# Smoothed Analysis of Algorithms: Why The Simplex Method Usually Takes Polynomial Time - PowerPoint PPT Presentation

PPT – Smoothed Analysis of Algorithms: Why The Simplex Method Usually Takes Polynomial Time PowerPoint presentation | free to download - id: 150d19-NzYyZ

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Smoothed Analysis of Algorithms: Why The Simplex Method Usually Takes Polynomial Time

Description:

### 1. Smoothed Analysis of Algorithms: Why The Simplex Method ... Trickier in 3d. 47. Future Research Simplex Method. Smoothed analysis of other pivot rules ... – PowerPoint PPT presentation

Number of Views:72
Avg rating:3.0/5.0
Slides: 49
Provided by: danielas7
Category:
Tags:
Transcript and Presenter's Notes

Title: Smoothed Analysis of Algorithms: Why The Simplex Method Usually Takes Polynomial Time

1
Smoothed Analysis of Algorithms Why The Simplex
Method Usually Takes Polynomial Time
Gaussian Perturbation with variance s2
• Shang-Hua Teng
• Boston University/Akamai

Joint work with Daniel Spielman (MIT)
2
Remarkable Algorithms and Heuristics
• Work well in practice, but
• exponential,
• contrived.
• Average case good,
• polynomial,
• meaningful?

3
Random is not typical
4
Smoothed Analysis of Algorithms
worst case maxx T(x) average case avgr
T(r) smoothed complexity maxx avgr T(xer)
5
Smoothed Analysis of Algorithms
• Interpolate between Worst case and Average Case.
• Consider neighborhood of every input instance
• If low, have to be unlucky to find bad input
instance

6
Complexity Landscape
7
Classical Example Simplex Method for Linear
Programming
max zT x s.t. A x y
• Worst-Case exponential
• Average-Case polynomial
• Widely used in practice

8
The Diet Problem
Minimize 30 x1 80 x2 20 x3 s.t. 30x1
10 x2 6 x3 ? 300 5x1
9x2 8x3 ? 50 1.5x1 2.5 x2
18 x3 ? 70 10x1
6 x3 ? 100
x1, x2, x3 ? 0
9
Linear Programming
max zT x s.t. A x y
Max x1 x2 s.t x1 1
x2 1 -x1 - 2x2 1

10
Smoothed Analysis of Simplex Method
G is Gaussian
11
Smoothed Analysis of Simplex Method
• Worst-Case exponential
• Average-Case polynomial
• Smoothed Complexity polynomial

max zT x s.t. aiT x 1, ai 1
max zT x s.t. (aisgi )T x 1
12
Perturbation yields Approximation
• For polytope of good aspect ratio

13
But, combinatorially
14
The Simplex Method
15
History of Linear Programming
• Simplex Method (Dantzig, 47)
• Exponential Worst-Case (Klee-Minty 72)
• Avg-Case Analysis (Borgwardt 77, Smale 82,
Haimovich, Adler, Megiddo, Shamir, Karp, Todd)
• Ellipsoid Method (Khaciyan, 79)
• Interior-Point Method (Karmarkar, 84)
• Randomized Simplex Method (mO(?d) )
• (Kalai 92, Matousek-Sharir-Welzl
92)

16
17
18
start
z
objective
19
Theorem For every plane, the expected size of
the shadow of the perturbed tope is poly(m,d,1/s )
20
Theorem For every z, two-Phase Algorithm runs in
expected time poly(m,d,1/s )
z
21
A Local condition for optimality
z
a2
z
0
Vertex on a1,,ad maximizes z iff z Î
a1
22
Primal a1T x ? 1 a2T x ? 1 amT x ?
1
Polar ConvexHull(a1, a2, am)
23
(No Transcript)
24
Polar Linear Program
z
max ? ?z Î ConvexHull(a1, a2, ..., am)
25
Initial Simplex
Opt Simplex
26
27
(No Transcript)
28
Count facets by discretizingto N directions, N ??
29
Count pairs in different facets
So, expect c Facets
30
Expect cone of large angle
31
Angle
Distance
32
Isolate on one Simplex
33
Integral Formulation
34
Example For a and b Gaussian
distributed points, given that ab
intersects x-axis Prob? lt
e O(e2)
a
?
b
35
a
?
b
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39
(No Transcript)
40
a
41
(No Transcript)
42
Change of variables
u
a
z
?
v
b
da db (uv)sin(q) du dv dz dq
43
Analysis For e lt e0, Pe lt e2
Slight change in q has little effect on ni
for all but very rare u,v,z
44
Distance Gaussian distributed corners
45
Idea fix by perturbation
46
Trickier in 3d
47
Future Research Simplex Method
• Smoothed analysis of other pivot rules
• Analysis under relative perturbations.
• Trace solutions as un-perturb.
• Strongly polynomial algorithm for linear
programming?

48
A Theory Closer to Practice
• Optimization algorithms and heuristics, such as