Simplex

1
Simplex
walk on the vertices of the feasible region
v current vertex if ? neighbor v of v with
better objective then move to v
2
Simplex
walk on the vertices of the feasible region
vertex feasible point defined by a
collection of d inequalities
neighbors vertices sharing
d-1 of the inequalities
3
Simplex
v current vertex if ? neighbor v of v with
better objective then move to v
max cT x Ax ? b x ? 0
assume v (0,...,0)T
? i such that ci gt 0 iff v is not optimal
4
Simplex
v current vertex if ? neighbor v of v with
better objective then move to v
max cT x Ax ? b x ? 0
v (0,..,xi,..,0)T
Make xi as big as possible stopper
aj x bj
5
Simplex
v current vertex if ? neighbor v of v with
better objective then move to v
max cT x Ax ? b x ? 0
v (0,..,xi,..,0)T
Make xi as big as possible stopper
aj x bj
xi bj aj x
Substitute
6
Simplex
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
Is (x,y)(0,0) optimal?
7
Simplex
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
Lets increase y as much as we can.
8
Simplex
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
substitute z1-(y-x)
9
Simplex
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
substitute z1-(y-x) z ? 0
y ? x-z1
10
Simplex
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
y ? x-z1
11
Simplex
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
Is (x,z)(0,0) optimal?
12
Simplex
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
Lets increase x as much as we can.
13
Simplex
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
substitute w1-(x-z)
14
Simplex
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
substitute w1-(x-z) w ? 0
x ? 1z-w
15
Simplex
max 3x-z1 2x-z ? 3 z ? 0 z ? 1 x-z ? 1 x ? 0 z-x
? 1
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
x ? 1z-w
16
Simplex
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
Is (z,w)(0,0) optimal?
17
Simplex
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
Lets increase z as much as we can.
18
Simplex
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
substitute u1-(z-2w)
19
Simplex
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
substitute u1-(z-2w) u ? 0
z ? 12w-u
20
Simplex
max 2z-3w4 z-2w ? 1 z ? 0 z ? 1 w ? 0 w-z ? 1 w
? 2
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
z ? 12w-u
21
Simplex
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
Is (u,w)(0,0) optimal?
22
Simplex
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
Lets increase w as much as we can.
23
Simplex
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
substitute v2-(2w-u)
24
Simplex
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
substitute v2-(2w-u) v ? 0
w ? 1u/2-v/2
25
Simplex
max w-2u6 u ? 0 u-2w ? 1 2w-u ? 2 w ? 0 u-w ?
2 w ? 2
max 7-3u/2-v/2 u ? 0 v ? 3 v ? 0 v-u ? 2 uv ?
6 u-v ? 2
w?1u/2-v/2
26
Simplex
max 7-3u/2-v/2 u ? 0 v ? 3 v ? 0 v-u ? 2 uv ?
6 u-v ? 2
Is (u,v)(0,0) optimal?
27
Simplex
max 7-3u/2-v/2 u ? 0 v ? 3 v ? 0 v-u ? 2 uv ?
6 u-v ? 2
7
YES
Is (u,v)(0,0) optimal?
28
Simplex
(u,v)(0,0)
w ? 1u/2-v/2 1
z ? 12w-u 3
x ? 1z-w 3
y ? x-z1 1
(x,y)(3,1)
29
Simplex
(x,y)(3,1)
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
is an optimal solution
30
Simplex geometric view
(x,y)(3,1)
max 2xy xy ? 4 y-x ? 1 x-y ? 2 y ? 2 x ? 0 y ? 0
31
Getting the first point
min 1T z A x z b x ? 0 z ? 0
min cT x Axb x ? 0
wlog b ? 0
32
(No Transcript)
33
Points, lines
point (x,y)
line (x1,y1),(x2,y2)

2 points
34
Line as a point and a vector
point (x,y)
x1t (x2-x1),y1t (y2-y1)
line (x1,y1),(x2-x1,y2-y1)

point and a vector
35
Is point on a line?
point (x,y)
xx1t (x2-x1) yy1t (y2-y1)
line (x1,y1),(x2,y2)
36
Is point on a line?
point (x,y)
t (x2-x1)x-x_1 t (y2-y1)y-y_1
line (x1,y1),(x2,y2)
37
Is point on a line?
point (x,y)
is on
line (x1,y1),(x2,y2)
if and only if
0
38
Is point on a line?
0 for x on the line
gt0
lt0
39
Line segment
xx1t (x2-x1) yy1t (y2-y1)
t ? 0,1
line segment (x1,y1),(x2,y2)
40
Do two line segments intersect?
a1(x1,y1), a2(x2,y2) a3(x3,y3), a4 (x4,y4)
a3
L2
L1
a2
a4
a1
a1 and a2 on different sides of L2 a3 and a4 on
different sides of L1 or endpoint of a segment
lies on the other segment
41
Many segments, do any 2 intersect?
(a1,b1) (a2,b2) ... (an,bn)
O(n2) algorithm
42
Many segments, do any 2 intersect?
O(n log n) algorithm
assume no two points have the same
x-coordinate no 3 segments intersect at one
point
43
Sweep algorithm
44
Sweep algorithm
sort points by the x-coordinate
45
Sweep algorithm
events insert segment delete segment
46
Sweep algorithm
will find the left-most intersection point the
lines are neighbors on the sweep line
47
Sweep algorithm
sort the endpoints by x-coord ? p1,...,p2n T?
empty B-tree for i from 1 to 2n do if pi is
the left point of a segment s INSERT s
into T check if s intersects prev(s) or
next(s) in T if pi is the right point of a
segment s check if prev(s) interesects
next(s) in T DELETE s from T
48
Area of a simple polygon
(x3,y3)
(x2,y2)
(x1,y1)
49
Area of a simple polygon
(x1,y1),...,(xn,yn)
50
Area of a simple polygon
(x1,y1),...,(xn,yn)
(xn1,yn1)(x1,y1)
R0 for i from 1 to n do RR(yi1yi)(xi1-xi
) return R/2
51
Convex hull
smallest convex set containing all the points
52
Convex hull
smallest convex set containing all the points
53
Jarvis march
(assume no 3 points colinear)
s
find the left-most point
54
Jarvis march
(assume no 3 points colinear)
s
find the point that appears most to the right
looking from s
55
Jarvis march
(assume no 3 points colinear)
s
find the point that appears most to the right
looking from p
p
56
Jarvis march
(assume no 3 points colinear)
57
Jarvis march
(assume no 3 points colinear)
58
Jarvis march
(assume no 3 points colinear)
s ? point with smallest x-coord p ? s repeat
PRINT(p) q ? point other than p for i from
1 to n do if i ? p and point i to the
right of line (p,q) then q ? i p?
q until p s