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, D
*C"D+,! !
L7M4! 7 zO+,sz +o+,s7";BN";Bo! =67"A x;!-.+4+,
:
x !-u7*- .LwMN!
7zO
+sz!
+";9/0{*,m*C!-7*(!A !
9B :9uy!o¬.!(+w">+sz!
9
9B
" B"
0! 0! %
7
"B"
LMN!$sz A !
!+( ¬=!+,s7";B +C/x";!d x;
!
9
x !/" 0 MN!9 B,!(+d+,s!x; BF+w!=6z +,"5+,
t!(*(L. O>+s7";B
7
==
9
uy!(+FMN!
+";9T"D
O"D 1O7O7"D+";+C+,sz +L)+sz!k96zx"D 5uy!(+FMN!
";BgMo"D+,s7"
}
sz!-
+s! 7 x;!
L+sz!>H xD"z! *C"D+FE 9"D+FE Ns!B!g+FM4&Bs) 0;+,Bqsz A=!k !
+sz!K9-6*,*(!-.+/ z xE:B"B 24B";O7!-* ]9
, +s!k0{*, 0 , =
B"5x;!q95z6+C +";&E=";!
0;*- !k";B$+s!TBC !
Ns";B47*( A !
LYMosz!(*(! O ";B 24zBF+*( ".+CBq9 u !Tuz67"xD+q0;*( +sz!
,
,
,
,
, # !
,
7 J
, 19
67
D 1O+,s!Bsz67x;O!(* #Ns!B!
*(!+*- Bx +";) x=) *,+,BdMos7";9Cs *(!o+w! B"xEkO!
1*C" "z xxE
!
')
Bg97*(.O69!
"
15.2 Lagrangian form of the equations of motion r/BKB,!
=6) +,!-*," xz +*C"
+,s!B! *C!&B!
=6z +C!(*C7";
9.1*(O"z +C!
rN0;+C!(*9
";B
k6xD+"DxE."7 >+,s!]> +C!(*CL
0 19-+Ld"D+>xDEs) 1Bk*( 7¬]
O!(y!-zO7!
| +*C"
+C!(*Cª
*(!
2w6*C";6zBxEL)
N9
1J:Lz J
1O|B!+sz!/ +*," ! L z1+
!
0;*C!!
*C" ";O|uy.OE *(!
6x; *4A=!-x;.9-"D+FE>A !9(+,*q 1BNMo 1BoO7!
==
) 1 !
+C>1u+C "
*C 6!-.+ O
x!-0;+&Mo"D+s
? B" >24*CxDx *CE"7 /+s!4*(!
E=*(B,9
+,"D!$".+,!-*,A x"Bo+s!
:L +sz!/90;*(!
B6*,7*C";B" ) xE +,";9g*(!
Theorem 15.2.
+/ / I 7 1 L 1 / I0I / 3
$
,: *5 7 J 56879/ 0/2- I 1 /3 : *5 5 $7@/ 5 " / 56 7@/ /2-
$
+ / 8 / D1 1 I 7 1 L 1 / II I 1 3P3 5 1 " P5 6 " +/ I 31 L 1 I 5N7R5 1 $1 */Tx *, =!
:0C
+,s!oA *C" +";) x.!(+,s.OyL
2 Q 2 A :0C ; C 2 " 2 A 2 : L=+sz!"5y1*,+C 1.+w! .7*(!
UC zO , = C Nsz!
« *- 7 =!q!=6z +,"B
2 A 2 A 2 Q KC 2 , . 2 " K Ns";BKx; BF+ BF+C!("7 ! =6z +,"; "D ,: = / ";BK!-*C!B"5x;!
+*( zBF0{*, +";9 1uy!
9G6x;!(*,H «y 1 *- 7 =!
2 A 2
1* L! =67"A x;!-.+,xDE L
2
A
2 A
2
, =
Mos";9Cs";Bo! 6"A1 1x!";B,B"7 50;*(
, ,R
+> 7y! *CB
"7"D+";0o+sz!O";B,9(*(!(+,"Dt +";#0{*,T67x 1!
+,s!
O!(0;H
+s!-*C!-0{1*(!5uy!-z! 9A !9(+C1* 2
Mos7";9Cs
1BKMN!-xx4 B$+s!567"D+ *(! =67"D*(!-!-.+q0{1*
2
: =
+sz!x">"D+k10
C
)
:L
0{1*k 1:EA x6!510)C(ª :r/BgB,!
2
B,!
