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I.3
Introduction to the theory of convex conjugated function
Epigraph
RE:
)()( xExD
ofepi ,
E : set
Epigraph, is the set
)(, xRExepi
epi
lower semicontinuous
],(: E )()( xExD
)())(inf(lim xyNyxN
Assume that E is a topological space.
Define
is called lower semicontinuous (l.s.c) at x if
)()(inflim xyxy
i.e. for any
there is a neighborhood N of x such
that
0
Ν)()( yxy
f is l.s.c on E if f is l.s.c at each point of E
Exercise
)(xEx
is l.s.c on E if and only if
is open
is closed
epi
is l.s.c on E if and only if
is closed in E x R
is l.s.c on E if and only if
is closed
1
21
are l.s.c
then so is
If and 2
Iii
)(sup)( xx iIi
is l.s.c
then the upper envelope of
If
is a family of l.s.c functions on E
Iii i.e. the function definded by
is l.s.c
)(min)()(inf 0 xxxExEx
is attained
If E is compact and
is l.s.c on E ,then
)(inf xEx
i.e.
Convex
1,0, tEyx
is convex if Def
Suppose E is a vector space (real)
)()1()())1(( ytxtyttx
],(: E
epi
is convex if and only if
is convex in E x R
is a convex function then
is convex.
If
for R the set
Converse statement is not truein genernal
see next page
counter example
1
21
are convex
then so is
If and 2
Iii
is convex
then the upper envelope of
If
is a family of convex functions on E
Iii
Conjugated function
*
such that Given
Assume that E is a real n.v.s
)(D
],(: E
Define the conjugated function
of by EfxxffEx
,)(,sup)(*
proposition 1-9
then
is convex ,l.s.cSuppose
*
00
00
00
000
,
,,
,),(:,
,
.
,
sec
)(),(
kxf
epixkxf
kxfxkf
strictlyxandepigsepaaratin
eqtheofHhyperplaneclosedaisThere
REinxandepito
BanachHahnofformgeometricondApply
xDx
kxx
k
f
k
f
Dxk
xxk
f
kxx
k
f
xxk
fk
xkxf
k
xk
kxfxkxf
Dxxkxf
Dx
)(,sup)(
)()(,
)(,
)(,
)(,
0
0))((
,)(,
)()(,
)(
*
00
0000
RE :*
Def
)(,sup)( *** fxfxEf
Theorem I.10 (Fenchel-Moreau)
then
is convex ,l.s.cSuppose
**
ExxxHence
xx
xxfxf
fxfx
xxffce
ExFor
ExxxthatshowTo
furtherSupposeStep
Ef
Ef
Ef
Ex
)()(
)()(sup
)(,,sup
)(,sup)(
,)(,sup)(sin
)()()1(
0:1
**
***
*
**
0,),()12(
)(,)13(
,,)12(
..,,
9..
