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Lecture 17 - Linear Programs with Mixed Constraints

# Lecture 17 - Linear Programs with Mixed Constraints -...

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Unformatted text preview: Lecture 17 Linear Programs with Mixed Constraints UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, February 20, 2009 Recap: the phase-1 LP We are finding a feasible point for the constraints Ax ≥ b , x ≥ 0. The phase-1 LP is the linear program: minimize θ ∈ R , x ∈ R n θ subject to a T 1 x + θ ≥ b 1 a T 2 x + θ ≥ b 2 . . . . . . a T m x + θ ≥ b m θ ≥ UCSD Center for Computational Mathematics Slide 2/28, Friday, February 20, 2009 Recap: the phase-1 LP In matrix form, the phase-1 LP is: minimize ¯ x ∈ R n +1 ¯ c T ¯ x subject to ¯ A ¯ x ≥ ¯ b , ¯ x ≥ with m general constraints, n + 1 simple bounds n + 1 variables ⇒ m + n + 1 constraints and n + 1 variables UCSD Center for Computational Mathematics Slide 3/28, Friday, February 20, 2009 Solving the phase-1 LP Simplex iterates: x θ ! x 1 θ 1 ! ··· x ‘ θ ‘ ! Working sets: ¯ A ¯ A 1 ··· ¯ A ‘ ↑ last The optimal vertex has n + 1 constraints in the working set. If a feasible point exists, then θ ‘ = 0 ⇒ constraint θ ≥ 0 enters the working set at iteration ‘- 1. If no feasible point exists, phase-1 will terminate with θ ‘ > ⇒ θ ‘ is the smallest shift that makes the constraints feasible. UCSD Center for Computational Mathematics Slide 4/28, Friday, February 20, 2009 Assume that the constraints are feasible, i.e., θ ‘ = 0. The last working set is ¯ A ‘ ¯ x ‘ = ¯ b ‘ , where ¯ A ‘ includes rows of ( A e ) rows of I n +1 row e T n +1 (just added) Assume that e T n +1 entered as the last row of ¯ A ‘ , then ¯ A ‘ = A ‘ z 1 ! and ¯ b ‘ = b ‘ ! where A ‘ and b ‘ are rows of A and b , and z has elements 0 or 1. UCSD Center for Computational Mathematics Slide 5/28, Friday, February 20, 2009 From the previous slide: ¯ A ‘ = A ‘ z 1 ! and ¯ b ‘ = b ‘ ! If ¯ A ‘ ¯ x ‘ = ¯ b ‘ and ¯ x ‘ ≥ 0 then A ‘ z 1 ! x ‘ θ ‘ ! = b ‘ ! ⇒ A ‘ x ‘ + z θ ‘ = b ‘ θ ‘ = 0 ⇒ A ‘ x ‘ = b ‘ so that A ‘ defines an initial working set for phase 2....
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Lecture 17 - Linear Programs with Mixed Constraints -...

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