{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

bmhicks_homework_4

bmhicks_homework_4 - Brandy Hicks MTHSC 974.001 Homework 4...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Brandy Hicks September 19, 2007 MTHSC 974.001 - Homework # 4 1. Let f and g be convex functions in the modern sense. Let x be an interior point of the domain of f + g . Prove: (a) D - ( f + g )( x ) D - f ( x ) + D - g ( x ) Proof: Let f,g conV ( R ). Then let x 0 int(dom( f + g )), so x 0 int(dom( f )) and x 0 int(dom( g )), Hence dom( f + g )( x 0 ) = dom f ( x 0 ) dom g ( x 0 ) , Now as we proved in class, if x 0 int(dom( f )), then D - f ( x 0 ) = lim x x 0 f ( x ) - f ( x 0 ) x - x 0 = sup x<x 0 f ( x ) - f ( x 0 ) x - x 0 < similarly D + f ( x 0 ) = lim x x 0 f ( x ) - f ( x 0 ) x - x 0 = inf x>x 0 f ( x ) - f ( x 0 ) x - x 0 > -∞ Then, since x 0 int(dom( f + g )) and dom( f + g ) = dom f dom g D - ( f + g )( x 0 ) = lim x x 0 ( f + g )( x - x 0 ) x - x 0 = sup x<x 0 ( f + g )( x - x 0 ) x - x 0 Clearly since sup x<x 0 ( f + g )( x ) sup x<x 0 f ( x ) + sup x<x 0 g ( x ) then D - f ( x 0 ) + D - g ( x 0 ) D - ( f + g )( x 0 ) sup x<x 0 ( f + g )( x ) - ( f + g )( x 0 ) x - x 0 = D - ( f + g )( x 0 ) D - ( f + g )( x ) D - f ( x ) + D - g ( x ) 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(b) D + ( f + g )( x ) D + f ( x ) + D + g ( x ) Proof: Let f,g conV ( R ). Then let x 0 int(dom( f + g )), so x 0 int(dom( f )) and x 0 int(dom(
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

bmhicks_homework_4 - Brandy Hicks MTHSC 974.001 Homework 4...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online