Re: [Help-glpk] function

[Michael Hennebry]

On Fri, 17 Aug 2012, Mate Hegyhati wrote:

> Hi Zvonko,
>
> maybe something like this?
>
> R >= 0
> R <= P - PMIN*X
> R <= PMAX*X - P

This fails for X=0.
R >= 0
PMAX >= P >= 0       // might be redundant
if X==1 P >= PMIN    // might be redundant
if X==1 R <= P - PMIN
if X==1 R <= PMAX - P
if X==0 P <= 0
if X==0 R <= 0
Add binary Y
if Y==0 R >= PMAX - P
if Y==1 R >= P - PMIN
Note that if X==0 then Y==1.
It won't hurt to put it in explicitly:
Y >= 1-X            // might be redundant

Use small M for the conditional constraints, e.g.:
if Y==1 R >= P - PMIN
R >= P - PMIN - M*(1-Y)
This is correct for Y==1 or M big enough.
Make M as small as possible and still big enough.
Since it works for Y==1 assume Y==0 and X==1:
R >= P - PMIN - M
M >= P - PMIN - R <= PMAX - PMIN - 0
Any M >= PMAX - PMIN is valid
Use PMAX - PMIN
Simply picking a huge M is the wrong way to go.
It generates notoriously loose linear constraints
and causes numerical problems.
R >= P - PMIN - (PMAX - PMIN)*(1-Y)
R >= P + (PMAX-PMIN)*Y - PMAX

As Mate Hegyhati noted, you do not necesarily
need Y and the constraints that involve it.
If, for example,
one is minimizing a functions that only has one local minimum,
one does not need upper bounds on the function.

> This should be enough, if R is not represented anywhere else, and it
> should be as much as possible in the objective.
>
> Otherwise I think You need an additional binary variable, Y,
> representing whether P is closer to PMIN or to PMAX.
>
> P <= (PMIN + PMAX) / 2 + (1-Y) * (PMAX-PMIN)/2
> P >= (PMIN + PMAX) / 2 - (1-Y) * (PMAX-PMIN)/2
>
> And then You can add these to the above constraints:
>
> R >= P - PMIN - Y * PMAX
> R >= PMAX - P - (1-Y)*PMAX
>
> Hope this is what You are looking for, best regards,
>
> Mate
>
> On 08/17/2012 10:02 AM, Zvonko Bregar wrote:
> > Hi,
> >
> > My name is Zvonko Bregar.
> >
> > Please help me with the following problem.
> >
> > There is a real variable P having disjunct definition area {0} + [PMIN, PMAX].
> > There is also a binary variable X helping in this definition.
> > If X = 0 then P = 0
> > And if X = 1 then PMIN < P < PMAX.
> > So variables X an P work simultaneously.
> > This should be a rather standard construction.
> >
> > The question is to construct another variable, say R, from X an P
> > Such that
> > If X = P = 0 then also R = 0
> > Else R = min {P-PMIN,PMAX-P}
> >
> > In other words, R measures the minimal distance of P to its boundaries PMIN and 
> > PMAX.
> > But if P = 0 then R must be set to 0 too.
> >
> > For example, if PMIN = 200 and PMAX = 400, than the required function should be something 
> > like the "upside down" absolute value function:
> >
> > R =  - | P - 300 | + 100  if 200 < P < 400
> >
> > But also R = 0 if P = 0.

--
Michael   address@hidden
"On Monday, I'm gonna have to tell my kindergarten class,
whom I teach not to run with scissors,
that my fiance ran me through with a broadsword."  --  Lily