Chiang_Ch13

# Chiang_Ch13 - 1 Ch 13 Further Topics in Optimization 13.1...

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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 Maximum-value 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 non-negativity restrictions • Step 2: Effect of inequality constraints • Interpretation of the Kuhn-Tucker 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 λ λ λ λ λ λ 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 λ * 1 * 1 2 1 1 λ λ * 1 * 1 2 1 1 λ * 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 λ ,x x g r λ ,x x f Z x x x x x x x x x x λ λ λ λ λ λ λ λ 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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## This note was uploaded on 11/09/2011 for the course ECON 101 taught by Professor Richards during the Spring '11 term at Cambrian College.

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Chiang_Ch13 - 1 Ch 13 Further Topics in Optimization 13.1...

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