Unformatted text preview: f is convex if and only if its domain is convex and H ( x ) ” ∀ x ∈ dom f . Hint: Use the ﬁrst order condition for convexity. You may ﬁrst consider a function f : R → R and then extend the solution to f : R n → R . 4. Consider the following functions. First check these functions for convexity/concavity using Jensen’s inequality. Then verify your answer by plotting these functions in MATLAB. You may use the surf command. (a) f ( x ) = e x1 on R , (b) f ( x 1 ,x 2 ) = x 1 x 2 on R 2 ++ , (c) f ( x 1 ,x 2 ) = 1 / ( x 1 x 2 ) on R 2 ++ , (d) f ( x 1 ,x 2 ) = x 1 /x 2 on R 2 ++ , (e) f ( x 1 ,x 2 ) = x 2 1 /x 2 on R × R ++ , (f) f ( x 1 ,x 2 ) = x α 1 x 1α 2 where 0 ≤ α ≤ 1 R 2 ++ . 1...
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 Spring '11
 CenkToker
 Operations Research, Optimization, Convex function

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