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MIT6_235S10_hw04

# MIT6_235S10_hw04 - 6.253 Convex Analysis and Optimization...

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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 Problem 2 (a) Show that if f 1 : R 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. Problem 3 Let X 1 , . . . , X r , be nonempty subsets of R n . Derive formulas for the support functions for X 1 + + X r , conv ( X 1 ) + + conv ( X r ), j r =1 X j , and conv r j =1 X j . · · · · · · 1

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Problem 4 Consider a function φ of two real variables x and z
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MIT6_235S10_hw04 - 6.253 Convex Analysis and Optimization...

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