{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw5_sol - 10-725 Optimization Spring 2008 Homework 5...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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, α, β ) = 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 α i ( || y i || 2- t i ) + m X i =1 β i t i L ( x, y, t, u, v, α, β ) = 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 α i q y T i y i- t i + m X i =1 β i t i α i ≥ ∀ i β 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 + α i y i p y T i y i = 0 u i = α i y i p y T i y i || u i || 2 = q u T i u i = v u u t α i y i p y T i y i ! T α i y i p y T i y i ! || u i || 2 = α i v u u t y T i y i p y T i y i 2 !...
View Full Document

{[ snackBarMessage ]}

### Page1 / 13

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online