O7!77"D+,"B
2
2
2 A 0 2 A 2 2 A 2 A 2 2 3*L:" 6) +,!-*,"; x !(u*( :L 2 2 2 % 34 2 2 1O 2 2 34 2 2 A A
zO>y!(*,0{1*C>"7
B,!K x !-u7*- ";9o 7"Dz67x +";>"D:A=xA."7
+";9+,s!A=! 7 67x *K!-.+,67 9
2
9-x;! *o+sz N + MN!Ksz A=!A=!(*CE0{!-M A *C"; u x;!O! x Mo"D+sYLz 1!-xEL=+sz!$¬: Mo 2 L= 1O«y *( 17 !wT67xD+"DH )L 2 L 2 2 L +s!N6¬: Mo =z 6 +,!(*C7";
=6) +,!-*,"B
zx"!-* IgB"5zxD"D0E +,s!oz1+( +";>0)Mos) +30{xDx; M/BL*(!-x uy!-x+s!B!NA *C" ux;!B3 1Bx";BF+C!
uy.OE
+,!-*,B$10+,s!B!kA *," uzx;!
Q
k"B
"
w";Bo9-x *C"!
o! =+Ld9"x"D+6O! +C+s!>B,!->"DH "5x";9-"D+qO";B,9(*(!(+,"Dt +";#10/G6x!-* B
!=6z +,"B J CrNxE."7 B";O7!KMo"D+s
O +sz!/*C" s.+ds) 0{!!x *C !-*q+,"D!5B+,!-YkNsz!Tx; BF+Kx"Dz!k0
=
,: /";B
10/+,s!| 7*C:" +,""BT! . y!
zO o+,!Ks M , 7 ,
0 9(+L4 xx
5";B5B" "9 .+,xEx;!
*(!- "zB3yB"D+"A !$O7!77"D+C!
,: "D+C!(*- +"A !+,!-*,Bo0%=
8:!9(+"; : k zO";Bw¬: Mo&+,kuy!
1* F6B+IB,! =6!
+sz6 s+,s7";BN";BN+4*C A=!
uy!(++,!(*
L7+,s!
9Cs) 7 =!+,s!g"z!(*,+"
s M
>"Oy".+/*C67x;!
2 A 2 A 2 Q C
2 Q C C 2 2 2 2 " 2 2
"+,s!>x 1B+
!
IR !-+ ¤6O7!-*CB+C 1O" #10K+,s!
2
B,9 x *>+C!(*C
: O> 0;+C!(*N O7O" +,s!40{1*(9
2
, 2
" " 2
+CK"
2 L/ = u) 19,¬
+,]+s!# z xE=+";9
+L T67xD+," zxE=" q5+,s!Nx;!(0;+wMo" +,s
*C6+"z!/ " z67x +,"BL1+sz!Kk6xD+"Dx";!(* |";B40{67zO+CTuy!
Mos";9Cs#";Bg ..O"O7";9 +,"104Mosz +
.Q
,
Mo"xxsz y!- "D#+sz!5O";B9-*C!-+,!T9 1B,!5rox;B,L)+sz! ";B M = C -Lu +sR0{* +,s!B,9 x *k 1O
=6) +,!-*," +,!(*C
*CO!(*k0N+s!!z!(+g+C1* = 6! x";!"7
0{*,
) 1 !
==
A=!
CL:+,s!K!-" =!-:A x6!
|B, +,";B !B
9 1B,!$0 ,
Q
¤ >C
Q
Q
C Q
2
A=!MN!5O7s) A !k 1 !=zx"9
J CL=+,s!/7xE
+C!(*C
>+,s!o*C" s.+3s) O>B";O7!
==
" "B+,! 1O zO
C (LMos7";9Cs&"0 19-+q!-A x6z +,!
++( 19Cs7!
, o";Bq+s) ++,s!T9
=6z +C!(*C7";
*,";!
,
!
1O&+s:6zB
&
+s) +
LyB"z9
==
+,!-*,Bw10)+sz!/1*C";!-.+C +";=6z +C!(*C7";YL +s7";B
A !9(+C1*
uy!7*(!-A.";6B
O" !(*(B0;*(
$
O" , ,
&
Ns!#! =67"A x;!-z9
&
":A xDA=!$-0/
!
0{1*Cm";BoO7!-*,"A=!"R0 19-+L3 zORB6z9Cs
+";YL/B"9!|+sz! G67x;!(*,H « *- 7 =!&!=6z +,"B7xE
+ !+,
3 1 N5 1 N/ 40 +,s!y+,!-.+,"
.O7!
