constrained optimization and kuhn-tucker...
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![Page 1: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/1.jpg)
Author Name
Constrained Optimization and Kuhn-Tucker Conditions
Joseph Tao-yi Wang2019/5/23
(Calculus 4, 18.4)
1
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 2: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/2.jpg)
Author Name
Theorem 18.4 (Several Inequality Constraints)• Suppose f, g1,…, gk be C1 functions on Rn
• Let solve max. problem
• Notation: Constraints g1,…, gk0binds
• Constraints gk0+1,…, gk do not binds
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 3: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/3.jpg)
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Theorem 18.4 (Several Inequality Constraints)• Binding constraints g1,…, gk0
satisfies NDCQ
if its Jacobian matrix has maximum rank k0
• Or, row vectors
are linearly independent
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 4: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/4.jpg)
Author Name
Theorem 18.4 (Several Inequality Constraints)• Row vectors
are linearly independent if
implies
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 5: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/5.jpg)
Author Name
Theorem 18.4 (Several Inequality Constraints)For
• There exists such that
a)
b)
c)
d)
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 6: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/6.jpg)
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Theorem 18.7 (Kuhn-Tucker)• Suppose f, g1,…, gk be C1 functions on Rn
• Let solve max. problem
• NDCQ satisfied if has maximum rank
Binding constraints Positive xj
where
• Exists such that7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 7: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/7.jpg)
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Theorem 18.7 (Kuhn-Tucker)For
A.
B.
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 8: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/8.jpg)
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Theorem 18.7 (Kuhn-Tucker)• Let • Binding constraints g1,…, gk0
satisfies NDCQ if the following matrix has maximum rank k0
• Or, row vectors
(1st n0 elements of ) are linearly independent7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 9: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/9.jpg)
Author Name
Theorem 18.7 (Kuhn-Tucker)• Row vectors (1st n0 elements of )
are linearly independent if
implies
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 10: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/10.jpg)
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Exercise 18.14 (Generalize Example 18.9)
• NDCQ?
• FOC?
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 11: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/11.jpg)
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Exercise 18.14 (Generalize Example 18.9)
• NDCQ?
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 12: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/12.jpg)
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Exercise 18.14 (Generalize Example 18.9)
FOC:
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 13: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/13.jpg)
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Exercise 18.14 (Generalize Example 18.9)
Solution:
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 14: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/14.jpg)
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Ex: Sales-Maximizing Firm with Advertising• Suppose R(y, a), C(y) are C1 functions
satisfying C’(y)>0, R(0, a)=0,
• Firms choose y, a from R+ to maximize revenue R(y, a), without letting profit drop below m > 0
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 15: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/15.jpg)
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Ex: Sales-Maximizing Firm with Advertising
• Suppose C’(y)>0, R(0, a)=0,
1. Show that the constraint binds, so the firm will maintain minimum profit
2. Show that output (if positive) is larger than profit-maximizing output
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 16: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/16.jpg)
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Ex: Sales-Maximizing Firm with Advertising
• Wait, does NDCQ always hold? No!
= 0 if MR = MC and advertising MR = 1
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 17: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/17.jpg)
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The Meaning of the Multiplier
Joseph Tao-yi Wang2019/5/23
(Calculus 4, 19.1)
17
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 18: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/18.jpg)
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Theorem 19.1 (Single Equality Constraint)• Consider
• Let f, h be continuously differentiable (C1 )
• For any fixed value a, let
be the solution which satisfies NDCQ.
– (Implicit Function Theorem applies!)
• Suppose are functions of
• Then,
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
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Theorem 19.2 (Several Equality Constraints)For
• Let f, h1,…, hm be C1 functions on Rn
• For , is
the solution with Lagrange Multipliers
which satisfies NDCQ
• Suppose are functions of , then(j=1,…,m)
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 20: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/20.jpg)
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Theorem 19.3 (Several Inequality Constraints)For
• Let f, g1,…, gk be C1 functions on Rn
• For , is
the solution with Lagrange Multipliers
which satisfies NDCQ
• Suppose are functions near , then(j=1,…,k)
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 21: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/21.jpg)
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Ex: Limited Resources, Profit-maximizing FirmFor
• Firm provide services 1,…,n at levels x1,…, xn
• To maximize profit f(x1, …, xn) by allocating inputs 1,…,k at levels g1,…, gn
– Inputs 1,…,k constrained by
– Addition profit for adding 1 more unit of input j
= firm’s WTP for adding 1 more unit of input j
7/3/2019 Envelope TheoremJoseph Tao-yi Wang
![Page 22: Constrained Optimization and Kuhn-Tucker Conditionshomepage.ntu.edu.tw/~josephw/Calculus4_19S_18-6_Kuhn...Kuhn-Tucker Conditions Joseph Tao-yi Wang 2019/5/23 (Calculus 4, 18.4) 1 7/3/2019](https://reader033.vdocument.in/reader033/viewer/2022052101/603a9a5b796cb960a042fc79/html5/thumbnails/22.jpg)
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Exercise 18.14 (Generalize Example 18.9)
7/3/2019 Envelope TheoremJoseph Tao-yi Wang