8-ConvexFunctionals

8-ConvexFunctionals - MATHEMATICS 116 FALL 2007 CONVEXITY...

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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #8 (Convex Functionals) Last modified: December 2, 2008 Reading. Luenberger, Chapter 7, sections 7.8 and 7.10-7.12. We are skipping the starred sections 7.9 and 7.13. Lecture topics. 1. From local to global In single-variable calculus, a useful theorem is that if function(al) f ( x ) satisfies f 00 ( x ) on the interval [ a,b ], then any place where f ( x ) = 0 is not merely a local extremum, but a global minimum. Sketch a graph to illustrate this theorem. Let x = αb +(1- α ) a . Apply the mean-value theorem to f on the intervals [ a,x ] and [ x,b ]. Suppose that f 00 ( x ) > 0 on [ a,b ]. Apply the mean-value theorem to f to show that f is convex on [ a,b ] . 1 2. Convex functionals Suppose that C (generalizing [ a,b ]) is a convex subset of a linear vector space X . Then a real-valued functional f ( x ) is convex if f ( αx 1 + (1- α ) x 2 ) ≤ αf ( x 1 ) + (1- α ) f ( x 2 ) . Strange example 1: Define f ( x ) = 1 if x = 0, f ( x ) = x 2 if x > 0. This functional is convex on [0 , ∞ ]. Sketch a graph to illustrate. Strange example 2: Check that on L 2 [0 , 1] the functional f ( x ) = Z 1 ( x 2 ( t ) + | x ( t ) | ) dt is convex. 2 3. Global results This is Luenberger, page 191, Proposition 1. Suppose that f is a convex functional defined on convex subset C of a normed space X . Define the infimum μ = inf x ∈ C f ( x ) . Prove that the subset Ω where f ( x ) = μ is convex. Prove that if x is a local minimum of f , then it is also a global minimum. Prove that the set above the graph of f , [ f,C ] = { ( r,x ) ∈ R × X : X ∈ C ; r ≥ f ( x ) } , is convex. 3 4. Conjugate convex functionals – an example Let C be the convex set [0 , 2] and let f ( x ) be the functional f ( x ) = x 2 , defined just on the set C . Any element of the dual space can be represented as < x,x * > = mx ....
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8-ConvexFunctionals - MATHEMATICS 116 FALL 2007 CONVEXITY...

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