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The simplex algorithm

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Title: The simplex algorithm

1
The simplex algorithm

Section 29.3
Presented By Doug Nickerson
11/20/07

2
The simplex algorithm

Classical method for solving linear programs
(LPs)
Worst case running time is not polynomial
However, its quite fast in practice (most cases)
Provides insight into linear programs via its
geometrical interpretation as well as similarity
to Gaussian elimination
Could be called Gaussian elimination for
inequalities

3
The simplex algorithm

Iterates through several equivalent slack forms
of the linear program
Each iteration- increases the objective value
by at least 0- selects one non-basic and one
basic variable to swap- rewrites the LP in a new
(but equivalent) slack form which has the 2
variables swapped
A basic solution- results from setting all
non-basic variables to value 0- can be easily
obtained from the slack form- always corresponds
to a vertex of the simplex
Hence the simplex algorithm is simply walking
from vertex to vertex on the simplex

4
An example of the simplex algorithm

Lets run the simplex algorithm on an example
LP
Any setting of the non-basic variables (x1, x2,
x3), will yield and values for the basic
variables (x4, x5, x6)
The basic solution here is (x1, x2, , x6)
(0,0,0,30,24,36) and objective value z
3010200

Standard form
Slack form
5
An example of the simplex algorithm

Goal is to rewrite a new slack form which has a
basic solution that increases the objective value
z.
Note, a basic solution is a feasible basic
solution if the basic variables satisfy the
non-negativity requirements.
Choose a non-basic variable that has a positive
coefficient in the obj. value expression say x1
in this case
Increasing x1 increases z, but one of the
constraints will limit x1 more than the others
in this case, the last one

6
An example of the simplex algorithm

Now, weve determined the 2 variables to swap x1
will become a basic variable and x6 will become
non-basic.
Solve the most restrictive constraint for x1
Then substitute that into the other constraints
and the objective value equation to get the new
slack form

7
An example of the simplex algorithm

Simplify and we have the result of the first
iteration
Note the constant in the objective value
expression 27-This is the new objective value
for the new basic solution (x1, x2, , x6)
(9,0,0,21,6,0)
This procedure is called pivoting and is at the
heart of the simplex algorithm

Initial slack form
After x1 and x6 were swapped
8
An example of the simplex algorithm

Now we repeat- select a non-basic variable with
a positive coefficient in the z-equation we
call it the entering variable- determine which
constraint is the tightest and call the
basic variable of that constraint the leaving
variable- perform the pivot to get the new
slack form
New basic solution (33/4, 0, 3/2, 69/4, 0, 0)
gives new objective value z 111/4 27.75 gt 27

9
An example of the simplex algorithm

Finally, we only have one choice of the entering
variable
Final slack form
New basic solution (8, 4, 0, 18, 0, 0) gives new
objective value z 28 gt 27.75

10
Pivoting

Fundamental procedure of simplex algorithm
Input (N, B, A, b, c, v, l, e)- N and B are
lists of the non-basic and basic variables- A is
the matrix of constraint coefficients- b is the
vector of constants in the constraints- c is the
vector of coefficients in the objective
equation- v is the constant in the objective
equation- l and e are the indices of the leaving
and entering vars.

11
Pivoting

Output (N, B, A, b, c, v) which describes
the new slack form having variables xe and xl
exchanged.
Pseudo-code part 1 (page 795)
These lines determine the coefficients of the new
constraint with xe as the basic variable.
It essentially solves the equation for xe

Ex N1,2,3, e1, l6) x636 4 x1 x2
2 x3 b6 a61x1 a62x2 a63x3 x1
b6/a61 a62/a61 x2 a63/a61 x3
x6/a61 b1 a12x2 a13x3 a16x6
PIVOT(N, B, A, b, c, v, l, e) be ? bl / ale
for each j N e do aej ? alj /
ale ael ? 1 / ale
12
Pivoting

Pseudo-code part 2
These lines determine the coefficients of the
remaining constraints (the ones where the basic
variable remains).
It performs the substitution of xe and resulting
simplification (collecting coefficients).

Ex N1,2,3, B4,5,6, e1, l6) x430
x1 x2 3 x3 b4 a41x1 a42x2
a43x3 bi aie x1 aij x2 aij
x3 x1be aejx2 aejx3 aelx6 x4 bi
aie () aij x2 aij x3 (bi- aiebe)
(aij-aie aej)x2 (aij-aie aej)x3
aieaelx6
for each i B l do bi ? bi aie
be for each j N e do
aij ? aij aie aej ail ? - aie ael
13
Pivoting

Pseudo-code part 3
The first 4 lines determine the coefficients of
the objective expression by doing the
substitution of xe and resulting simplification
(collecting coefficients).
The last 3 lines simply swap the e and l indices
in the sets N and B and return the new slack form.

Ex N1,2,3, B4,5,6, e1, l6) case i
4 z 0 3 x1 x2 2 x3 v c1x1
c2x2 c3 x3 v ce() cjx2 cj
x3 x1be aejx2 aejx3 aelx6 z (v - ce
be) (cj- ce aej)x2 (cj- ce aej)x3
ce aelx6
v ? v ce be for each j N e do cj
? cj ce aej c l ? - ce ael N(N-e) U
l B(B-l) U e return (N, B, A, b, c,
v)
14
Pivoting

Time complexity of PIVOT- Contains 2 loops over
j and a double nested loop over i and j plus
some other O(1) operations- Each j-loop iterates
over N-e which has N-1n-1- The i-loop
iterates over B-l which has B-1m-1-
Therefore T(n,m) 2(n-1) (n-1)(m-1) O(1)
nm n m
O(1)- Clearly nm dominates, hence O(nm)
Lemma 29.1

15
The formal simplex algorithm