Theorem Proof:
Theorem:
Consider the simplex method for finding a better BFS. Then the following are
true:
1.
2.
Our new B = B remove r add k will also be a basis.
When our BFS is non-degenerate, our new z-value gets smaller in the
next iteration.
Theorem
Theorem Proof:
Theorem: Consider an LP that is in standard form. The LP is feasible iff it has a
BFS. (ie it will have an extreme point when it is feasible!)
(Remember, standard form forces Rank(A) = k = number of rows)
Proof:
We first start by choosing a
Theorem Proof:
Theorem: Consider an LP in standard form. All Extreme points are BFS.
Proof:
We start with a BFS x* and show that it is an extreme point.
We will want to take advantage of the following theorem.
Corollary: Let S be a set of the form S cfw_x
Theorem Proof:
Theorem: Consider an LP that has an extreme point and has an optimal solution
(ie not unbounded). Then one of the extreme points must be an optimal
solution.
Proof:
We will assume once more that our LP is in semi-FM form:
Min
cTx
s.t Ax b
L
Theorem Proof:
Theorem: Let S be a set of the form S cfw_x R n | Ax b . Let x* be a feasible solution, and
let A= represent the matrix where x* holds as equality from A. x* is an extreme
point iff Rank(A=) = n.
Proof:
First we start with proving the forwa
Theorem Proof:
Theorem:
Consider an LP problem P and its dual problem D. Let the objective function
for P be cTx and the objective function for D be bTy. Then the following hold
true:
1.
2.
3.
4.
5.
cT x bT y
When x and y are feasible in their respective
Theorem Proof:
Theorem: The intersection of two convex sets is convex.
Proof:
Let A and B be two convex sets, and let C be the intersection of A and B.
Consider any two points in C : x and y.
Let z be any choice of lambda such that: z = x + (1-)y where
0
Theorem Proof:
Fundamental Theorem for LPs:
Any linear program falls in one of the following 3 categories:
1.
2.
3.
The LP is unbounded, and has no solution.
The LP is infeasible, and has no solution.
The LP has an optimal solution.
To prove this theorem,
Theorem Proof:
min
Theorem: Consider the LP given:
( LP1)
c1 x1 c2 x2 . cn xn c
st
c1,1 x1 c1, 2 x2 . c1,n xn b1
.
cm,1 x1 cm, 2 x2 . cm ,n xn bm
Solving the above LP will result in the exact same solution when solving the LP below
min
z
c1 x1 c2 x2 . cn
Theorem Proof:
Theorem:
A system of linear inequalities is feasible if and only if the Fourier-Motzkin Elimination
Algorithm does not produce any contradictory inequalities.
Proof:
If a system of inequalities is feasible
There is an x* that satisfies all
Theorem Proof:
Theorem: Solving the Auxiliary Problem using the Simplex Method will
have two outcomes:
1.
The optimal solution will be 0 (thus all of the s variables are 0) and the
resulting BFS (not including the s-variables) will be a BFS in the origina
Theorem Proof:
Theorem:
1. G is a tree iff it is connected and has |A| =|N| - 1.
2. If G is a tree, then for each a,b in N, there is a unique path from a to b.
3. If G is a tree, then adding 1 arc will produce 1 unique cycle.
Proof:
2. We will prove 2 tru
Names (print):
_
_
_
_
Student Numbers:
_
_
_
_
Tutorial #1
Instructions: You may work in groups of up to 4 people answering the questions below. The tutorial is marked
out of 1 (either 0 or 1). If you get at least half of the answers correct, you will re
Test #1: (
/30)
Name (print): _
Student Number:
Part A: Short Answer (1 Mark Each)
Question 1: Which of the following inequalities can be implied by
a)
b)
c)
e)
None of the above
_
d)
Question 2: The variables in the dual correspond to _ in the primal pro
Theorem Proof:
Theorem:
Given any node incidence matrix of a connected network whose b-vector sums to
0, then we can remove any row and the resulting matrix (and remove the
corresponding b-value from b) and the new LP will have an A with full row rank and