On Apr 26, 3:09 am, Ray Vickson <RGVick...@shaw.ca> wrote: > On Thursday, April 25, 2013 6:32:44 PM UTC-7, anal...@hotmail.com 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? > > 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).