Fo DA.

Gradient

L I 5° Descent

C functions )

A domain sara"

Goat minimum

f ;yzd→lR

¥h¥÷.minfk##nerd ¥÷gspa

Mdm.de ,smack. . . .ae )

F = arsmiy f Cd )examples at Rd Maki )

flat SSECH.gl ,Ma) - It Cgi - airs )

?

Properties & Petn of Functions-

Local Neighborhood for

aeRIdEudidesnballBrCd-fpElRdIlk-pllEr3EdE@s.m

. . . " ⇒ an sofa .ie .tk small

value r 20.

-

loaalmiimm of f ! a point AEIRD

so for all p E Brca )

He aIfi:c.se/Fi:Iasf C p ) a for )aximum

"

fcp ) ap # a

glohalmi-nmm.ae/Rdso ftp.fk )

maximumfor all PER

fcp )EfG ) all pelted

f

" ÷:?÷./÷÷!strict -5 global minimumnot

strict

king

continuation f if fer angieRd

F sufficiently small 820 I radius ro

so p E Broca) has Ifk) - fcp)/ ES

v:*ftp.lcfcqfcpl?

continuous not continuous

t¥ -

saddlepoint.ae/Rd so in Brca )

↳ Gtx'

somep.EU/fCp)7fCAsome p ,

'Pa

'

ftp.lcfk )

inflection but no path from p ,to Pe without passingPoint through a or some p

'

s .t . ftp. ) Cfb )

Most AEIRD are regolarpointssit . F p , p

'

E Brca )

I. so fcp ) > fca ) > fcp ' )

and a not saddle point

÷÷÷÷÷:maximum

- i

.÷÷i¥:÷iµ°

lp.qfx-dptCtNgldeiR3qGffoPYq@i8D.QX-dp.q( x) 4=0 P 2 f IRA

Ip,qCo) - 8 f is convex

lIgCd)§fq, ,

lp.gl ) - P if Hashtagscon psgmbiratoo. lp.gl 's ) -

Ptf¥?!÷?g!÷#⇐,

lp.gl 2) lp.co ( - D flap -14 - Ng ) EA Kitty

fitted - I is conies ifFor all pegE Rd and all d c- Coil ]

f- ( dptcl - d) g) I dflp ) c- G . d) ffg)

" t.int " "

propertiesofcoauexEuncfvfhcn-fcd-gcaandf.gg

convex → h convex

-

hCG - max Ifla ,gH} thenf.gconuex-shhco.nu#Aug local minimum ef convex f is

bf also a globs) minimum

Gradients d dimension

- d differential if Tuf always

directional derivatives definedf differentiable

function f : IR d- IR

using"

direction"

u END u - lo . .ua.

. . 0dBHull - I

$mabla$

fu f = Inigo fkthY¢ ith cord

U = fei, ez . . . ed ) Ei I ( o

,0

,on

. .

,0,1

,a

,- . . O )

DEAD = Oeifcx ) = fca )

gradient Of ← 8¥ e. t ¥ .eu. - =L If.dz?i.i-iod#)vfiRd-s1pd

Exampte a- ( x, g. E) EIR

?e- 2.71

. . .

flx , g. E) = 3×2 - 2g3 - zxeZA - Az 93

Of

Tx= Gx - Zet ¥= - Gg

? Its - zxezBf= ( Gx - Zet

,-6g '

,-2×5 )

Offs,

-2, 1) = ( 18 - 2e

,-24

,

- be )

§ as← steer::'t

-

recover directorial derivative

Do flat Afca,

u )

which director u is Tuft ) largest ?

→ u= DfH Of Call