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K!%$ S -" +"0+)6'+ "!" M%1+ 1+ &!%"$0 1+ Rn3 Q8(.3('$&&( #$.3(4( 1+0 &!%"$0 1+ S.
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#!"M+;+ 1+ &!%"$0 1+ S3
5" &$'78*2( P +0$ -" +"0+)6'+ 1+ &!%"$0 1+ Rn +"R+"1/2 &*/ '8%"$+/0+#$%!" 18-"
"!)6/+ ?"% 1+ 1+)%<+0&*#+0 1+ Rn3 G8-"+ )*"%7/+ 2,-%M*'+"$+. P +0$ '8+"0+)6'+ 1+0
0!'-$%!"0 18-" 0(0$7)+ 18%"2R*'%$20 '%"2*%/+0. #8+0$<9<1%/+ P = {x ∈ Rn| Ax ≤ b}. !S
A +0$ -"+ )*$/%#+ 9 m '%R"+0 +$ n #!'!""+0. +$ b -" M+#$+-/ 1+ m #!)&!0*"$+03 4!-0 &!-M!"0 *'!/0 1%/+ ,-+ '+ 0(0$7)+ Ax ≤ b #*/*#$2/%0+ '+ &!'(71/+ P 3
!"#$"%& '()*$+$%,$(-&
! !"#$! % "#$ %! &'()*+," -',!./ 01!#12 %! &'()*+," P ⊆ Rn "#$ %! &'()$'&" #1
"$ #"%("3"!$ #41( "51#$" l, u ∈ Rn $"(# 6%" l ≤ x ≤ u, &'%, $'%$ x ∈ P / 78, "5"3&("2
(4"!9"('&&" :'!9"5" +4%! !'3-," ;!1 +" &'1!$# "#$ %! &'()$'&"/
! &'()*+," P +" Rn "#$ +" &'(%)*'!) +2 !'$." &'(+ P ,2 #1 (" !'3-," 38513%3
+" &'1!$# +" P 8<!"3"!$ 1!+.&"!+8!$# "#$ .=8( > d + 1/ ! &'()*+," "#$ +" "%')% &'(%)*'!) #1 &'(+P ,= n/
! &'1!$ x ∈ Rn "#$ %! !')$ %-$./(% +4%! &'()*+," P #41( !" &"%$ #4.:,1," :'33"
:'3-1!81#'! :'!9"5" +48%$,"# &'1!$# +" P /
!" 1!.=8(1$. ?'% 0!)$.1')$%@ aTx ≤ α2 'A a "#$ %! 9":$"%, > n :'3&'#8!$"# "$ α
%! #:8(81,"2 "#$ +1$" 21"'&% &'%, %! &'()*+," P #1 "((" "#$ 9.,1;." &8, $'%# ("# &'1!$# +" P 2 #'1$ P ⊆ {x ∈ Rn | aTx ≤ α}/ B$8!$ +'!!. %! &'1!$ x∗ ∈ P2 !'%# +1,'!# 6%"
(41!.=8(1$. aTx ≤ α "#$ *%..3% ?,"#&/ "405%@ &'%, x∗ #1 aTx∗ = α ?,"#&/ aTx∗ < α@/ C'1$
aTx ≤ α%!" :'!$,81!$" 98(1+" &'%, (" &'()*+," P / 0(',# (" #'%#D"!#"3-(" F ⊆ P +.;!1
&8, F = {x ∈ P | aTx = α} "#$ %!" 610% +" P / E! +1$ 8(',# 6%" (8 F8:" F "#$ +.;!1"
?'% 1!+%1$"@ &8, (8 :'!$,81!$" aTx ≤ α/ C12 +" &(%#2 F 6= ∅ "$ F 6= P 8(',# F "#$ %!"
