v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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256 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.6.1.0.4 Exercise. Order of composition.Real function f =x −2 is convex on R + but not predicted so by results inExample 3.6.1.0.3 when g=h(x) −1 and h=x 2 . Explain this anomaly. The following result for product of real functions is extensible to innerproduct of multidimensional functions on real domain:3.6.1.0.5 Exercise. Product and ratio of convex functions. [61, exer.3.32]In general the product or ratio of two convex functions is not convex. [230]However, there are some results that apply to functions on R [real domain].Prove the following. 3.20(a) If f and g are convex, both nondecreasing (or nonincreasing), andpositive functions on an interval, then fg is convex.(b) If f , g are concave, positive, with one nondecreasing and the othernonincreasing, then fg is concave.(c) If f is convex, nondecreasing, and positive, and g is concave,nonincreasing, and positive, then f/g is convex.3.6.2 first-order convexity condition, real functionDiscretization of w ≽0 in (498) invites refocus to the real-valued function:3.6.2.0.1 Theorem. Necessary and sufficient convexity condition.[61,3.1.3] [132,I.5.2] [391,1.2.3] [42,1.2] [330,4.2] [306,3]For real differentiable function f(X) : R p×k →R with matrix argument onopen convex domain, the condition (conferD.1.7)f(Y ) ≥ f(X) + 〈∇f(X) , Y − X〉 for each and every X,Y ∈ domf (607)is necessary and sufficient for convexity of f . Caveat Y ≠ X and strictinequality again provide necessary and sufficient conditions for strictconvexity. [199,B.4.1.1]⋄3.20 Hint: Prove3.6.1.0.5a by verifying Jensen’s inequality ((497) at µ= 1 2 ).

3.6. GRADIENT 257f(Y )[ ∇f(X)−1]∂H −Figure 77: When a real function f is differentiable at each point in its opendomain, there is an intuitive geometric interpretation of function convexity interms of its gradient ∇f and its epigraph: Drawn is a convex quadratic bowlin R 2 ×R (confer Figure 160, p.680); f(Y )= Y T Y : R 2 → R versus Y onsome open disc in R 2 . Unique strictly supporting hyperplane ∂H − ⊂ R 2 × R(only partially drawn) and its normal vector [ ∇f(X) T −1 ] T at theparticular point of support [X T f(X) ] T are illustrated. The interpretation:At each and every coordinate Y , there is a unique hyperplane containing[Y T f(Y ) ] T and supporting the epigraph of convex differentiable f .

3.6. GRADIENT 257f(Y )[ ∇f(X)−1]∂H −Figure 77: When a real function f is differentiable at each point in its opendomain, there is an intuitive geometric interpretation of function convexity interms of its gradient ∇f and its epigraph: Drawn is a convex quadratic bowlin R 2 ×R (confer Figure 160, p.680); f(Y )= Y T Y : R 2 → R versus Y onsome open disc in R 2 . Unique strictly supporting hyperplane ∂H − ⊂ R 2 × R(only partially drawn) and its normal vector [ ∇f(X) T −1 ] T at theparticular point of support [X T f(X) ] T are illustrated. The interpretation:At each and every coordinate Y , there is a unique hyperplane containing[Y T f(Y ) ] T and supporting the epigraph of convex differentiable f .

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