Chapter 4: Interior Point Method

1
Chapter 4 Interior Point Method

• Starts at feasible point
• Moves through interior of feasible region
• Always improves objective function
• Theoretical interest

2
The Karmarkars Method

• The LP form of Karmarkars method
• minimize z CX
• subject to
• AX 0
• 1X 1 (1)
• X 0
• This LP must also satisfy
• satisfies AX 0 (2)
• Optimal z-value 0 (3)
• Where X (x1, x2, ., xn)T , A is an m x n matrix

3
The Karmarkars Method

• Suppose the LP is in the form (1) - (3)
• Step 1 k 0, start with the solution point
  and compute
• Step 2 stop if CXk lt e, else go to Step 3
• Step 3
• Define
• Compute
• Where

4
How to Transform any LP to the Karmarkars Form

• Step 1 set up the dual form of the LP
• Step 2 apply the dual optimal condition to form
  the combined feasible region form
• Step 3 Convert the combined feasible region to
  the homogeneous form AX 0
• Add the sum of all variables M constraint
• Convert this constraint to form
• Introduce new dummy variable d2 1 to the system
  to convert the system to AX 0 and 1X M 1
• Step 4 convert the system to the form (1)-(3)
• Introduce the set of new variables xj (M 1)xj
  to convert the system to the form AX 0 and 1X
  1
• Introduce new dummy variable d3 to ensure (2)
  and (3)

5
Examples

• Example 1 convert the following LP to the
  Karmarkars LP
• Maximize z 3x1 x2
• Subject to 2x1 x2 2
• x1 2x2 5
• x1, x2 0
• Example 2 Perform one iteration of Karmarkars
  method for the following LP
• Minimize z 2x1 2x2 3x3
• s.t. - x1 2x2 3x3 0
• x1 x2 x3 1
• x1, x2, x3
  0