Linear Programming: Chapter 4 - Efficiency and Complexity, Study notes of Linear Programming

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Linear Programming: Chapter 4

Efficiency

Robert J. Vanderbei

October 17, 2007

Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb

Efficiency

Question: Given a problem of a certain size, how long will it take to solve it?

Two Kinds of Answers:

• Average Case. How long for a typical problem.
• Worst Case. How long for the hardest problem.

Average Case.

• Mathematically difficult.
• Empirical studies.

Worst Case.

• Mathematically tractible.
• Limited value.

Klee–Minty Problem (1972)

maximize

∑^ n

j=

2 n−j^ xj

subject to 2

∑^ i−^1

j=

2 i−j^ xj + xi ≤ 100 i−^1 i = 1, 2 ,... , n

xj ≥ 0 j = 1, 2 ,... , n.

Example n = 3:

maximize 4 x 1 + 2x 2 + x 3 subj. to x 1 ≤ 1 4 x 1 + x 2 ≤ 100 8 x 1 + 4x 2 + x 3 ≤ 10000 x 1 , x 2 , x 3 ≥ 0.

A Distorted Cube

Constraints represent a “minor” dis- tortion to an n-dimensional hyper- cube:

0 ≤ x 1 ≤ 1 0 ≤ x 2 ≤ 100 ...

0 ≤ xn ≤ 100 n−^1.

Case n = 3:

1

100

10000

96

9992 9592 9600

Case n = 3:

Now, watch the pivots...

Exponential

Klee–Minty problem shows that:

Largest-coefficient rule can take 2 n^ − 1 pivots to solve a problem in n variables and constraints (thereby visiting all 2 n^ vertices of the distorted cube).

For n = 70, 2 n^ = 1. 2 × 1021.

At 1000 iterations per second, this problem will take 40 billion years to solve. The age of the universe is estimated at 13. 7 billion years.

Yet, problems with 10,000 to 100,000 variables are solved routinely every day.

Worst case analysis is just that: worst case.