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$$ \begin{array}{ll} \underset {x, y} {\text{minimize}} & f(x,y) := | a x - b y | \\ \text{subject to} & 0 \le x \le C_1 \\ & 0 \le y \le C_2 \end{array} $$

What is the minimum? I found a similar question, Minimum of $|az_x-bz_y|$, but their $x$ and $y$ are integers. Thus, I guess the question is very different.

I started by computing partials: \begin{align} \frac{df}{dx}=\frac{a(ax-by)}{|ax-by|}=a\, {\mathrm{sign}(ax-by)}\\ \frac{df}{dy}=\frac{b(by-ax)}{|ax-by|}= b\, {\mathrm{sign}(by-ax)}\\ \end{align}

Having difficulty proceeding further with the Lagrange method. Is there a simpler way?

Rodrigo de Azevedo
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user173737
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1 Answers1

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There is a simpler way:

$f(x,y) \ge 0$ and $f(0,0)=0$ so the minimum is $0$.

You also have $f\left(x,\dfrac{a}{b}x\right)=0$, which may provide other ways of achieving the minimum. At these points, there may not be a well-defined derivative.

Henry
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  • Are there other minimums. For example, if we exclude point $(x,y)=(0,0)$. – user173737 Sep 04 '14 at 13:14
  • Also, do you think there is an analytical solution. Like Lagrange method??? – user173737 Sep 04 '14 at 13:23
  • There are other minima when $\left(x,\dfrac{a}{b}x\right)$. Your derivatives are telling you that when $ax \gt by$ then reduce $x$ or increase $y$, and when $ax \lt by$ then reduce $y$ or increase $x$. – Henry Sep 04 '14 at 14:03