Linear Programming Chapter 6.14-7.3 Bjrn Morn 1 Simplex Method - - PowerPoint PPT Presentation

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Linear Programming Chapter 6.14-7.3 Bjrn Morn 1 Simplex Method - - PowerPoint PPT Presentation

May 14, 2023 461 likes •788 views

Linear Programming Chapter 6.14-7.3 Bjrn Morn 1 Simplex Method with Upper Bounds Optjmality Conditjons Simplex Method 2 Interior Point Methods 1 Simplex Method with Upper Bounds Optjmality Conditjons Simplex Method 2 Interior Point

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Linear Programming

Chapter 6.14-7.3

Björn Morén

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1 Simplex Method with Upper Bounds

Optjmality Conditjons Simplex Method

2 Interior Point Methods

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1 Simplex Method with Upper Bounds

Optjmality Conditjons Simplex Method

2 Interior Point Methods

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Linear Programming Björn Morén December 15, 2016 3 / 22

Problem Formulatjon

  • 1. Upper bounds as constraints
  • 2. Consider upper bounds implicitly
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Linear Programming Björn Morén December 15, 2016 4 / 22

Basic Solutjon

Divide variables if they are basic, at lower bound, or at upper bound (xB, xL, xU)

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Linear Programming Björn Morén December 15, 2016 5 / 22

Dual Problem

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Linear Programming Björn Morén December 15, 2016 6 / 22

Optjmality Conditjons and Dual Feasibility

Reduced costs are defined as ¯ cj = πA,j

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Linear Programming Björn Morén December 15, 2016 6 / 22

Optjmality Conditjons and Dual Feasibility

Reduced costs are defined as ¯ cj = πA,j

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Linear Programming Björn Morén December 15, 2016 7 / 22

Updated Simplex Method

  • 1. Check optimality conditions
  • 2. Select entering variable
  • 3. Minimum ratio test (and select exiting variable)
  • 4. Do pivot step
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Linear Programming Björn Morén December 15, 2016 8 / 22

Updated Simplex Method

  • 1. Check optimality conditions
  • 2. Select entering variable
  • 3. Minimum ratio test (and select exiting

variable)

  • 4. Do pivot step
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Linear Programming Björn Morén December 15, 2016 9 / 22

Optjmality Conditjons

Solution is optimal if

  • 1. For all xj at lower bound, xj = lj reduced cost

¯ cj ≥ 0

  • 2. For all xj at upper bound, xj = uj reduced cost

¯ cj ≤ 0

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Linear Programming Björn Morén December 15, 2016 10 / 22

Minimum Ratjo

If the entering variable xs is at lower bound:

  • 1. xs = θ
  • 2. xB = ¯

g − θ ¯ A,s

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Linear Programming Björn Morén December 15, 2016 11 / 22

Minimum Ratjo

If the entering variable xs is at upper bound:

  • 1. xs = -θ
  • 2. xB = ¯

g+θ ¯ A,s

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Linear Programming Björn Morén December 15, 2016 12 / 22

Phase I

To find a feasible solution, a standard Phase I method can be used.

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1 Simplex Method with Upper Bounds

Optjmality Conditjons Simplex Method

2 Interior Point Methods

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Linear Programming Björn Morén December 15, 2016 14 / 22

History

Dikin 1967 was first. Khachiyan 1979, ellipsoid algorithm, solves Linear Programs in polynomial time. Work by Karmakar 1984 made it popular. Also solves Linear Programs in polynomial time, more effective than the ellipsoid algorithm.

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Linear Programming Björn Morén December 15, 2016 15 / 22

Interior Point

interior point if

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Linear Programming Björn Morén December 15, 2016 15 / 22

Interior Point

x interior point if

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Linear Programming Björn Morén December 15, 2016 16 / 22

Relatjve Interior Point

relative interior point if

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Linear Programming Björn Morén December 15, 2016 16 / 22

Relatjve Interior Point

x relative interior point if

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Linear Programming Björn Morén December 15, 2016 17 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 17 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 18 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 18 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 19 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 19 / 22

Phase I - examples

Find a solution from the following Phase I problem

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Linear Programming Björn Morén December 15, 2016 20 / 22

Rounding Procedure

Theorem 1 (7.1). if x is sufficiently close to the

  • ptimal objective function value. An optimal basic

feasible solution can be found by using the purification routine.

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Linear Programming Björn Morén December 15, 2016 21 / 22

General Algorithm

  • 1. Find search direction
  • 1. Solving approximation problems e.g affine scaling

method, Karmarkars projective scaling method

  • 2. Solving a problem with optimality conditions,

nonlinear equations, e.g primal-dual path following IPM

  • 2. Determine step length
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Linear Programming Björn Morén December 15, 2016 21 / 22

General Algorithm

  • 1. Find search direction
  • 1. Solving approximation problems e.g affine scaling

method, Karmarkars projective scaling method

  • 2. Solving a problem with optimality conditions,

nonlinear equations, e.g primal-dual path following IPM

  • 2. Determine step length
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Linear Programming Björn Morén December 15, 2016 22 / 22

Observatjons

  • Number of iterations grow very slowly with

problem size, almost constant

  • Time per iteration increases with problem size
  • Extra steps required to get a BFS (purification

routine)

  • Useful on large scale problems
  • Simplex method is good to start with for

understanding

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Questjon?

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