1. Show that log-sum-exp f(x) = log(e1 ++ en) is convex [10 points]. 2. (Jensen's inequality.)...

  

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1. Show that log-sum-exp f(x) = log(e1 ++ en) is convex [10 points]. 2. (Jensen's inequality.) Use the definition of a concave function f, to show that m f(; > @if(x) () i=1 where 3. Find the minimizer of a = 1, a; 20 [10 points]. for some given d [10 points]. min r>0:2=1x=1 m 72 dx + x log xi i=1 1. Show that log-sum-exp f(x) = log(e1 ++ en) is convex [10 points]. 2. (Jensen's inequality.) Use the definition of a concave function f, to show that m f(; > @if(x) () i=1 where 3. Find the minimizer of a = 1, a; 20 [10 points]. for some given d [10 points]. min r>0:2=1x=1 m 72 dx + x log xi i=1