Lecture 20 - Duality Theory I

Lecture 20 - Duality Theory I - Lecture 20 Duality Theory I...

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Unformatted text preview: Lecture 20 Duality Theory I UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, February 27th, 2009 Recap: Simplex method for standard form Solve B T = c B ; z N = c N- N T ; ( z N ) s = min( z N ); if ( z N ) s then stop ; Solve Bp B =- a s ; i = ( x B ) i- ( p B ) i if ( p B ) i < 0; + if ( p B ) i 0; t = min { i } ; = t ; if = + then stop ; x B x B + p B ; ( x B ) t ; Exchange index t of B with index s of N ; UCSD Center for Computational Mathematics Slide 2/29, Friday, February 27th, 2009 Getting feasible How do we get a feasible basic solution of Ax = b ? Example: A = 2 3 1 1 1 2 1- 1 ! , b = 5 3 ! UCSD Center for Computational Mathematics Slide 3/29, Friday, February 27th, 2009 2 x 1 + 3 x 2 + x 3 + x 4 = 5 x 1 + 2 x 2 + x 3- x 5 = 3 x 1 , x 2 , x 3 , x 4 , x 5 Add positive shifts x 6 and x 7 : 2 x 1 + 3 x 2 + x 3 + x 4 + x 6 = 5 x 1 + 2 x 2 + x 3- x 5 + x 7 = 3 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 The basis B = { 6 , 7 } defines the basic solution x B = 5 3 . UCSD Center for Computational Mathematics Slide 4/29, Friday, February 27th, 2009 Minimize the sum of infeasibilities minimize x R 7 x 6 + x 7 2 x 1 + 3 x 2 + x 3 + x 4 + x 6 = 5 x 1 + 2 x 2 + x 3- x 5 + x 7 = 3 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 If a feasible point exists, both x 6 and x 7 will be zero (i.e., nonbasic) at the end of phase 1. two other variables will be basic phase 1 solution is an initial feasible basic solution for phase 2 UCSD Center for Computational Mathematics Slide 5/29, Friday, February 27th, 2009 If the first constraint had been of the form 2 x 1 + 3 x 2 + x 3 + x 4 =- 5 we would have included the shift x 6 as 2 x 1 + 3 x 2 + x 3 + x 4- x 6 =- 5 but we still use + x 6 in the sum of infeasibilities. UCSD Center for Computational Mathematics Slide 6/29, Friday, February 27th, 2009 General case To find an initial vertex for Ax = b , x we define shifts v = x n +1 x n +2 . . . x n + m UCSD Center for Computational Mathematics Slide 7/29, Friday, February 27th, 2009 General case Define shifted constraints A V x v !...
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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 20 - Duality Theory I - Lecture 20 Duality Theory I...

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