610% .! .%/ C1 F "#$ &,'&," "$ 6%" dim(F ) = dim(P ) − 12 8(',# F "#$ %!" 610%$$%/ !" 1!.=8(1$. "#$ +1$" %**%)$'%""% &'%, %! &'()*+," P #1 "((" 1!+%1$ %!" F8:"$$" &'%, P/ !" 1!.=8(1$. "#$ +1$" .%&!)&1)$% +8!# %! #)#$*3" Ax ≤ b +.;!1##8!$ %! &'()*+," P2 #1 (" #'%#D#)#$*3" '-$"!% > &8,$1, +" Ax ≤ b "! #%&&,138!$ :"$$" 1!.=8(1$. +.;!1$ (" 3G3" &'()*+," P /
!" #$$%&'() $&*+,-%.*)/ 0,1(&-) -) '&2$)3 )1 4%.56
'()0)513
C'1"!$ P %! &,'-(*3" +4'&$131#8$1'! :'3-1!8$'1," "$ S (4"!#"3-(" +" #"# #'(%$1'!#/ H" &,'-(*3" P #4.:,1$ 8(',#
min {cx | x ∈ S}
'A c "#$ %! 9":$"%, :'I$ 8##':1. 8%5 98,18-("# +% &,'-(*3"/ J'!#1+.,'!# (4"!9"('&&" :'!9"5" conv(S) +"# #'(%$1'!# +" P/ H" &,'-(*3" P "#$ .6%198("!$ 8% &,'=,833" (1!.81,"
min {cx | x ∈ conv(S)}
H41 .!05% !"#3&.1"%2 1!$,'+%1$" &8, K+3'!+# LMNO +8!# (" :8+," +% &,'-(*3" +" :'%&(8="#2 :'!#1#$" > +.:,1," (" &'()*+," conv(S) &8, %! #)#$*3" +41!.=8(1$.# (1!.81,"#2
!" #$$%&'() $&*+,-%.*)/ 0,1(&-) -) '&2$)3 )1 4%.5'()0)513 6
!"#$!$% !&$'& (# ) *+(,"# P - (! .'*(/%&*$ 01/$ ) *2 !""# (&$.!& #3 4#5& )# "#% 0*$5 0# .'*/0 # (# ) *+(,"# P #$ /%&(&'!$% (#' !(2* &%6"#' 0# (! ) *2 !""!%&*$ (&$.!& # 789: ;;: ;<=3 >!))#(*$' ?/# (! .'*(/%&*$ 01/$ ) *2 !""# (&$.!& # )#/% @% # #A#5%/.# #$ %#")' )*(B$*"&!( 7;;: ;<=3
C$# 0#'5 &)%&*$ 5*")(,%# 0/ )*(B,0 # )#/% 5#)#$0!$% 5*")* %# /$ $*"+ # #D)*E $#$%&#( 01&$.2!(&%.'3 F# ) *+(,"# 01*)%&"&'!%&*$ '/ (# )*(B%*)# conv(S) $# )#/% 0*$5 @% # .'*(/ 5*""# /$ ) *2 !""# (&$.!& # !B!$% #D)(&5&%#"#$% %*/%#' '#' 5*$% !&$%#'3 4#)#$0!$%: /$ $*"+ # .0/&% 0# 5#' &$.2!(&%.' )#/% @% # '/G'!$% )*/ .'*/0 # (# ) *+(,"# - (1!&0# 01/$# !"#$%& %& '$()&*3 4#%%# ".%6*0# '1!))/&# '/ (# ) *+(,"# '/&H!$%: !))#(. )+$,-. & %& *!)/+/"0$13
!"#$%&' I*&% C /$# 5(!''# 01&$.2!(&%.' H!(&0#' )*/ conv(S)3 F# ) *+(,"# 0# '.)!E !%&*$ !''*5&. - C 5*$'&'%#: .%!$% 0*$$. /$ )*&$% x ∈ Rn: - 0.5&0# '& x '!%&'J!&% %*/%#'
(#' &$.2!(&%.' 0# C #% '&$*$: - % */H# /$# &$.2!(&%. 0# C H&*(.# )! x3
F# ) *+(,"# 0# '.)! !%&*$ #'% /$ 0#' )*&$%' #''#$%&#(' 0# (1!)) *56# )*(B.0 !(#3 K$ #A#%: L M%'56#(: F*'H-'N #% I56 &OH# 7P9= *$% "*$% . ?/1/$ ) *+(,"# 01*)%&"&'!%&*$ 5*"+&$!%*& # '/ /$ #$'#"+(# 0# 5*$% !&$%#' C )#/% @% # .'*(/ #$ %#")' )*(B$*"&!( '& #% '#/(#"#$% '& (# ) *+(,"# 0# '.)! !%&*$ !''*5&. - C )#/% @% # .'*(/ #$ %#")' )*(BE $*"&!(3 Q&$'&: /$# ".%6*0# 0# 5*/)#' )# "#% 0# .'*/0 # /$ ) *+(,"# 01*)%&"&'!%&*$ 5*"+&$!%*& # #$ %#")' )*(B$*"&!( '& (1*$ '!&% .'*/0 # #$ %#")' )*(B$*"&!( (# ) *E +(,"# 0# '.)! !%&*$ )*/ /$ 'B'%,"# 01&$.2!(&%.' 5! !5%. &'!$% (# )*(B,0 # !''*5&.3 K$ #A#%: (! ".%6*0# 0# 5*/)#' )# "#% 0# .'*/0 # /$ ) *+(,"# P 5*""# /$# '.?/#$5# 0# ) *2 !""#' (&$.!& #': 56!5/$ 5*$%#$!$% /$ $*"+ # !&'*$$!+(# 0# 5*$% !&$%#'3 R*/ 5#(!: (! ".%6*0# 5*""#$5# )! .'*/0 # /$ ) *2 !""# (&$.!& # 5*$%#$!$% /$ '*/'E #$'#"+(# 0# 5*$% !&$%#' 0# conv(S)3 S*%*$' )! x∗ (! '*(/%&*$ *)%&"!(# *+%#$/#3 K$
