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Lecture3 - f we can consider the fol-lowing linear program...

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Lecture 3: Handling Nonlinearity August 27, 2010
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Nonlinear Programs Is this a linear program?
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Converting nonlinear programs to linear programs This is an example of a nonlinear program. A mathematical program that is not linear is called a nonlinear program But some nonlinear programs can be converted to linear programs in terms of new variables
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Handling nonlinearity
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Why does this work?
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Handling Minimax (Maximin) Objective Functions Consider the following mathematical program: min u = max { 3 x 1 + 4 x 2 , 5 x 1 + 2 x 2 } s.t. x 1 + x 2 4 2 x 1 + x 2 5 This is a Minimax problem. The objective function is a max function of some linear functions. This is a nonlinear program since the objective is not linear. However, we can convert it to a linear program by introducing a new decision variable f to represent the objective function value. 18
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Handling Minimax (Maximin) Objective Functions After introducing the new variable
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Unformatted text preview: f , we can consider the fol-lowing linear program: min u = f s.t. x 1 + x 2 ≥ 4 2 x 1 + x 2 ≤ 5 3 x 1 + 4 x 2 ≤ f 5 x 1 + 2 x 2 ≤ f If ( x * 1 ,x * 2 ,f * ) is an optimal solution of the linear program, then ( x * 1 ,x * 2 ) is an optimal solution of the minimax problem, and f * is the optimal value of the minimax problem. Also, we have f * = max { 3 x * 1 + 4 x * 2 , 5 x * 1 + 2 x * 2 } . All minimax problems can be converted to linear programs, it is similar for maximin problems. 19 Handling absolute values: another way ● Observe that |z| = max {z, – z} ● So the problem min |x 1 – x 2 | s.t. x 1 + x 2 ≥ 4 becomes min max {x 1 – x 2 , x 2 – x 1 } s.t. x 1 + x 2 ≥ 4 ● Now it can be handled as a minimax program ● This works when the original problem is a minimization...
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