A Friendly Smoothed Analysis of the Simplex Method Daniel Dadush - - PowerPoint PPT Presentation

A Friendly Smoothed Analysis of the Simplex Method

Daniel Dadush (CWI) Sophie Huiberts (CWI) Highlights of Algorithms, June 2018

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 2

Linear Programming (LP) and the Simplex Method

maximize cTx subject to Ax ≤ b

◮ d variables ◮ n constraints

Linear time in practice.

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 3

Linear Programming (LP) and the Simplex Method

maximize cTx subject to Ax ≤ b

◮ d variables ◮ n constraints

Linear time in practice. Exponential in worst case.

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 4

Linear Programming (LP) and the Simplex Method

maximize cTx subject to Ax ≤ b

◮ d variables ◮ n constraints

Linear time in practice. Exponential in worst case. Average case analysis?

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 5

Random is Not Typical

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 6

Random is Not Typical

Smoothed Complexity (Spielman-Teng ’ 01)

Worst case, σ = 0
Smoothed analysis, σ variable

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 7

Results: smoothed complexity bounds

◮ c ∈ Rd, ¯

A ∈ Rn×d, ¯ b ∈ Rn. Rows of ( ¯ A, ¯ b) have norm at most 1.

◮ ˆ

A, ˆ b: entries iid N(0, σ).

◮ A = ¯

A + ˆ A, b = ¯ b + ˆ b.

◮ Smoothed Linear Program:

maximize cTx subject to Ax ≤ b.

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

SLIDE 8

Results: smoothed complexity bounds

◮ c ∈ Rd, ¯

A ∈ Rn×d, ¯ b ∈ Rn. Rows of ( ¯ A, ¯ b) have norm at most 1.

◮ ˆ

A, ˆ b: entries iid N(0, σ).

◮ A = ¯

A + ˆ A, b = ¯ b + ˆ b.

◮ Smoothed Linear Program:

maximize cTx subject to Ax ≤ b. Works Expected Number of Pivots Spielman-Teng ’04

O(n86d55(1 + σ−30))

Vershynin ’09 O(d3 ln3 nσ−4 + d9 ln7 n) Dadush-H., ’18 O(d2√ ln nσ−2 + d5 ln3/2 n)

Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method