Random Simplex for operation research.ppt

1. A Randomized Polynomial-
Time Simplex Algorithm for
Linear Programming
CS3150 Course Presentation

2. Linear Programming
Example:
･Find the maximum value of p = 3x - 2y + 4z
･subject to ･4x + 3y - z >= 3
･x + 2y + z<=4 ･
x >= 0, y >= 0, z >= 0

3. Linear Programming
• Objective Function
max zT x
• Constraints
s.t. A x  y

4. Simplex Method - Intuition
Objective:
– Min C = 3x + 4y
Constraints:
– ･3x - 4y <= 12,
– ･x + 2y >= 4
– ･x >= 1, y >= 0.

5. Simplex Method - Intuition
max zT x
s.t. A x  y
• Worst-Case:
exponential
• Average-Case:
polynomial
• Widely used in
practice

6. Shadow Vertices

7. Shadow vertex pivot rule
objective
start z

8. Complexity Landscape

9. Complexity Landscape of
Perturbed Problem

10. Some issues
Problem
max zT x
s.t. A x  y
– The perturbed
problem is no longer
the original problem
we want to solve!
Solution
– Reduce the original
problem to another
problem, where the
perturb won’t affect
solution.
– Is this set of constraints
bounded?
A’w<=1

11. Intuition of the Algorithm
Since the right hand side won’t affect
solution, we want to carefully choose it
so that the Shadow-Vertex Simplex will
run poly-time with high probability.

12. Intuition of the Proof
P is the polytope of A’w<=1
Case 1:
– The polytope P is in k-near-isotropic
position

13. K-near-isotropic position

14. Intuition of the Proof
Case 1:
– The polytope is in k-near-isotropic position
–
Case 2:
– The polytope is not in k-near-isotropic position

15. K-near-isotropic Case
Upper bound of total shadow length
(Shadow Size).
Lower bound expected length of each
edge.
Number of edges of the shadow is poly
in size w.h.p.

16. K-near-isotropic position

17. Randomization
Each of vector is a independently
exponentially distributed random
variable with expectation
Project onto a random plane

18. None-K-near-isotropic Case
By Running the shadow vertex for a
limited amount of time we can either:
– Find the optimal
– Or find a way to eliminate bad events
w.h.p.

19. None-K-near-isotropic Case

20. K-near-isotropic Case
Upper bound of the total shadow length.
Lower bound the expected length of
each edge.
Number of edges of the shadow is poly
in size w.h.p.

21. Upper Bound of Shadow Size
A’w<=1
A’w<=1+r

22. Shadow Size in Case 1
The expected shadow size is at most:

23. Upper Bound of Shadow Size

24. K-near-isotropic Case
Upper bound of the total shadow length.
Lower bound the expected length of
each edge.
Number of edges of the shadow is poly
in size w.h.p.

25. Expected Edge Length
The Expected Edge Length is at least:

26. Case 1 Main Theorem
The expected number of edges is at
most

27. None-K-Isotropic Case
The expected shadow size inside any
given ball is small

28. None-K-near-Isotropic Case
Upper bound of the total shadow length
within the given ball.
Lower bound the expected length of
each edge within the given ball.
Number of edges of the shadow is poly
in size w.h.p.

29. Outline of the Algorithm
Run Shadow Vertex Simplex on the
Randomized Input
– If Find Optimal then halt
Else transform the coordinates and run
shadow vertex simplex on the
transformed inputs
Algorithm will halt in poly-time w.h.p.

30. Summery and Intuitions
Deterministic algorithms run exponential time
on some “bad” inputs
By introducing some randomness into the
algorithm fixed the problem.
The Randomized algorithm run poly time on
all inputs with high probability.
Start with something strict, which is easy to
prove the poly-bound, eliminate the bad
events in poly-time.