How to optimize $f(x,y)=\sum_{i=1}^Mx_{i}y_{i}$, given that $\sum_{i=1}^Mx_i=a>0$, $\sum_{i=1}^My_i=b>0$, $x_i > 0,y_i> 0$ for all $i$.

Question

I need to optimize

$f(x,y)=\sum_{i=1}^Mx_{i}y_{i}$

such that

$\sum_{i=1}^Mx_i=a>0$

$\sum_{i=1}^My_i=b>0$

$x_i > 0,y_i> 0$ for all $i$.

1. I have a feeling that the point $x_i=\frac{a}{M}$ and $y_i=\frac{b}{M}$ achieves the optimal, but haven't found a way to prove/disprove it.
2. Can this be formulated as a standard convex optimization problem?

Answer 1

$$0<\sum_{i=1}^Mx_iy_i\leq\sum_{i=1}^Mx_i\sum_{i=1}^My_i=ab$$ For $x\rightarrow(a,0,...,0)$ and $y\rightarrow(0,b,0,...,0)$ we get the infimum

and for $x\rightarrow(a,0,...,0)$ and $y\rightarrow(b,0,...,0)$ we get the supremum.