# 2 2 2 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 1 2 1 1 2 2 1 2 2

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2 2 2 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 1 2 1 1 2 2 1 2 2 1 1 Use definition to argue that ( ) defined on is a convex function Let , , (0,1) ( (1 ) ) (1 ) (1 ) ( ) 2 (1 )( ) (1 ) ( ) ( ) 2 (1 )( f x x x R x y R f x y x y x y x x x y x y y y f x x y 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 ) (1 ) ( ) ( ) (1 )( ) (1 ) ( ) ( (1 )) ( ) ( (1 ) (1 ) ) (y) ( ) (1 ) ( ) x y f y f x x y x y f y f x f f x f y 50
𝑓 isconcaveiff െ𝑓 isconvex; 𝑓 isstrictlyconcaveiff െ𝑓 isstrictlyconvex. Alinearfunctionisbothconcaveandconvex If 𝑓 and 𝑔 arebothconcave(orconvex),then 𝑓 ൅ 𝑔 isalsoconcave(convex) If 𝑓 and 𝑔 arebothconcave(orconvex),andoneofthenisstrictlyconcave(strictly convex),then 𝑓 ൅ 𝑔 isstrictlyconcave(strictlyconvex) 𝑓: 𝑆 → 𝑅 isconcave,thentheupperlevelset ሼ𝑥 ∈ 𝑆|𝑓ሺ𝑥ሻ ൒ 𝑡ሽ isconvex, ∀𝑡 ∈ 𝑅 𝑓: 𝑆 → 𝑅 isconvex,thenthelowerlevelset ሼ𝑥 ∈ 𝑆|𝑓ሺ𝑥ሻ ൑ 𝑡ሽ isconvex, ∀𝑡 ∈ 𝑅 51 Properties
𝑓 concaveiff for 𝑘 ൒ 2 𝑓 𝜆 𝑥 ൅ ⋯ ൅ 𝜆 𝑥 𝜆 𝑓 𝑥 ൅ ⋯ ൅ 𝜆 𝑓 𝑥 where 𝜆 ,…, 𝜆 ൒ 0 , 𝜆 ൌ 1 ௜ୀଵ , 𝜆 𝑥 ൅ ⋯ ൅ 𝜆 𝑥 iscalled convexcombination of 𝑥 , … , 𝑥 Moregenerally, Jensen’sinequality :If 𝑓 isconcave,then 52 Properties(continued) ( ) ( ) ( ) , or ( ) ( ) where ( ) is a density function such that ( ) 1, and is the random variable with density function f xg x dx f x g x dx f EX E f X g g x dx X g   
Theorem (First‐ordercharacterizationofconcave(convex)functions): Let 𝑓: 𝑆 → 𝑅 be 𝐶 functiondefinedonanopen,convexset 𝑆 ,then 1. 𝑓 isconcave ⇔ 𝑓 𝑣 ൑ 𝑓 𝑢 ൅ 𝛻𝑓ሺ𝑢ሻ ∙ ሺ𝑣 െ 𝑢ሻ forall 𝑢, 𝑣 ∈ 𝑆 .Inotherwords,the curveisalwaysbelowanytangentplane 2. 𝑓 isstrictlyconcave ⇔𝑓 𝑣 ൏ 𝑓 𝑢 ൅ 𝛻𝑓 𝑢 𝑣 െ 𝑢 forall 𝑢, 𝑣 ∈ 𝑆 and 𝑢 ് 𝑣 . 3. 𝑓 isconvex ⇔ 𝑓 𝑣 ൒ 𝑓 𝑢 ൅ 𝛻𝑓ሺ𝑢ሻ ∙ ሺ𝑣 െ 𝑢ሻ forall 𝑢, 𝑣 ∈ 𝑆 .Inotherwords,the curveisalwaysaboveanytangentplane 4. 𝑓 isstrictlyconvex ⇔𝑓 𝑣 ൐ 𝑓 𝑢 ൅ 𝛻𝑓 𝑢 𝑣 െ 𝑢 forall 𝑢, 𝑣 ∈ 𝑆 and 𝑢 ് 𝑣 . 53