1.225, 11/19/02
Lecture 6, Page 3
Optimization: Mathematical Programs
General formulation (
n
variables,
m
constraints)
:
:
decision variables
:
A constraint
Notes:
)
,...,
,
(
min
2
1
n
x
x
x
z
≥
≥
≥
m
n
m
n
n
b
x
x
x
g
b
x
x
x
g
b
x
x
x
g
)
,...,
,
(
)
,...,
,
(
)
,...,
,
(
2
1
2
2
1
2
1
2
1
1
M
b
x
x
x
g
b
x
x
x
g
b
x
x
x
g
b
x
x
x
g
b
x
x
x
g
n
n
n
n
n
−
≥
−
≥
⇔
=
−
≥
−
⇔
≤
)
,...,
,
(
and
)
,...,
,
(
)
,...,
,
(
)
,...,
,
(
)
,...,
,
(
2
1
2
1
2
1
2
1
2
1
Objective function
Feasible set
Subject to (s.t.):
)
,...,
,
(
2
1
n
x
x
x
j
n
j
b
x
x
x
g
≥
)
,...,
,
(
2
1
)
,...,
,
(
z(x)
)
,...,
,
(
ax
2
1
2
1
n
n
x
x
x
f
Min
x
x
x
f
M
−
=
=
1.225, 11/19/02
Lecture 6, Page 4
Types of Mathematical Programs (MPs)
Types of Mathematical Programs (MPs)
Linear programs (LPs)
: objective function is linear, and
constraints are linear
Non-linear programs (NLPs)
: objective function is linear.
(constraints are usually linear. Otherwise, there might be more than
one optimal solution (finding such a solution can be a very time
consuming task))
If decision variables are further constrained to take integer values, a
linear program is an
integer program
If decision variables are constrained to take 0/1 values: an integer
program is an
0/1 integer program
If some, but not all, variables are constrained to take integer values: a
linear program is called a
mixed integer program
2