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

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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

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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
  • Worst case bad,
  • 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
Shadow Vertices
17
Another shadow
18
Shadow vertex pivot rule
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
cone(a1,,ad )
a1
22
Primal a1T x ? 1 a2T x ? 1 amT x ?
1
Polar ConvexHull(a1, a2, am)
23
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24
Polar Linear Program
z
max ? ?z Î ConvexHull(a1, a2, ..., am)
25
Initial Simplex
Opt Simplex
26
Shadow vertex pivot rule
27
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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
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37
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38
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39
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40
a
41
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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
    Newtons Method, Conjugate Gradient, Simulated
    Annealing, Differential Evolution, etc.
  • Computational Geometry, Scientific Computing and
    Numerical Analysis
  • Heuristics solving instances of NP-Hard
    problems.
  • Discrete problems?
  • Shrink intuition gap between theory and practice.
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