hw5_sol - 10-725 Optimization, Spring 2008: Homework 5...

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Unformatted text preview: 10-725 Optimization, Spring 2008: Homework 5 Solutions Due: Wednesday, April 23, beginning of the class (Acknowledgement: The solutions for Q1 and Q2 are adapted from Prasanna Veglagapudi) 1 Conjugate functions [Han, 20 points] Derive the conjugates of the following functions (4 points each): (a) Max function. f ( x ) = max i =1 ,...,n x i on R n . F SOLUTION: f * ( y ) = , i y i = 1 , y i i , otherwise (b) Sum of largest elements. f ( x ) = r i =1 x [ i ] on R n . (where x [ i ] denotes the i-th largest element in x ). F SOLUTION: f * ( y ) = , i y i = r, y i i , otherwise (c) Power function. f ( x ) = x p on R ++ , where p > 1. F SOLUTION: f * ( y ) = y y p 1 / ( p- 1)- y p p/ 1- p , y > , y (d) Geometric mean. f ( x ) =- ( Q n i =1 x i ) 1 /n on R n ++ . F SOLUTION: f * ( y ) = , Q i (- y i ) (1 /n ) n y i < i , otherwise (e) Negative generalized logarithm for second-order cone. f ( x, t ) =- log( t 2- x T x ) on { ( x, t ) R n R |k x k 2 < t } . 1 F SOLUTION: f * ( y, s ) =- 2 + log 4 s 2- y T y , s < s 2 > y T y s R , y R n , otherwise 2 Dual of SOCP [Han, 20 points] Show that the dual of the SOCP minimize f T x s . t . k A i x + b i k 2 c T i x + d i , i, = 1 , . . . , m with variables x R n can be expressed as maximize m X i =1 ( b T i u i + d i v i ) s . t . m X i =1 ( A T i u i + c i v i ) + f = 0 k u i k 2 - v i , i = 1 , . . . , m, with variables u i R n , v i R , i = 1 , . . . , m . The problem data are f R n , A i R n i n , b i R n i , c i R n and d i R , i = 1 , . . . , m . Derive the dual in the following two ways. (a) [10 points] Introduce new variables y i R n i and t i R and equalities y i = A i x + b i , t i = c T i x + d i , and derive the Lagrange dual. F SOLUTION: First, restate the problem with new variables y i and t i : min x,y,t f T x s.t. y i = A i x + b i t i = c T i x + d i || y i || 2 t i t i Next, solve for the Lagrangian: L ( x, y, t, u, v, , ) = f T x + m X i =1 u T i ( A i x + b i- y i ) + m X i =1 v i ( c T i x + d i- t i ) + m X i =1 i ( || y i || 2- t i ) + m X i =1 i t i L ( x, y, t, u, v, , ) = f T x + m X i =1 u T i ( A i x + b i- y i ) + m X i =1 v i ( c T i x + d i- t i ) + m X i =1 i q y T i y i- t i + m X i =1 i t i i i i i Use the derivatives of the Lagrangian to get dual constraints: 2 L x = f + m X i =1 ( A T i u i + c i v i ) = 0 m X i =1 ( A T i u i + c i v i ) + f = 0 L y i =- u i + i y i p y T i y i = 0 u i = i y i p y T i y i || u i || 2 = q u T i u i = v u u t i y i p y T i y i ! T i y i p y T i y i ! || u i || 2 = i v u u t y T i y i p y T i y i 2 !...
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hw5_sol - 10-725 Optimization, Spring 2008: Homework 5...

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