mthsc810-lecture08-2x2

mthsc810-lecture08-2x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 8 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 20, 2011 Reading for today: Chapters 3.3 and 3.4, textbook Reading for Sep. 22: Chapters 3.5, textbook Midterm exam : Next Tuesday, 2:00pm-3:15pm, in class 1: procedure SIMPLEX( A , b , c ) 2: Find an initial basis B , N 3: loop 4: Reduced costs: c c N- c BB 1 N 5: if c 0, STOP : optimum found 6: Select j N : c j < j is the entering variable 7: d B - B 1 A j 8: If d B 0, STOP : problem unbounded 9: argmin i N : di < n- xi di o = exiting var. 10: B B { j } \ { } ; N N \ { j } { } 11: end loop 12: end procedure Dictionaries and tableaus Associate the objective function with a new variable, z := n i = 1 c i x i . Once a basis B is chosen, a dictionary describes the basic variables as affine functions of the non-basic variables: x B = B 1 b- B 1 N x N z = c B B 1 b + ( c N- c BB 1 N ) x N Tableaus and dictionaries are equivalent. Example Given the problem in standard form max { c x : A x b , x } , max 5 x 1 + 4 x 2 + 3 x 3 s . t . 2 x 1 + 3 x 2 + x 3 5 4 x 1 + x 2 + 2 x 3 11 3 x 1 + 4 x 2 + 2 x 3 8 x 1 , x 2 , x 3 , rewrite it in standard form: min- 5 x 1- 4 x 2- 3 x 3 s . t . 2 x 1 + 3 x 2 + x 3 + x 4 = 5 4 x 1 + x 2 + 2 x 3 + x 5 = 11 3 x 1 + 4 x 2 + 2 x 3 + x 6 = 8 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ....
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mthsc810-lecture08-2x2 - MthSc 810: Mathematical...

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