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# sol3 - ISyE 7682 Fall 2010 3rd Homework Problem 0.1 For a...

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ISyE 7682 Fall 2010 – 3rd Homework Problem 0.1 For a set S ⊆ ℜ n , show the following equivalences: a) S is closed if, and only if, I S is a lower semi-continuous function; b) S is convex if, and only if, I S Conv( n ) ; c) S is a nonempty closed convex set if, and only if, I S C onv ( n ) . Solution . a) Since epi I S = S × [0 , ), it follows that S is closed if, and only if, epi I S is closed, and the latter condition is equivalent to the lower semi-continuity of I S , in view of Proposition 2.3.3. b) S is closed if, and only if, epi I S = S × [0 , ) is convex, and the latter condition is equivalent to the convexity of I S , in view of Proposition 2.1.6. c) In view of a) and b), S is a closed convex set if, and only if, I S is a lower semi-continuous convex function. Moreover, the nonemptyness of S is equivalent to the properness of I S . Since C onv ( n ) is exactly the set of all proper lower semi-continuous convex functions, c) follows. Problem 0.2 Show the following statements: a) for given functions f 1 , f 2 : n ¯ , we have f 1 f 2 if, and only if, epi f 1 epi f 2 ; in particular, f 1 = f 2 if, and only if, epi f 1 = epi f 2 ; b) for given function f : n ¯ and a family of functions f λ : n ¯ , λ Λ , we have f = sup λ Λ f λ if, and only if, epi f = λ Λ epi f λ .

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