Linear programming

1
Linear programming
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
2
Linear programming
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
x1
x2
3
Linear programming
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
x1
x2
4
Linear programming
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
x1
feasible solutions
x2
5
Linear programming
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
x1
optimal solution x11/2, x23/2
x2
6
Can you prove it is optimal ?
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 x1 ?
0 x2 ? 0
x1
optimal solution x11/2, x23/2
x2
7
Can you prove it is optimal ?
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5 4x1
4x2 ? 8
x1
optimal solution x11/2, x23/2
x2
8
Can you prove it is optimal ?
maximize x1 x2 x1 3x2 ? 3 3x1 x2 ? 5
x1x2 ? 2
x1
optimal solution x11/2, x23/2
x2
9
Another linear program
maximize x1 x2 x1 2x2 ? 3 4x1 x2 ? 5 x1 ?
0 x2 ? 0
10
Another linear program
maximize x1 x2 x1 2x2 ? 3 4x1 x2 ? 5 x1 ?
0 x2 ? 0
x11, x21, optimal ?
11
Another linear program
maximize x1 x2 x1 2x2 ? 3
3 4x1 x2 ? 5 1 x1 ? 0 x2 ? 0
7x1 7x2 ? 14
x11, x21, optimal !
12
Systematic search for the proof of optimality
maximize x1 x2 x1 2x2 ? 3
y1 4x1 x2 ? 5 y2 x1 ? 0 x2 ? 0
13
Systematic search for the proof of optimality
maximize x1 x2 x1 2x2 ? 3
y1 4x1 x2 ? 5 y2 x1 ? 0 x2 ? 0
y1 ? 0 y2 ? 0
14
Systematic search for the proof of optimality
maximize x1 x2 x1 2x2 ? 3
y1 4x1 x2 ? 5 y2 x1 ? 0 x2 ? 0
y1 ? 0 y2 ? 0
min 3y15y2
y1 4y2 ? 1 2y1y2 ? 1
15
Systematic search for the proof of optimality
max x1x2 x1 2x2 ? 3 4x1 x2 ? 5 x1 ?
0 x2 ? 0
min 3y15y2
y1 4y2 ? 1 2y1y2 ? 1
y1 ? 0 y2 ? 0
dual linear programs
16
Systematic search for the proof of optimality
max x1x2 x1 2x2 ? 3 4x1 x2 ? 5 x1 ?
0 x2 ? 0
min 3y15y2
?
y1 4y2 ? 1 2y1y2 ? 1
y1 ? 0 y2 ? 0
dual linear programs
17
Linear programming duality
max x1x2 x1 2x2 ? 3 4x1 x2 ? 5 x1 ?
0 x2 ? 0
min 3y15y2

