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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 Linear Programming

  1. 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

  2. Linear Programming (LP) and the Simplex Method maximize c T x 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

  3. Linear Programming (LP) and the Simplex Method maximize c T x 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

  4. Linear Programming (LP) and the Simplex Method maximize c T x 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

  5. Random is Not Typical Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

  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

  7. Results: smoothed complexity bounds ◮ c ∈ R d , ¯ A ∈ R n × d , ¯ b ∈ R n . 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: c T x maximize subject to Ax ≤ b . Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

  8. Results: smoothed complexity bounds ◮ c ∈ R d , ¯ A ∈ R n × d , ¯ b ∈ R n . 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: c T x maximize subject to Ax ≤ b . Works Expected Number of Pivots � O ( n 86 d 55 (1 + σ − 30 )) Spielman-Teng ’04 O ( d 3 ln 3 n σ − 4 + d 9 ln 7 n ) Vershynin ’09 O ( d 2 √ ln n σ − 2 + d 5 ln 3 / 2 n ) Dadush-H., ’18 Daniel Dadush (CWI), Sophie Huiberts (CWI) A Friendly Smoothed Analysis of the Simplex Method

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