week1_handout-1

5x1 2 subject to x1 5 x1 0 1 12 19 tricks for lps

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Unformatted text preview: ive Functions minimize max{2x1 − 1, x1 , .5x1 + 2} subject to x1 ≤ 5 x1 ≥ 0 (1) 12 / 19 Tricks for LPs - Minimax Objective Functions minimize subject to d x1 ≤ 5 x1 ≥ 0 2x 1 − 1 ≤ d x1 ≤ d . 5x 1 + 2 ≤ d (2) 13 / 19 Tricks for LPs - Absolute Values minimize |x1 + x2 | subject to |x1 | ≤ 6 (3) 14 / 19 Tricks for LPs - Absolute Values minimize subject to x3 x1 + x 2 ≤ x 3 − x1 − x2 ≤ x3 x1 ≤ 6 − x1 ≤ 6 x3 ≥ 0 (4) Note: Whenever you add a variable in place of an absolute value, remember to constrain that variable to be greater than or equal to zero! 15 / 19 Standard Form This will be covered in lecture on Tuesday. ￿ Turn all inequality constraints into equalities (add slack variables!) ￿ Constrain all variables to be ≥ 0 (add variables to split unconstrained variables) ￿ Make the objective function minimize (multiply a maximizer by -1) 16 / 19 Example: Standard Form Convert to standard form: maximize s.t. x1 + x2 x1 + 2 x 2 ≤ 4 x2 ≥ 2 x2 ≥ 0 17 / 19 Example: Standard Form In standard form: minimize s.t. − x1 − x2 (x1 − x3 ) + 2x2 + x4 = 4 x2 − x 5 = 2 ( x1 , x2 , x3 , x4 , x 5 ) ≥ 0 x3 is the negative part of (previously unconstrained) x1 x4 is the slack variable for...
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This note was uploaded on 12/10/2013 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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