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 R , x R n subject to a T 1 x + b 1 a T 2 x + b 2 . . . . . . a T m x + b m 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 ! x 1 1 ! x ! Working sets: A A 1 A last The optimal vertex has n + 1 constraints in the working set. If a feasible point exists, then = 0 constraint 0 enters the working set at iteration - 1. If no feasible point exists, phase-1 will terminate with > 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., = 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 ! = b ! A x + z = b = 0 A x = b so that A defines an initial working set for phase 2....
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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

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