Optimisation
achievingthe best possible outcomeundergiven conditions
Is minimax Time costbenefitincome admissable set
t
Optimisation problem can bewritten as minflat X E F EIR
IR t variable decision
f x cost f IR
Apoint xe F is a globalminimiser if the
fly f ye F
strictglobalminimiserif f G fly V EF x
y y
Apoint X EF is a localminimiserif I p 0
suchthat
flatly fI yet Il x lap
yl
localminimiser
s givesstrict
Let f IR R Assumethat f is continuousC
SupposethatF is compactThenthereexists aglobalminimiser forfin F
set F iscompactif it isclosedandbounded
set F isclosedifitsboundarybelongsto FF EF
set F isboundedifit'scontainedin an n sphereofsomeradius R Ocentredattheorigin
Unconstrained problems F IR admissablesetis NOTboundedcompact
Gradientvector 78 If If is definedand continuous Cg
34,114 then f is C continuous
s s 27 It N't isdefinedandcontinuous
Hessian matrix
pay 4 o
4
IIe
then f is C continuous
Eta If If
, If admissableset is NOTcompact e X thentheexistence
g EIR
of a localglobal minimiser is notguaranteed
Given f IR IR that is continuous CY
IsetL off is any non emptysetdescribedby fW
y
If I
LEGER
XoEIR suchthat L f Xo is compact
FED
FEITH
then R has aglobalminimisen which
filk 9 a L CLa a
is contained in L flat nested set
levelset Xo
whichlevelsetsare compact
L flat flxo A It depends onthefunction f
fifty'itge fat flat
j fi
all levelsets
for
off are compact i
we have
anysequence xx
him11 111 0 him f xx a
Descentdirectionmm
Given
f IR IRcontinuous C
the direction delk descentdirection
is a
for f at xx if I 8 0suchthat
f x tdd f xx H de 0,81
Given
f IR IRcontinuous C
a.IE It j
LAICEX
a
I ascentdirection Itf xx d O INCONCLUSIVE
achievingthe best possible outcomeundergiven conditions
Is minimax Time costbenefitincome admissable set
t
Optimisation problem can bewritten as minflat X E F EIR
IR t variable decision
f x cost f IR
Apoint xe F is a globalminimiser if the
fly f ye F
strictglobalminimiserif f G fly V EF x
y y
Apoint X EF is a localminimiserif I p 0
suchthat
flatly fI yet Il x lap
yl
localminimiser
s givesstrict
Let f IR R Assumethat f is continuousC
SupposethatF is compactThenthereexists aglobalminimiser forfin F
set F iscompactif it isclosedandbounded
set F isclosedifitsboundarybelongsto FF EF
set F isboundedifit'scontainedin an n sphereofsomeradius R Ocentredattheorigin
Unconstrained problems F IR admissablesetis NOTboundedcompact
Gradientvector 78 If If is definedand continuous Cg
34,114 then f is C continuous
s s 27 It N't isdefinedandcontinuous
Hessian matrix
pay 4 o
4
IIe
then f is C continuous
Eta If If
, If admissableset is NOTcompact e X thentheexistence
g EIR
of a localglobal minimiser is notguaranteed
Given f IR IR that is continuous CY
IsetL off is any non emptysetdescribedby fW
y
If I
LEGER
XoEIR suchthat L f Xo is compact
FED
FEITH
then R has aglobalminimisen which
filk 9 a L CLa a
is contained in L flat nested set
levelset Xo
whichlevelsetsare compact
L flat flxo A It depends onthefunction f
fifty'itge fat flat
j fi
all levelsets
for
off are compact i
we have
anysequence xx
him11 111 0 him f xx a
Descentdirectionmm
Given
f IR IRcontinuous C
the direction delk descentdirection
is a
for f at xx if I 8 0suchthat
f x tdd f xx H de 0,81
Given
f IR IRcontinuous C
a.IE It j
LAICEX
a
I ascentdirection Itf xx d O INCONCLUSIVE
