mthsc810-lecture08-2x2

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

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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: ℓ ← argmin i ∈N : di < n- xi di o ⊲ ℓ = exiting var. 10: B ← B ∪ { j } \ { ℓ } ; N ← N \ { j } ∪ { ℓ } 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 Programming...

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