Standard Form Linear Programs
max cx
s.t.
Ax = b
x 0
symmetric form
max
s.t.
⇥
3 2.5
2
⇤
x
max
3
2
4.44
0
6 0
6
6.67 7
6
7x  6
4 4
4
2.86 5
3
6
x
standard form
3
100
100 7
7
100 5
100
s.t.
⇥
3 2.5 0 0 0 0
2
4.44
0
1 0
6 0
6.67 0 1
6
4 4
2.86 0 0
3
6
0 0
x
0
⇤
0
0
1
0
x
3
2
0
6
0 7
7x = 6
4
0 5
1
3
100
100 7
7
100 5
100
“slack variables”
0
bly
em
ass
truck assembly
e
gin
en
x1 (thousands of cars)
basic feasible solutions
car assembly
me
tal
sta
m
pi n
g
x2 (thousands of trucks)
e62: lecture 8
2
6
6
6
6
6
6
4
0
0
100
100
100
100
1
3 2
7
7
7
7
7
7
5
6
6
6
6
6
6
4
22.5
0
0
100
10
32.5
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
22.5
3.5
0
76.6
0
11.5
3 2
7
7
7
7
7
7
5
“nonbasic variables”
6
6
6
6
6
6
4
20.4
6.5
9.4
56.6
0
0
3
7
7
7
7
7
7
5
20161020
Identification of a BFS
max cx
s.t.
Ax = b } M constraints
x 0 } N constraints
•
BFS = intersection of N linearly independent constraint boundaries
select N-M nonbasic
variables B.
N linearly
independent constraints?
no vertex
no vertex
•
vertex
feasible?
compute BS
Solving for a BFS given B
Ax = b
xB = 0
e62: lecture 8
2
4.44
6 0
6
4 4
3
0
6.67
2.86
6
2
1
0
0
0
0
1
0
0
0
0
1
0
3
2
0
6
0 7
7x = 6
4
0 5
1


x3
=
x5
3
100
100 7
7
100 5
100
0
0
2
6
6
6
6
6
6
4
22.5
3.5
0
76.6
0
11.5
3
7
7
7
7
7
7
5
20161020
Edges
•
Def. adjacent vertices: share N-1 constraint boundaries
An edge is a part of the line that satisfies these constraints
•
•
•
Swap one constraint to switch between adjacent vertices
In a standard form LP, swap one nonbasic variable
2
6
6
6
6
6
6
4
e62: lecture 8
22.5
3.5
0
76.6
0
11.5
3
7
7
7
7
7
7
5
adjacent
3
2
6
6
6
6
6
6
4
20.4
6.5
9.4
56.6
0
0
3
7
7
7
7
7
7
5
20161020
Traversing an Edge
2
4.44
6 0
6
4 4
3
2
6
6
6
6
6
6
4
e62: lecture 8
22.5
3.5
0
76.6
0
11.5
0
6.67
2.86
6
1
0
0
0
0
1
0
0
0
0
1
0
3
2
0
6
0 7
7x = 6
4
0 5
1


x3
=
x5
2
3
6
6
6
6
6
6
4
7
7
7
7
7
7
5
4
3
100
100 7
7
100 5
100
0
20.4
6.5
9.4
56.6
0
0
3
7
7
7
7
7
7
5
20161020
Reduced Profits
•
Def. reduced profit = objective value increase per unit
nonbasic variable increase
2
4.44
6 0
6
4 4
3
0
6.67
2.86
6
1
0
0
0
0
1
0
0
0
0
1
0
3
2
0
6
0 7
7x = 6
4
0 5
1


x3
=
x5
3
100
100 7
7
100 5
100
0
x3 !
x !y
cx ! cy
reduced profit =
e62: lecture 8
5
cy
cx
20161020
Algorithm and Termination
compute
reduced profits
maximum
positive?
no
terminate
yes
traverse
maximizing edge
•
Finite time termination
There are a finite number of vertices
•
e62: lecture 8
6
20161020
Optimality of Result
convex hull of
adjacent vertices
e62: lecture 8
7
20161020
Degeneracy and Cycling
•
Def. degeneracy = when there is a zero-valued basic variable
When there are more than N active constraints at a BFS
BFS lies at intersection of N+1 constraint boundaries
When one of the basic variables takes value 0
•
•
•
•
Def. cycling = keep swapping basic variables without changing
solution
•
Anticycling: logic to prevent cycling
e62: lecture 8
8
20161020