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lecture09

# lecture09 - Yinyu Ye MS&E Stanford Applications of KKT...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 1 Applications of KKT Conditions and Nonlinear Optimization Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 2 Recall Portfolio Management Let r denote the expected return vector and V denote the co-variance matrix of an investment portfolio, and let x be the investment proportion vector. minimize x T V x subject to r T x π, e T x = 1 , x 0 , where e is the vector of all ones. This is a quadratic program .

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 3 Allow Call Option minimize x T V x subject to r T x π, e T x = 1 . Optimality Conditions: 2 V x - λ r - μ e = 0 , λ 0 , λ ( r T x - π ) = 0 . μ = 2 x T V x - λ · π. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 4 Solution to linear equations Try case λ > 0 first 2 V - r - e r T 0 0 e T 0 0 x λ μ = 0 π 1 . If λ < 0 , then try case λ = 0 2 V - e e T 0 x μ = 0 1 and verify that r T x > π .
Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 5 Portfolio Example r 1 = 0 . 0825 , r 2 = 0 . 1125 , V = 0 . 0139 0 . 0056 0 . 0056 0 . 0261 π = 0 . 1 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 6 Knapsack Maximization maximize j f j ( x j ) subject to j p j x j w, x j 0 , j Optimality Conditions: λ - f 0 j ( x j ) p j 0 , λ 0 , x j λ - f 0 j ( x j ) p j = 0 , λ ( w - X j p j x j ) = 0 , j. Implications: λ = f 0 j ( x j ) p j , x j > 0; and λ f 0 j ( x j ) p j , x j = 0 Invest only to the projects whose marginal return per dollar is above a threshold λ .

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 7 Recall Fisher’s Exchange Market Buyers have money ( w i ) to buy goods and maximize their individual utility functions ; Producers sell their goods for money. The equilibrium price is an assignment of prices to goods so as when every buyer buys an maximal bundle of goods then the market clears , meaning that all the money is spent and all goods are sold.
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lecture09 - Yinyu Ye MS&E Stanford Applications of KKT...

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