Specialized Linear Program

[anal...@hotmail.com]

Max sum over j c(j).x(j)
St
sum over j a(i,j) . x(j) <= b(i) for i = 1,2,..m.
x(j) >= 0 for j = 1,2,...n.

All a's,b's and c's are >= 0.

This LP has the property that x feasible implies x' feasible whenever
0<=x'(j) <= x(j) for all j.

Are these problems any easier than general LPs?

[quasi]

analyst41 wrote:
>
>Max sum over j, c(j).x(j)
>St
> sum over j, a(i,j).x(j) <= b(i) for i = 1,2,..m.

> x(j) >= 0 for j = 1,2,...n.
>
>All a's,b's and c's are >= 0.
>
>This LP has the property that x feasible implies x' feasible

>whenever 0 <= x'(j) <= x(j) for all j.

>
>Are these problems any easier than general LPs?

Not as far as I can see.

The only difference is that you have a known vertex of the
feasible region, namely the origin, and that the feasible region
is entirely in the first orthant.

Given any LP with a known vertex of the feasible region, a
change of coordinates can be used to achieve another LP
in the form you specify above.

quasi

[quasi]

quasi wrote:
>analyst41 wrote:
>>
>>Max sum over j, c(j).x(j)
>>St
>> sum over j, a(i,j).x(j) <= b(i) for i = 1,2,..m.
>> x(j) >= 0 for j = 1,2,...n.
>>
>>All a's,b's and c's are >= 0.
>>
>>This LP has the property that x feasible implies x' feasible
>>whenever 0 <= x'(j) <= x(j) for all j.
>>
>>Are these problems any easier than general LPs?
>
>Not as far as I can see.
>
>The only difference is that you have a known vertex of the
>feasible region, namely the origin, and that the feasible region
>is entirely in the first orthant.

Forget the first orthant part -- that's automatic just from
x(j) >= 0 for all j.

>Given any LP with a known vertex of the feasible region, a
>change of coordinates can be used to achieve another LP
>in the form you specify above.

I'm not so sure about the above.

Maybe there's some advantage after all.

quasi

[anal...@hotmail.com]

On Apr 26, 3:09 am, Ray Vickson <RGVick...@shaw.ca> wrote:

> They are not much easier than a general LP; we do not need to worry about whether or not it is feasible, since the all-slack solution x(j) = 0 for all j is certainly a basic feasible solution, which means that we can start the simplex method right away (avoiding Phase I, for example). However, the pathological examples that show exponentiality of the the simplex method are precisely of that form, so your example is already of worst-case-type.- Hide quoted text -
>
> - Show quoted text -

There doesn't seem to be an obvious way to reduce a general LP to this
form. Even if it turns out that this form is as hard as the general
case, perhaps you can get approximate solutions a lot easier (for
example, in so-called Online LP problems in which the columns get
revealed one by one).