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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?

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  • $\begingroup$ Thank you @Dr.MV, may I ask how you numerically approximated the maximum? What about the minimum? $\endgroup$ – Alvey Mar 16 '15 at 23:18
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Hint: Note that the minimum must occur on the boundary, because if you get a negative result and you are not on the boundary, then you can scale the answer up by a constant factor and get on the boundary with a lower objective value, because your objective function is a homogeneous polynomial. So Lagrange multipliers are probably the way to go here, unless there's some trick hidden in the structure of the terms in the objective.

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