hm3 - 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 ) . 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 . Problem 0.3 Let f : n ¯ be given. Show that f is strongly convex with modulus β > 0 if, and only if, ˜ f := f ( β/ 2) b · b 2 Conv( n ). Problem 0.4 Let f 1 , f 2 Conv( n ) be such that dom f 1 dom f 2 n = . a) Show that cl f 1 + cl f 2 cl ( f 1 + f 2 ). b) Show that if ri (dom f 1 ) ri (dom f 2 ) n = then cl f 1 + cl f 2 = cl ( f 1 + f 2 ). c) Give an example to show that the equality in b) fails if ri (dom
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