Linear ProgrammingChapter 6.14-7.3

Björn Morén

1 Simplex Method with Upper BoundsOptimality Conditions

Simplex Method

2 Interior Point Methods

1 Simplex Method with Upper BoundsOptimality Conditions

Simplex Method

2 Interior Point Methods

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Problem Formulation

1. Upper bounds as constraints

2. Consider upper bounds implicitly

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Basic Solution

Divide variables if they are basic, at lower bound, or at

upper bound (xB, xL, xU )

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Dual Problem

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Optimality Conditions and Dual Feasibility

Reduced costs are defined as cj = πA,j

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Optimality Conditions and Dual Feasibility

Reduced costs are defined as cj = πA,j

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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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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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Optimality Conditions

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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Minimum Ratio

If the entering variable xs is at lower bound:

1. xs = θ

2. xB = g − θA,s

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Minimum Ratio

If the entering variable xs is at upper bound:

1. xs = -θ

2. xB = g+θA,s

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Phase I

To find a feasible solution, a standard Phase I method

can be used.

1 Simplex Method with Upper BoundsOptimality Conditions

Simplex Method

2 Interior Point Methods

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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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Interior Point

x interior point if

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Interior Point

x interior point if

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Relative Interior Point

x relative interior point if

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Relative Interior Point

x relative interior point if

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Phase I - examples

Find a solution from the following Phase I problem

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Phase I - examples

Find a solution from the following Phase I problem

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Phase I - examples

Find a solution from the following Phase I problem

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Phase I - examples

Find a solution from the following Phase I problem

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Phase I - examples

Find a solution from the following Phase I problem

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Phase I - examples

Find a solution from the following Phase I problem

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Rounding Procedure

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

optimal objective function value. An optimal basic

feasible solution can be found by using the purification

routine.

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General Algorithm

1. Find search direction

1. Solving approximation problems e.g affine scalingmethod, Karmarkars projective scaling method

2. Solving a problem with optimality conditions,nonlinear equations, e.g primal-dual pathfollowing IPM

2. Determine step length

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General Algorithm

1. Find search direction

1. Solving approximation problems e.g affine scalingmethod, Karmarkars projective scaling method

2. Solving a problem with optimality conditions,nonlinear equations, e.g primal-dual pathfollowing IPM

2. Determine step length

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Observations

• 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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