Simplex Example with n=1 and m=1

Simplex Example with n=1 and m=1 PhL 11.26.04 This subject appears in Scheid chapter 27. I got sufficiently confused that I am breaking out this separate document for more details. I need to understand the general idea in a very simple case. 1. Simplest case I can think of. n = 1 and m = 1 Let n =1 and our volume is a segment of the x1 axis between 0 and 5. We have m = 1 conditions in addition to x1 0 which is this: x1 5. We introduce a slack variable x2 and say x1 + x2 = 5. This is a line in 2D space. The line is our "boundary" in E2 , the extended space of E1 to include the slack variable. [ In the extended space E2 where n+m = 1+1=2, the feasible region is now the above triangle. ] We want to start off at a "vertex" of our space. Our choices are x1 = 0 or x1 = 5. Let's go with the latter. Then we have x = (5,0) // position vector in E2 to start off with, [ also on original region ] Now, we want to move in a [Scheid's ] k direction, which here can only be the x2 direction so k = 2. If we want to stay on the boundary. We want to go distance p in the x2 direction, so we are going to have x' = (5 - , p) // to be found below This must still be on the line, so we know that (5-) + p = 5, so = p. Thus, our point is x' = (5 - p, p) = ( x1', x2') What is our matrix form of the boundary conditions? [ 1 1 ] = 5 = [ v11 v12 ] = [ v1 v1 ] We have m=1 independent column vectors and it is just v1 = 1. Our H = x1c1 , that is all here is. [OK to here] So, we now want to "head off" in the x2 direction but we want to stay on the boundary [ in E2], so we wander up the line shown. Eventually we will hit a point where x1' = 0, and that is the top dot. We know that p = 5 will take us to that point. At that dot, we know that H = x1c1 = 0. If we are doing a max problem, this is surely smaller than the point we started with. Now look at some other Scheid equations: (3) vj = v1j v1 j = 1,2 which really says in our case v1 = v1 and v2 = v1 (4) hj = v1j c1 - cj j = 1,2 h1 = 0 h2 = c1 - c2 I now realize that this Scheid equation is "illogical" because there is no c2 in the problem, yet he writes it here. For the moment, I will pretend c2 exists and write this out as shown to the right above: (5) If p = 0, his equation says: x1 v1 = 5 or x1 = 5 If p > 0 we get instead: (x1 - pv12) v1 + p v2 = 5 or (x1 - p) + p = 5 What is the meaning of this equation? I used to thing we were going off in the x2 direction by amount p, but that is completely wrong. So my entire Scheid has now collapsed into nothingness! (6) (x1 - pv12) c + pc2 = x1 c1 - p [ c1 - c2 ] [ See official Scheid notes on this stuff. ]