Linear Programming Lecture Notes - Penn State Personal Web Server

3. Matrices and Linear Programming Expression
You will recall from your matrices class (Math 220) that matrices can be used as a short
hand way to represent linear equations. Consider the following system of equations:
⎧
a11x1 + a12x2 + · · · + a1nxn = b1
⎪⎨ a21x1 + a22x2 + · · · + a2nxn = b2
(3.12)
.
⎪⎩
am1x1 + am2x2 + · · · + amnxn = bm
Then we can write this in matrix notation as:
(3.13) Ax = b
where Aij = aij for i = 1, . . . , m, j = 1, . . . , n and x is a column vector in R n with entries
xj (j = 1, . . . , n) and b is a column vector in R m with entries bi (i = 1 . . . , m). Obviously,
if we replace the equalities in Expression 3.12 with inequalities, we can also express systems
of inequalities in the form:
(3.14) Ax ≤ b
Using this representation, we can write our general linear programming problem using
matrix and vector notation. Expression 2.1 can be written as:
⎧
⎪⎨ max z(x) =c
(3.15)
⎪⎩
T x
s.t. Ax ≤ b
Hx = r
For historical reasons, linear programs are not written in the general form of Expression
3.15.
Definition 3.12 (Canonical Form). A maximization linear programming problem is in
canonical form if it is written as:
⎧
⎪⎨ max z(x) =c
(3.16)
⎪⎩
T x
s.t. Ax ≤ b
x ≥ 0
A minimization linear programming problem is in canonical form if it is written as:
⎧
⎪⎨ min z(x) =c
(3.17)
⎪⎩
T x
s.t. Ax ≥ b
x ≥ 0
Definition 3.13 (Standard Form). A linear programming problem is in standard form
if it is written as:
⎧
⎪⎨ max z(x) =c
(3.18)
⎪⎩
T x
s.t. Ax = b
x ≥ 0
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