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Unformatted text preview: 1 Ch. 13 Further Topics in Optimization 13.1 Nonlinear Programming and Kuhn Tucker Conditions 13.2 The Constraint Qualification 13.3 Economic Applications 13.4 Sufficiency Theorems in Nonlinear Programming 13.5 Maximumvalue Functions and the Envelope Theorem 13.6 Duality and the Envelope Theorem 13.7 Some Concluding Remarks 2 13.1 Nonlinear Programming and Kuhn Tucker Conditions Step 1: Effect of nonnegativity restrictions Step 2: Effect of inequality constraints Interpretation of the KuhnTucker conditions Example 1 & 2 3 ( 29 ( 29 ( 29 * * * x ) (x f x x ) (x f x ) (x f x x ) (x f ) (x f x x ) (x f ) (x f x x ) (x f x ) (x f x x ) (x f m equilibriu ) (x f x x x ) (x f x f Z 1 * 1 * 1 * 1 1 1 * 1 * 1 * 1 1 1 * 1 * 1 1 1 * 1 * 1 1 1 * 1 * 1 * 1 1 st 1 * 1 * 1 1 st 1 to Z of ce independen or dependence a either reveals (6) cond. T K : Conclusion and , , ) 6 together (3) and (2) Eqs. from Z of ce independen an reveal 4) (2 conds. T K : Conclusion and , , ) 5 range feasible of boarder at the min or max local min or max boundary Boarder and , , ) 4 range) feasible in neighbors n (lower tha minimum boundary Local and , , ) 3 range) feasible in neighbors an (higher th maximum boundary Local on Z of dependence a reveals (1) cond. T K : Conclusion and , , ) 2 range feasible w/in the is max local maximum interior Familiar slackness ary complement value derivative 1 such that slackness ary complement called product their is and variable, endogenous an of value m equilibriu the is , respect to with Z of derivative 1 is Let 1) Polya solution" the Examining " conditions maximum local : step1 conditions Tucker  Kuhn and g programmin Nonlinear 13.1 = = = = = = = = < = = = 4 Y X Y X Y X Figure 1 Figure 3 Figure 2 ( 29 ( 29 ( 29 slackness ary complement e nonnegativ marginal 409) (p. maximum a for conditions Tucker  Kuhn constraint Multiple conditions Tucker  Kuhn and g programmin Nonlinear 13.1 2 1 =  = =  = i i j j j j j Z x g r Z Z x x h g f Z i i j i x j j x x x x 5 ( 29 ( 29 [ ] ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 and , , ) 5 and , , ) 4 and , , ) 3 and , , ) 2 ) 1 Slackness ary Complement : 1 Step conditions Tucker  Kuhn and g programmin Nonlinear 1 . 13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * 1 * 1 2 1 1 * 1 * 1 2 1 1 * 1 * 1 2 1 * 1 * 1 2 1 2 1 2 2 2 1 1 1 2 1 = = = = = = = = < = = = = + + = Z ,x x g r Z or Z ,x x g r Z Z x x h g f Z or Z x x h g f Z ,x x h r ,x x g r ,x x f Z x x x x x x x x x x 6 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 [ ] ( 29 ( 29 ( 29 ( 29 3 , 2 , 1 2 , 1 ) 9 and , , ) 8 and , , ) 7 ) 6 ) 5 , for solve . . , , ) 4 . . ) 3 . . ) 2 maximize ) 1 Polya obtained" be cannot connection immediate an if problem auxilliary an Find " , for solve . . ) 4 . . ) 3 . . ) 2 maximize 1) 405 404 pp. (maximum) s constraint inequality of Effect : 2 Step conditions...
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 Spring '11
 richards

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