Question

I'm writing a computer program and I need to fit some IF..ELSE condition into mathematic model, so I can't use regular programming constructs. For example, how would I turn this into mathematic equation (or inequation):

if (3ax + 5by + 8cz >= 320) then
    w = 1.0;
else
    w = 0.8;

I understand how to express the domain of w to accept only values 0.8 and 1.0 by using:

(w - 1.0) * (w - 0.8) = 0

But I haven't got a clue how to transform the above IF statement. Is it even possible?

P.S. I'm rather new to math overflow, so I'm not sure which tags to assign to this question. Please feel free to re-tag appropriately.

Answer 1

You can express piecewise functions by using the unit step function. Your example would be:

$ w = 0.8 + 0.2 \times \textrm{unit_step}(3 a x + 5 b y + 8 c z - 320) $

Note that the unit step is usually defined to have a value of $ \frac{1}{2} $ at 0, but this is not generally an issue.

It may be more convenient to replace the unit step with the signum function, or with the function $ \mathrm{s}(x) = \frac{x}{|x|} $ which is equal to it almost everywhere; this is left as an exercise to the reader.

Answer 2

In the example you give, $w$ is simply a function of $a$,$x$,$b$,$y$,$c$, and $z$ (i.e., $w = f(a,x,b,y,c,z)$), so would the following be what you are looking for:

$f(a,x,b,y,c,z) = \begin{cases}1 & \text{if }3ax + 5by + 8cz \geq 320 \\ 0.8 & \text{otherwise}\end{cases}$

I would usually write $f$ as a multi-part definition, but cannot seem to figure out how to do that on mathoverflow (in Latex I would usually use an array).

Answer 3

You can use indicators or characteristic functions of sets. For a set $A$, you can define:

${\bf 1}_A(x)$ equals $1$ if $x\in A$, and equals $0$ if $x\notin A$.

Or, for a condition $A$ you can define ${\bf 1}_A$ to be $1$ if $A$ holds and $0$ if it does not. Then you can write things like

$f(x)=3\cdot {\bf 1}_{x\le 2}+10\cdot {\bf 1}_{x>2}$

for a function that is equal to $3$ for $x\le 2$ and to $10$ for $x>2$

Answer 4

There are many conventions. My favorite is the Iverson bracket where [foo] is 1 if foo is true and 0 otherwise.

For a slightly more general notation you can write $f(x)=\begin{cases}1,&\text{foo is true}\\0,&\text{foo is false}\end{cases}$

The characteristic function (or indicator function) is also used, as are various step functions, e.g., the Heaviside step function H or the sign function sgn.