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Given an arbitrary number of variables $\epsilon_i$ that can be picked from a domain $[0, W]$ and some inequality relation between all the variables $G(\epsilon_1, \epsilon_2, ...)$, is there some way to use Mathematica to find the smallest possible value for $W$ such that $G$ holds true?

Example: Somewhere in the problem I'm solving I get the following relation between the values of a diagonal matrix perturbation with values $\{ \epsilon_1 ... \epsilon_4\}$ that must hold if I want certain properties of the perturbed eigenvectors to be true:

$\epsilon_1 > a(b - c \epsilon_2 - d\epsilon_3 + f\epsilon_4)$

where $a, b, c, d, f$ are positive constants. Is there a way to find the smallest bound on $\epsilon_i$, given these constants, so that I could pick values of $\epsilon_i$ that satisfy this inequality? Thanks.

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It would be nice to give an example for the constants $a$, $b$, $c$, $d$ and $f$. Here I use some random values:

SeedRandom[1]
{a, b, c, d, f} = RandomReal[10, 5];

You can use NMinimize to minimize $w$:

NMinimize[
    {
    w,
    ϵ1>a(b - c ϵ2 - d ϵ3+f ϵ4) && {ϵ1, ϵ2, ϵ3, ϵ4} ∈ Cuboid[{0,0,0,0}, {w,w,w,w}]
    },
    {w, ϵ1, ϵ2, ϵ3, ϵ4}
]

{0.112595, {w -> 0.112595, ϵ1 -> 0.112595, ϵ2 -> 0.112595, ϵ3 -> 0.112595, ϵ4 -> 0.}}

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