)(,
sec
)()()1(
)()(
)()()2(
0**
0
0**
0
00**
000**
**
kthenDxchooseIn
xkxf
Epixkxf
tsRandRkEf
Ipropofprooftheasthenexiststhere
xxBandEpiAwithThm
BanachHahnofformgeometricondtheApply
xxbythen
ExsomeforxxthatSuppose
oncontraditibyExxxthatshowTo
tioncontradicixkxf
haveweLetting
xkxf
xfxk
kx
k
f
k
fx
k
fx
kk
f
kxx
k
f
xkxf
DxFor
,)(,
,0
)()(,
,)()(
,
,)(
)(,
)()(,)12(
)(,0
0**
0
0**
0
00**
0
*00
**
*
)()(
)()(,sup
)(,)(,sup
)(,sup)(
)()(1
0,,..,
)(,)()(
),(
)(:2
0*
0*
0*
0
0*
0
*
***
**
0*
0
*0
fff
fxxff
fxfxxf
xxff
andCalculate
ExxxStep
andcslconvexis
Exfxfxx
bydefineDfLet
caseGeneralStep
Ex
Ex
Ex
)()(
)(,)()(,)(
),()(
)(,)(
)(,)(,sup
)()(,sup
)(,sup)(
**
0*
00*
0**
**
0*
0**
0*
00*
0
0*
0*
***
xx
fxfxfxfx
xxSince
fxfx
fxfffxff
fffxf
fxfx
ExFor
Ef
Ef
Ef
Example
**
xx )(
10
1)(*
fif
fiff
epi )(x
)()(
sup,sup)(,sup)(
10
1)(
1sup
sup
)(,sup
)(
**
1,1,
***
*
*
xx
x
xfxffxfx
fif
fiff
xf
xxf
xxf
f
fEffEfEf
Ex
Ex
Ex
Lemma I.4
then
is convex , Let
IntC
EC
IntCC
then IntC is convex
If
Theorem I.11
Ex 0
are convex and suppose that
and
such that
Suppose
there is
)(,)( 00 xx and
is continuous at 0x
see next page
)()(max ** ffEf
)()(inf xxEx
)()(sup ** ffEf
Observe
)(x usually appears for constrain(1)
)()()()( ** ffxx (2)
see next page
)()()()()2()1(
)2()(,)(
)(,)(
)1()(,)(
)(,)(
,
**
*
*
*
*
ffxx
xxff
xxff
xxff
xxff
EfExFor
The proof of Thm I.11
see next page
.
)(,
)(
,
)()(
11.,
)2(
)()(sup)()(inf
0
**
**
convexisBandBthen
xaRExBandIntCAto
ThmBanachHahnofformGeometricfirstApply
xatcontinuousisIntC
thenepiCLet
RanowSuppose
Efbffwith
holdIThmandbthenaIf
aorRaEither
baObserve
ffbandxxaPutEfEx
.
,,),(
,
..,,
)(Re
.
,
)()(
,,
sensebroadinBandCseparates
kxfx
whereequationof
hyperplanethetsRandRkEfthen
CAmember
sensebroadinBandAseparatesHthen
sensebroadBinandAseparatewhich
HHyperplanecloseda
BAHence
Bx
xax
thenAxif
BAverifyTo
0
0
,
)1,0(,,
)1,0(,
0)(),(
)(,
)(,
0
00Re0:
0,)14(
,,)15(
,,)14(
0
00
00
00
000
0
f
f
fxf
Bzzfxf
Bzzxf
smallsufficientifDxB
CDxxf
CDxxf
kassumeandioncontradictbyArgue
kfcallkClaim
kthenandxxchooseIn
Bxkxf
Cxkxf
)()(
)()()#(#)(#
)#(#)(
)(,
)(,
))((,)15(
)(#)(
)(,
)(,)14(
**
**
*
*
k
f
k
fba
bak
f
k
fb
akk
f
ak
xxk
f
akxkxf
xakxfkk
f
kxx
k
f
xkxf
Example
**
xx )(
10
1)(*
fif
fiff
epi )(x
)()(
sup,sup)(,sup)(
10
1)(
1sup
sup
)(,sup
)(
**
1,1,
***
*
*
xx
x
xfxffxfx
fif
fiff
xf
xxf
xxf
f
fEffEfEf
Ex
Ex
Ex
Exercise
otherwise
pandpiff
n
iii 100
)( 1*
Epppf n ,,, 21 nRE
Exxxx n ,,, 21
nxxxx ,,,max)( 21
Example
xffI
KxK ,sup)(*
EK
Kxif
KxifxIK
0)(
Let be nonempty, close
and convex. Put
1
1, 0
fif
fifxf
Exxxx 00)(
0* ,sup)( xxxffEx
Let
000 ,,sup xfxxxxfEx
01
0
1
*0
1
**
00
,max,max
)(,max
)()(max
)()(inf
)()(inf
)(infinf),(
xfxf
fIxf
fIf
xIx
xIx
xxxKxdis
fKf
fKf
K
fEf
KEf
KEx
KKx
KxKx
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