!))(&?/!$% (#' ) *+(,"#' 0# '.)! !%&*$ !''*5&.' !/D 0&A. #$%#' 5(!''#' 01&$.2!(&%.' 0.TE $&''!$% conv(S): &( #'% )*''&+(# 0# '!H*& '& x∗ H. &T# %*/%#' (#' 5*$% !&$%#' 0# conv(S):
02&2: '& x∗ ∈ conv(S)3 I& %#( #'% (# 5!': !(* ' x∗ #'% (! '*(/%&*$ 0/ ) *+(,"#3 Q/% #"#$%:
(#' 5*$% !&$%#' H&*(.#' )! x∗ 2.$. .#' (* ' 0# (1!))(&5!%&*$ 0#' ) *+(,"#' 0# '.)! !%&*$
'*$% !O*/%.#' !/ ) *2 !""# (&$.!& #3 4# ) *5#''/' 0# .'*(/%&*$ #'% .).%. O/'?/1- 5# ?/# (! '*(/%&*$ *)%&"!(# !))! %&#$$# - conv(S)3
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
!"#$"%& '()*$+$%,$(-&
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D# /,-*#& #(# (,+"#*- "$* #(*- G = (V, E) (E V "$* !&.'& +0& %&' '$ &"' "* E !&.'& +0& %&' -,1"&'7
F+ e "$* )#" .,G*" ," +.#* '"); $(%%"*$ u "* v: . (,$ u "* v $",(#* .00" -$ "$ &2",! 3"!' '" e: "* #()$ -3,+,(#$ e = uv () e = {u, v}7 F+ u "$* )#" ";*,-%+*- '" e: . (,$ u 5,"$07 e6 "$* '+* 3.(3%&." < e 5,"$07 u67 H" %G%": '"); $(%%"*$ u "* v 2(,%.#* )#" .,G*" $(#* '+*$ -%4-(&."'7 F+ F ⊆ E "$* )# $()$/"#$"%& " '!.,G*"$: . (,$ V (F ) ,"0,-$"#*" !"#$"%& " '"$ ";*,-/ %+*-$ '"$ .,G*"$ '" F 7 F+ W ⊆ V "$* )# $()$/"#$"%& " '" $(%%"*$: . (,$ E(W ) '"#(*" !"#$"%& " '"$ .,G*"$ .I.#* "),$ '"); ";*,-%+*-$ '.#$ W 7 D# '$)'5/,-*#& H = (U, F ) '" G "$* )# B,.09" *" =)" U ⊆ V "* F ⊂ E7 D# $()$/ B,.09" H = (U, F ) '" G "$* '+* ($)6,-." $+ U = V 7 F(+* W ⊆ V : H = (W, E(W )) "$* '+* $()$/B,.09" '" G 3.%)3" 0., W "* $",. #(*- 0., G(W )7 F+ F ⊂ E 5,"$07 W ⊂ V 6: (# #(*",. 0., G\F 5,"$07 G\W 6 " B,.09" (&*"#) < 0.,*+, '" G "# $)00,+%.#* "$ .,G*"$ '" F 5,"$07 "$ $(%%"*$ '" W "* "$ .,G*"$ +#3+'"#*"$ < W 67 F+ F 5,"$07 W 6 "$* ,-')+* < )#" $") " .,G*" e 5,"$07 )# $") $(%%"* v6: #()$ -3,+,(#$ G\e 5,"$07 G\v67 F(+* W ⊆ V : ∅ 6= W 6= V : )# $()$/"#$"%& " '" $(%%"*$ '" V 7 J!"#$"%& " '"$ .,G*"$ .I.#* )#" ";*,-%+*- '.#$ W "* !.)*," '.#$ V \W "$* .00" - ($)*& "* #(*- δ(W )7 K#
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*+ {V1, . . . , Vp}& p ≥ 2& ,"% '$, #-%+%+!$ /, V & #1!-" δ(V1, . . . , Vp) ,"% 15,$",271, /,"
#-;%," e = uv %,11," <', u ∈ Vi& v ∈ Vj ,% i 6= j)
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