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I have an optimization problem with the following objective function in $(x,y)$ $$ A\log \left(\sum_{i=1}^n x_i\right)+\log\left(1-\frac{f}{n}\left(\sum_{i=1}^n\frac{x_i}{y_i}\right)\right) $$ where $x_i\geq 0$, $y_i\geq x_i$, and where $0<A<1$ and $0<f<1$ are constant. I would like to use Mathematica to verify its convexity.

I can compute the Hessian. Let's focus on the case with $n=2$:

F[x1_, x2_, y1_, y2_] = Log[x1 + x2] + Log[1 - 0.1*(x1/y1 + x2/y2)]
A := D[F[x1, x2, y1, y2], {{x1, x2, y1, y2}, 2}]

I can then compute the eigenvalues of A but they are a gigantic mess. Is there anyway I can test whether the eigenvalues are negative for a set of $(x,y)$? Any help would be greatly appreciated!

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