Multiplication of two variables, Linear or Nonlinear?

Question

This should be very easy but I am confused as I am getting different answers in different sources.

Say I have an equation:

$z = x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4 + x_5 y_5 $

where, $0 < x_k \leq 2, 0 \leq y_k \leq 1, k = 1,2,...5$

the question is $z$ linear or nonlinear? For me, it should be nonlinear as two variables are multiplied. But I have got some different answers too. So I need a better explanation. The reason I am asking this is I have an optimization problem as :

$\min \sum_{k} x_k y_k$

Now, I cant decide whether I should go for linear or nonlinear optimization for this problem.

Answer 1

Consider $x$ to be the vector $(x_1, x_2, x_3, x_4, x_5)$, and similarly for $y$, then $z = x \cdot y$, which as Matt Samuel has pointed out, is a bi-linear function. As with all inner products, its value is $\|x\|\|y\|\cos \theta$ where $\theta$ is the angle between the two vectors. Which should make solving your optimization problem easier.

Answer 2

If you optimize over both $x$ and $y$, this is a nonlinear optimization problem. If you fix either $x$ or $y$ and optimize with respect to the other variable, it is a linear optimization problem.