I'm trying to minimize the following function:
$f(x) = \frac{a \cdot x}{min(b \cdot x, c \cdot x)}$
where $x, a, b$ and $c$ are vectors of $n$ values. For instance, $x = (1, x_1, x_2, ..., x_n)$.
The first problem is the min() function but it's not the only one because I have a bunch of constraints:
$d \cdot x$ < 16
$\frac{e \cdot x}{f \cdot x} > -4$
$\frac{g \cdot x}{f \cdot x} > 98$
$\frac{h \cdot x}{i \cdot x} > 30$
where $d, e, f, g, h$ and $i$ are vectors of $n$ values.
How can I minimize my $f$ function? I thought to SGD but with the min() function and the constraints, I'm not sure it can work.
A minimum will either be a minimum of $\frac{a\cdot x}{b\cdot x}$ or a minimum of $\frac{a\cdot x}{c\cdot x}$, or lie on the hyperplane defined by $b\cdot x=c\cdot x$ where the two functions are equal.
This gives you three simpler minimization tasks; after you have solved them check which of the three outcomes is best.
Each constraint like $\frac{e\cdot f}{f\cdot x} > 98$ can be split into two cases based on the sign of $f\cdot x$ to give $e\cdot x \gtrless 98 f\cdot x$. Also splitting on $i\cdot x$, this gives you $4$ regions to consider for each of the three tasks, where each of the regions is defined by linear constraints only.