# What is the minimum of the following function in 8 variables, with the given constraints?

I need to minimize the following function: $f : \mathbb{R}^8 \to \mathbb{R}$, defined by \begin{align*}f(x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4)& =x_1x_2x_3x_4 + x_3 x_4 y_1 y_2 - x_2 x_3 y_1 y_4 + x_2 x_4 y_1 y_4 + x_1 x_3 y_2 y_4\\ &- x_1 x_4 y_2 y_4 + x_1 x_2 y_4^2 + y_1 y_2 y_4^2 \end{align*} with the constraint: $$x_1^2+x_2^2+x_3^2+x_4^2+y_1^2+y_2^2+y_3^2+y_4^2 \leq 1$$

I tried a few points which satisfy the constraint, and got a minimum of $\frac{-1}{16}$, but I am having trouble proving this rigorously.

I first tried letting Mathematica compute it but it would not terminate. Is there any quick way to do it rather than do calculus-style optimization in the interior or Lagrange multiplier on the boundary?

• Thank you @Dr.MV, may I ask how you numerically approximated the maximum? What about the minimum? Mar 16, 2015 at 23:18