I have a problem where I need to minimize the following function:
minimize: $\max(|F z - A x|, |B x - C y|, |D y - E z|)$
where A,B,C,D,E,F are constants and x and y are the variables to be defined. z in my specific case will always be 1:
minimize: $\max(|F - A x|, |B x - C y|, |D y - E|)$
I want to find the values for x and y that minimize the maximum diference between F and A; B and C; and D and E.
I know how to solve it using Linear Programming, but I was wondering if it can be solved using a closed form solution like least squares, replacing abs by squared and max by the sum. This approximation would lead to a simmilar result, but I'm having throuble modeling it:
minimize: $|F - A x|^2 + |B x - C y|^2 + |D y - E|^2$
The first derivative should be zero and the second, positive.
Thanks for any help.
I will assume that $A$, $B$, $C$ and $D$ are nonzero as otherwise the solution is trivial.
It is easy to show that there is an optimal solution where all three absolute values have the same value. That means you can get the optimal solution by solving eight $3 \times 3$ linear systems ($\pm(F-Ax)=t, \pm(Bx-Cy)=t, \pm(Dy-E)=t$ for all 8 combinations of $\pm$). This is essentially the simplex method for linear optimization, but easier yet often less efficient.