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Ü%T75 F Ý â X N I H S u _ ö È/²8 b "g  ï 8 º É ß. Theorem 1 A function f Rn!Ris convex if and only if the function g R!Rgiven by g(t) = f(x ty) is convex (as a univariate function) for all xin domain of f and all y2Rn (The domain of ghere is all tfor which x tyis in the domain of f) Proof This is straightforward from the de nition. U Z W l l Á Á Á X i X P } X i l o l ( l ( Ç î ì í õ r r î í X.

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Theorem 1 A function f Rn!Ris convex if and only if the function g R!Rgiven by g(t) = f(x ty) is convex (as a univariate function) for all xin domain of f and all y2Rn (The domain of ghere is all tfor which x tyis in the domain of f) Proof This is straightforward from the de nition. à ý í Ü æ Ñ w à ý F x x É _ ¿ í Ë & ´ æ F w ¾ I y !. B N g A e C X g K N { h J ŁA u I C 90 j ^ ϔM ^ L b ` N X A { f B ^ I E E Eetc 07,08 N f ڕ ͈ړ ܂ ̂ŁA @.

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De nition The Taylor series for f(x) centered at x= ais f(x) = X1 n=0 f(n)(a) n!. F is twice differentiable if domf is open and the Hessian ∇2f(x) ∈ Sn, ∇2f(x)ij = ∂2f(x) ∂xi∂xj, i,j = 1,,n, exists at each x ∈ domf 2ndorder conditions for twice differentiable f with convex domain • f is convex if and only if ∇2f(x) 0 for all x ∈ domf • if ∇2f(x) ≻ 0 for all x ∈ domf, then f is strictly convex. Title (03)draftindd Author Created Date 12/6/13 752 AM.

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