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v2006.03.09 - Convex Optimization

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206 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSNecessary and sufficient discretization (2.13.4.2.1) allows relaxation of thesemi-infinite number of conditions w ≽ 0 instead to w ∈ {e i , i=1... M} theextreme directions of the nonnegative orthant. Each extreme direction picksout an entry from the vector-valued function and its directional derivative,satisfying Theorem 3.1.1.5.1.The vector-valued function case (424) is therefore a straightforwardapplication of the first-order convexity condition for real functions to eachentry of the vector-valued function.3.1.1.7 Second-order convexity condition, vector-valued functionAgain, by discretization, we are obliged only to consider each individual entryf i of a vector-valued function f .For f(X) : R p →R M , a twice differentiable vector-valued function withvector argument on open convex domain,∇ 2 f i (X) ≽0 ∀X ∈ domf , i=1... M (427)S p +is a necessary and sufficient condition for convexity of f .Strict inequality is a sufficient condition for strict convexity, but that isnothing new; videlicet, the strictly convex real function f i (x)=x 4 does nothave positive second derivative at each and every x∈ R . Quadratic formsconstitute a notable exception where the strict-case converse is reliably true.3.1.1.8 second-order ⇒ first-order conditionFor a twice-differentiable real function f i (X) : R p →R having open domain, aconsequence of the mean value theorem from calculus allows compression ofits complete Taylor series expansion about X ∈ domf i (D.1.6) to threeterms: On some open interval of ‖Y ‖ so each and every line segment[X,Y ] belongs to domf i , there exists an α ∈ [0, 1] such that [259,1.2.3][27,1.1.4]f i (Y ) = f i (X)+ ∇f i (X) T (Y −X)+ 1 2 (Y −X)T ∇ 2 f i (αX +(1 −α)Y )(Y −X)(428)The first-order condition for convexity (417) follows directly from this andthe second-order condition (427).

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