x_i=y_j=1$. I posted the solution here.