Math 171A: Linear Programming
Lecture 20
LPs with Mixed Constraint Types
Philip E. Gill
c 2011
http://ccom.ucsd.edu/~peg/math171a
Wednesday, February 23rd, 2011
Recap: LP formulations
Problems considered so far:
ELP
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b
LP
minimize
x
c
T
x
subject to
Ax
≥
b
Now we consider a
mixture of constraint types
.
UCSD Center for Computational Mathematics
Slide 2/32, Wednesday, February 23rd, 2011
Linear programs with mixed constraints
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b

{z
}
equality constraints
,
Dx
≥
f

{z
}
inequality constraints
The dimensions are:
A
m
×
n
b
m
vector
D
m
D
×
n
f
m
D
vector
UCSD Center for Computational Mathematics
Slide 3/32, Wednesday, February 23rd, 2011
Optimality conditions
Example: Consider inequalities and just
one
equality constraint:
minimize
x
∈
R
n
c
T
x
subject to
a
T
x
=
b
,
Dx
≥
f
Write the equality constraint as two inequalities:
a
T
x
≥
b
and
a
T
x
≤
b
,
which is equivalent to
(

a
)
T
x
≥ 
b
If
x
*
is an optimal solution, then both
a
T
x
≥
b
and
(

a
)
T
x
≥ 
b
must be active at
x
*
.
UCSD Center for Computational Mathematics
Slide 4/32, Wednesday, February 23rd, 2011
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Optimality conditions
Suppose that
x
*
is a solution of the mixedconstraint problem:
minimize
x
∈
R
n
c
T
x
subject to
a
T
x
≥
b
,
(

a
)
T
x
≥ 
b
,
Dx
≥
f
This problem has all inequalities, so we can use existing theory.
Let
D
a
denote the matrix of active inequalities at
x
*
.
The full active set for the mixedconstraint problem is
a
T

a
T
D
a
x
*
=
b

b
f
a
UCSD Center for Computational Mathematics
Slide 5/32, Wednesday, February 23rd, 2011
Optimality conditions
From the previous slide:
”
A
a
” =
a
T

a
T
D
a
The optimality conditions
A
T
a
λ
a
=
c
, with
λ
a
≥
0 are:
c
=
a

a
D
T
a
λ
*
1
λ
*
2
z
*
a
,
with
λ
*
1
≥
0
,
λ
*
2
≥
0 and
z
*
a
≥
0
UCSD Center for Computational Mathematics
Slide 6/32, Wednesday, February 23rd, 2011
Optimality conditions
From the previous slide:
c
=
a

a
D
T
a
λ
*
1
λ
*
2
z
*
a
,
with
λ
*
1
≥
0
,
λ
*
2
≥
0 and
z
*
a
≥
0
c
=
a
λ
*
1

a
λ
*
2
+
D
T
a
z
*
a
=
a
(
λ
*
1

λ
*
2

{z
}
positive or negative
) +
D
T
a
z
*
a
=
a
π
*
+
D
T
a
z
*
a
,
with
π
*
=
λ
*
1

λ
*
2
UCSD Center for Computational Mathematics
Slide 7/32, Wednesday, February 23rd, 2011
Optimality conditions
From the previous slide:
c
=
a
π
*
+
D
T
a
z
*
a
,
with
z
*
a
≥
0
π
*
is the Lagrange multiplier for the constraint
a
T
x
=
b
.
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 Linear Programming, UCSD Center for Computational Mathematics, UCSD Center, Computational Mathematics

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