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**Unformatted text preview: **MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #7 (Calculus of Variations) Last modified: December 6, 2007 Reading. Luenberger, Chapter 7, sections 7.1-7.5 and 7.7. 1 Lecture topics. 1. Review of the two-variable case Let f ( u,v ) = u 2 v u 2 + v 2 with f (0 , 0) = 0 First we use the Gateaux approach, without introducing a norm into R 2 Calculate ∂f ∂u and ∂f ∂v Calculate the directional derivative at (0,0) along the vector h = (1 ,m ): δf ((0 , 0);(1 ,m )) = lim α → 1 α f ( α (1 ,m ))- f (0 , 0)) True or false?: the directional derivative in this case is a linear of its second argument (the increment) What is the only linear function of h , δf ( θ ; h ) that could possibly work in this case, given the values of the partial derivatives? Introduce the Euclidean norm, so r = || x || = √ u 2 + v 2 . Show that lim || x ||→ 1 || x || ( f ( x )- f (0 , 0)) 6 = 0 Example 1 on page 171, specialized to this case, says that if X = E 2 and f ( u,v ) is a functional with continuous partial derivatives, then δf ( x ; h ) = ∂f ∂u h 1 + ∂f ∂v h 2 . Is there any contradiction? 2 2. Gateaux and Frechet differentials Consider a transformation T : X → Y . The space Y has to be normed, because we need to take a limit to define the differential. Gateaux generalizes the concept of directional derivative from multivariable calculus. It defines a functional of the increment vector h at any point x ∈ X , provided the limit exists. δT ( x ; h ) = lim α → 1 α ( T ( x + αh )- T ( x )) Assuming that the limit exists for all h , is the Gateaux differential guaran- teed to be linear in h ? Frechet generalizes the concept of derivative from multivariable calculus. It requires X to be normed. It defines a continuous linear functional δT ( x ; h ) of the increment vector h at any point x ∈ X , provided the error in the linear approximation goes to zero faster than || h || . lim || h ||→ 1 || h || || T ( x + h )- T ( x )- δT ( x ; h )) || = 0 In the example on the preceding page, does the Frechet differential of f at (0,0) exist? 3 3. A few familiar thoerems The following results are proved in Luenberger. What are the corresponding theorems from multivariable calculus? • If T has a Frechet differential, it is unique. • If T has a Frechet differential at x , it has a Gateaux differential at x , and the two are equal. • If T has a Frechet differential at x , it is continuous at x . Prove this one. • If function f on E n has continuous partial derivatives, then it has the Frechet differential δf ( x ; h ) = n X i =1 ∂f ∂x i h i . Just beacuse f has a Gateaux differential at x , it is not necessarily even con- tinuous at x . For a counterexample, define f ( u,v ) = u 2 v u 4 + v 2 with f (0 , 0) = 0 and consider the sequence x i = { 1 i , 1 i 2 } 4 4. Differentials in infinite-dimensional spaces Consider the nonlinear functional f ( x ) = Z a x ( t ) 2 dt ....

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