
182 Introduction to Linear Optimization and Extensions with MATLAB
R
λ = (λ
1
1
, ..., λ
1
N
1
, λ
2
1
, ..., λ
2
N
2
, ..., λ
K
1
, ..., λ
K
N
K
)
T
µ = (µ
1
1
, ..., µ
1
l
1
, µ
2
1
, ..., µ
2
l
2
, ..., µ
K
1
, ..., µ
K
l
K
)
T
r
T
= ((b
0
)
T
, e
T
) = ((b
0
)
T
, (1..., 1)
T
),
where s is a vector of slack variables and e is the vector of dimension K with
all components equal to 1. Q
v
is a matrix such that the column associated
with λ
k
i
is
q
k
i
e
k
=
L
k
v
k
i
e
k
,
where e
k
is the kth unit vector. Q
d
is a matrix such that the column associated
with µ
k
j
is
q
k
j
0
=
L
k
d
k
j
0
.
The number of variables λ
k
i
and µ
k
j
can be extremely large for even moder-
ately sized problems since feasible sets of subproblems (linear programs) ...