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MIT6_235S10_hw04_sol

# MIT6_235S10_hw04_sol - 6.253 Convex Analysis and...

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6.253: Convex Analysis and Optimization Homework 4 Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 Let f : R n �→ R be the function n 1 f ( x ) = p | x i | p i =1 where 1 < p . Show that the conjugate is n 1 q f ( y ) = q | y i | , i =1 where q is defined by the relation 1 1 + = 1 . p q Solution . Consider first the case n = 1. Let x and y be scalars. By setting the derivative of xy (1 /p ) | x | p to zero, and we see that the supremum over x is attained when sgn( x ) | x | p 1 = y , which implies that xy = | x | p and | x | p 1 = | y | . By substitution in the formula for the conjugate, we obtain p 1 1 1 1 f ( y ) = | x | p p | x | p = (1 p ) | x | p = q | y | p 1 = q | y | q . We now note that for any function f : R n �→ ( −∞ , ] that has the form f ( x ) = f 1 ( x 1 ) + + f n ( x n ) , · · · where x = ( x 1 , . . . , x n ) and f i : R �→ ( −∞ , ], i = 1 , . . . , n , the conjugate is given by f ( y ) = f 1 ( y 1 ) + + f n ( y n ) , · · · where f i : R �→ ( −∞ , ] is the conjugate of f i , i = 1 , . . . , n . By combining this fact with the result above, we obtain the desired result. 1

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(a) Show that if f 1 : R Problem 2 n ( −∞ , ] and f 2 : R n �→ ( −∞ , ] are closed proper convex functions, with conjugates denoted by f 1 and f 2 , respectively, we have f 1 ( x ) f 2 ( x ) , x R n , if and only if f 1 ( y ) f 2 ( y ) , y R n . (b) Show that if C 1 and C 2 are nonempty closed convex sets, we have C 1 C 2 , if and only if σ C 1 ( y ) σ C 2 ( y ) , y R n . Construct an example showing that closedness of C 1 and C 2 is a necessary assumption.
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MIT6_235S10_hw04_sol - 6.253 Convex Analysis and...

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