y1 4y2 ? 1 2y1y2 ? 1
y1 ? 0 y2 ? 0
18
Linear programs
variables x1,x2,...,xn
linear function a1x1 a2x2 ... anxn
linear constraint equality a1x1 a2x2
... anxn b inequality a1x1 a2x2 ...
anxn ? b
19
Linear programs
variables x1,x2,...,xn
linear function a1x1 a2x2 ... anxn
linear constraint equality a1x1 a2x2
... anxn b inequality a1x1 a2x2 ...
anxn ? b
max/min of a linear function subject to
collection of linear constraints
20
Linear programs
variables x1,x2,...,xn
linear function a1x1 a2x2 ... anxn
max/min of a linear function subject to
collection of linear constraints
linear constraint equality a1x1 a2x2
... anxn b inequality a1x1 a2x2 ...
anxn ? b
Goal find the optimal solution (i.e., a
feasible solution with the maximum value of the
objective)
21
Linear programs
one of the most important modeling tools oil
industry manufacturing marketing
circuit design
very important in theory as well
22
Shortest path
23
Shortest path
ds 0 du ? ds 5 dv ? ds 6 dw ? du 3 dw ?
dv 1 dt ? dw 2 dt ? dv 4
max dt
24
Max-Flow
FLOW CONSERVATION CAPACITY CONSTRAINTS
? fu,v 0
v?V
fu,v ? c(u,v)
SKEW SYMMETRY
fu,v - fv,u
25
Max-Flow
objective ?
u?s,t
fu,v ? c(u,v)
fu,v fv,u0
26
Max-Flow
u?s,t
fu,v ? c(u,v)
fu,v fv,u0
27
Linear programming duality
maximize ? minimize constraint ? variable
equality ? unrestricted ? ?
non-negative variable ? constraint
unrestricted ? equality non-negative ?
?
28
Linear programming duality
max x1x2 x1x2x3x41 x12x3 ? 1 x22x4 ? 2 x1
? 0 x4 ? 0
maximize ? minimize constraint ? variable
equality ? unrestricted ? ?
non-negative variable ? constraint
unrestricted ? equality non-negative ?
?
29
Linear programming duality
max x1x2 x1x2x3x41 x12x3 ? 1 x22x4 ? 2 x1
? 0 x4 ? 0
y1 y2 ? 0 y3 ? 0
maximize ? minimize constraint ? variable
equality ? unrestricted ? ?
non-negative variable ? constraint
unrestricted ? equality non-negative ?
?
DONE
30
Linear programming duality
max x1x2 x1x2x3x41 x12x3 ? 1 x22x4 ? 2 x1
? 0 x4 ? 0
min y1 y2 2 y3
y1 y2 ? 0 y3 ? 0
DONE
maximize ? minimize constraint ? variable
equality ? unrestricted ? ?
non-negative variable ? constraint
unrestricted ? equality non-negative ?
?
DONE
31
Linear programming duality
max x1x2 x1x2x3x41 x12x3 ? 1 x22x4 ? 2 x1
? 0 x4 ? 0
min y1 y2 2 y3
y1 y2 ? 0 y3 ? 0
DONE
maximize ? minimize constraint ? variable
equality ? unrestricted ? ?
non-negative variable ? constraint
unrestricted ? equality non-negative ?
?
DONE
y1 y2 ? 1 y1 y3 1 y1 2y2 0 y1 2y3 ? 0
DONE
32
Linear programming duality
max x1x2 x1x2x3x41 x12x3 ? 1 x22x4 ? 2 x1
? 0 x4 ? 0
min y1 y2 2 y3
y2 ? 0 y3 ? 0
y1 y2 ? 1 y1 y3 1 y1 2y2 0 y1 2y3 ? 0
33
? ? and non-negativity
a1 x1 ... an xn ? b
a1 x1 ... an xn b y, y ? 0
a1 x1 ... an xn y b, y ? 0
34
? ?
a1 x1 ... an xn b
a1 x1 ... an xn ? b a1 x1 ... an xn
? b
a1 x1 ... an xn ? b -a1 x1 - ... -
an xn ? -b
35
optimization ? feasibility
max a1x1...anxn
a1x1...anxn ? P
binary search on P
36
Max-Flow
u?s,t
fu,v ? c(u,v)
fu,v fv,u0
37
Max-Flow
u?s,t
yu
zu,v ? 0
fu,v ? c(u,v)
fu,v fv,u0
wu,v
38
Max-Flow
min ? c(u,v)zu,v
u,v
u?s,t
yu
fu,v ? c(u,v)
zu,v
fu,v fv,u0
wu,v
zu,v ? 0
39
Max-Flow
min ? c(u,v)zu,v
u,v
u?s,t
u?s,t
yu

fu,v ? c(u,v)
zu,v
0

fu,v fv,u0
wu,v
zu,v ? 0
40
Max-Flow
min ? c(u,v)zu,v
u,v
u?s,t
yu zu,v wu,v 0
ys -1
zs,v ws,v 1
yt 0
zt,v wt,v 0
zu,v ? 0
41
Max-Flow
min ? c(u,v)zu,v
u,v
yu zu,v wu,v 0
ys -1
yt 0
zu,v ? 0
42
Max-Flow
min ? c(u,v)zu,v
u,v
yu zu,v wu,v 0
ys -1
yv zv,u wu,v 0
yt 0
zu,v ? 0
43
Max-Flow
min ? c(u,v)zu,v
u,v
yu zu,v wu,v 0
ys -1
yv zv,u wu,v 0
yt 0
yu - yv zv,u - zu,v
zu,v ? 0
44
Max-Flow
ys -1
yu - yv zv,u - zu,v
yt 0
zu,v ? 0
45
Max-Flow
ys -1
yu - yv zv,u - zu,v
yt 0
zu,v ? 0
46
Max-Flow
ys -1
yt 0
47
Max-Flow Min-Cut
ys -1
one more trick achieves yu ? -1,0
yt 0
min ? c(u,v)
S,s? S t?SC
u? S,v? SC