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mathNPP3-07-draft

mathNPP3-07-draft - ARE211 Fall 2007 LECTURE#23 THU PRINT...

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Preliminary draft only: please check for final version ARE211, Fall 2007 LECTURE #23: THU, NOV 15, 2007 PRINT DATE: AUGUST 21, 2007 (NPP3) Contents 6. Nonlinear Programming Problems and the Kuhn Tucker conditions (cont) 1 6.4. KT conditions and the Lagrangian approach 1 6.5. Interpretation of the Lagrange Multiplier 3 6.6. A worked solution to an NPP: S&B #18.18 (on the problem set) 5 6.7. Computing a solution to a NPP: a simple worked example 9 6.8. Computed solution to a NPP: ARE problem set example. 12 6.9. Second Order conditions Without Quasi-Concavity 19 6.9.1. One linear equality constraint 20 6.9.2. One nonlinear equality constraint 23 6. Nonlinear Programming Problems and the Kuhn Tucker conditions (cont) 6.4. KT conditions and the Lagrangian approach There’s an alternative way of writing the KT conditions, which may be more familiar. Set up the Lagrangian function, take its FOC’s, and look for a solution to them. The FOC for the Lagrangian will be identical to the KT conditions. Why bother with the Lagrangian? Absolutely no reason why you shouldn’t just look at the KT conditions, and write down the conditions explicitly for a 1
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2 LECTURE #23: THU, NOV 15, 2007 PRINT DATE: AUGUST 21, 2007 (NPP3) given problem. But most people find that approach a bit abstract. So we offer this less abstract route. Important to see that you end up at exactly the same place, though. The Lagrangian function for the general NPP is as follows: L ( x , λ ) = f ( x ) + λ ( b - g ( x )) = f ( x ) + m summationdisplay j =1 λ j ( b j - g j ( x )) Look for an ¯ x and ¯ λ 0 satisfying the following conditions ∂L x , ¯ λ ) /∂x i = 0; ∂L x , ¯ λ ) /∂λ j 0; ¯ λ j ∂L x , ¯ λ ) /∂λ j = 0 . Note well: this specification of the Lagrangian first order conditions differs from many textbooks, eg., Chiang. this books impose the condition ∂L x , ¯ λ ) /∂x i 0. The source of the difference is that they impose an additional restriction on their specification of the NPP, i.e., that x 0. My specification is more general: if you want x 0, include this as one of your constraints. To see that these are the same as the K-T conditions, do the derivatives one by one ∂f x ) /∂x i = m summationdisplay j =1 ¯ λ j · ∂g j x ) /∂x i The inner product you are calculating is precisely the inner product you calculate when you pre- multiply the λ T vector with the i ’th column of the Jacobian matrix. The following is very important for what follows: the maximized value of the objective function is identically equal to the value of the Lagrangian at the solution to the NPP. More formally: Theorem: At a solution ¯ x to the NPP, L x , ¯ λ ) = f x ).
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ARE211, Fall 2007 3 To see that this is correct note that L ( x , λ ) = f ( x ) + λ ( b - g ( x )). Each component of the inner product λ ( b - g ( x )) is zero, since either ( b j - g j ( x )) is zero or λ j is zero. Hence the entire second term on the right hand side of the Lagrangian is zero.
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