Programming
Standard Linear Form
Objective function
n
maximize
j=1
cj xj
n j=1
Constraints
subject to
Nonnegativity constraints
aij xj bi
for i = 1, 2, . . . , m
for j = 1, 2, . . . , n
xj 0
Polynomial time solution in n, m
Non-Standard Form
n
maximize
j=1 n
cj xj
subject to
j=1
aij xj bi
for i = 1, 2, . . . , m
for j = 1, 2, . . . , n
xj 0
Converting into Standard Form (1)
maximize -2x1 + 3x2 subject to
x1 + x2 x1 - 2x1 x1 =
minimize 2x1 - 3x2
7 4 0
Converting into Standard Form (2)
minimize 2x1 - 3x2 subject to x1 + x2 = 7
x1 - 2x1 x1 4 0
2x1 - 3x2 + 3x2
x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 = 7 4 0
x2 = x2 - x2
Converting into Standard Form (3)
x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 = 7 4 0
x1 + x2 - x2 x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 7 4 0
Converting into Standard Form (4)
x1 + x2 - x2 x1 + x2 - x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 7 4 0 x1 + x2 - x2 -x1 - x2 + x2 x1 - 2x2 + 2x2 x1 , x2 , x2 7 -7 4 0
Optimal Routes
Topology G = (V, E) K -- set of origin-destination flows k K, s -- source, t -- destination,
k k
dk -- demand
Set of given link weights {w | (i, j) E} X -- Fraction of flow k going over (i, j)
ij
k ij
minimize
kK (i,j)E
k wij Xij
subject to
k K, i = sk , tk ,
j:(i,j)E k Xij - k Xji = 0 j:(j,i)E
k,
k X sk j = 1 j:(sk ,j)E
k,
k Xjtk = 1 j:(j,tk )E
k i, j, k, Xij 0
minimize
kK (i,j)E
k wij Xij
...
1 wij Xij (i,j)E (i,j)E 2 wij Xij
...
k wij Xij (i,j)E
...
subject to
i = sk , tk ,
j:(i,j)E k Xij - k Xji = 0 j:(j,i)E
k Xij 0
k X sk j = 1 j:(sk ,j)E j:(j,tk )E
k Xjtk = 1
Interpretation
k X ij
Optimal solutions
{0, 1} {0, 1}
k X ij
k X ij
is the shortest path
k X ij
Multiple shortest paths exist
n
minimize
j=1 n
cj xj aij xj bi
subject to
j=1
for i = 1, 2, . . . , m
for j = 1, 2, . . . , n
xj 0
minimize subject to
cT x
Ax = b
x0
x0 is feasible if Ax0 = b and x0 0
Dual Problem
Primal maximize Problem cT x subject to Ax b x0 Dual Problem maximize yT b subject to y T A cT y0
If {xi} and {yi} are optimal, then the objective functions are equal
y i bi = i i
ci x i
Properties
If x and y are feasible and if c x = y b,
If x and y are feasible, then cTx yTb
T T
then x and y are optimal
If either problem has finite solution, so
does other; if either has unbounded solution, so does other
k wij Xij kK (i,j)E
...
1 wij Xij (i,j)E (i,j)E 2 wij Xij
minimize ...
k wij Xij (i,j)E
...
subject to
i = sk , tk ,
j:(i,j)E k Xij - k Xji = 0 j:(j,i)E
k Xij 0
k X sk j = 1 j:(sk ,j)E j:(j,tk )E
k Xjtk = 1
Example: Primal
minimize
(i,j)E k wij Xij
subject to
0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E
k Xij 0
i = sk , tk i = sk i = tk
minimize
(i,j)E
k wij Xij
subject to
0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E
k Xij 0
i = sk , tk i = sk i = tk
Primal Problem maximize cT x subject to Ax b x0
Dual Problem maximize yT b subject to y T A cT y0
minimize
(i,j)E
k wij Xij
subject to
0 k k 1 Xij - Xji = -1 j:(i,j)E j:(i,j)E
k Xij 0
i = sk , tk i = sk i = tk
Primal Problem maximize ...