«y *( 17 " 0 ,: duz6+oB67u !9(+d+Ck+s!qy1+C!-.+,"; 1x , ,3Ns!$ =!-!-*( 1xD"t!
+C1* =6!$"+,s7";Bo9 1B,! d0;*(b+s7";B4y1+C!-.+,"; 1xz";B40{67O&+CkE.";!-x;O " & " " 6) +,!-*,";#0{1*C&Ns!Ty+,!-.+," x3!-z!(*C E9Cs) 7 =!
*CO!(*".+,!- *( +,1*IMo"D+s 1O 7+"A=!+"!NBF+C!(IL +s!w*(!
6zB"
+,s!
lsode
BxDA=!(*/10 $9(+(
+,s!q!-!-*, E5 7y! *(B"
" 67*C!
1O
".+C!- *- +,"LIu67+5+s!|B+C uz"xD"t!
=
+s!|O7" *(!
BF+( u"x"D+FENs!
Ns7";B4".+,!-*C!BF+,"D g*Cux;!-
#| H
!.x";9-"D+
E
Ns!
B+*C67 x;!
9Cs B!
B+C E:B$6*," s.+/Mosz!-z!-A !-*
(
==
*C+C +,"D 0;*(!
"
J 67x *A !
=1J
' G
A *C"; +";) x
==
A *C"; +";) x
7*C:" +,! B+C u"x"Dt!
+,s*(!
!(+,s.O7B
67x *A !
x; *, =!
s M¤+s!q!-z!(*C E>10y+,s!$BF+( u"x"t!O>BF+C!(7y!-*NO!
J
M
Lsz! A.EBE.5H
B"T67x +C!Mo"D+s0{6*/O" !(*(!-.+w!-+sz=OB!(+*,";9 x4+,1VMosz!-
= B"T67x +,!ORMo"D+s
0{67*O" !(*(!-.+N"D.+C!- *( +,";B,9Cs!
BF+,"xDx3+,.|x; M
0{6*
O" )!-*C!
K
BF+( u"x"t!O A1 *," +,"z x
K
" K
" 6*(!
: :Gz!(*C E
0K+s!
=O!(*- +,!LNs! A.E
= BE.>!(+*,";9 1xo+,
= Mos!B"T67x +C!
" +,!OA *C" +";z x)BF+C!( !-*ou7*(! ¬BNO7 MoY
J
==
.O7!-*( +C!LYs! A.E#BE.>!(+*,";9 1x+,
= Mos!
+,";+C!
A !(*CxEB"D5zx;! =!
* E=*(B,9"!4sz
M
Mo" +,s
+s!-*
u .O7EkMo"D+s 1! . x";9-"D+
+,uy!> x;1u) 1xd!(*,*C*guy6O7Bg"Bk10B" 7"9 9!
0 *d+sz!ox"! *,"t!OT 77*(:" +";T108:!9(+";
:
9 L zOMos) +/9 &uy!qO!/+,k*(! 7 z!/+,s!-bMos!-&+sz!-E *(!q1+oO7!
+sz!- +,"9 xq7*( *- >>"7 Mos!-*C!
0679-+";
z +,6*- xxDE
x;! 1O¤+,
9
2ws) 7+C!(*& Mos!(*(!>"5) 9(+,B
zO&O*,E>0;*C";9(+,"| *C!K9
zO>+FMNkO7"D!
T« !-*( E=HF8:9Csz 6zO7!-* +FE y!4 xD+,!(*Cz +"A !B O
/ L / I 1 5N7 5" D1 7 1 /
"A +1y!(*- +";B5 *C!|".+*CH
O6z9 *(!47*(! !"D">"Dt +"; 0q ') 6 -/ - ".+,!(*CA $ x 65 5 =*( A !(*gS
*(! xB9 x *069-+";
9 1x9+,s!/".+C!(*CA1 1x
5
q!/0 +s*(!
.9"t +,";*Cux;!-
monotonic increase:
monotonic decrease:
Nsz!(*(!(0{*C!LM4!Kx;.1¬&0{1*/+,s!
k0{* V 0{1*
"|z!$O7"!-B";Y
65 L
5
>""DT67ªy"D.+q +o+s!gx MN!(*ouy67O
"D0Y+,s!K069-+";
!
7Hz!- . +"A !> +
*- s#Mo"D+sRx uy!-x/L +sz!
+sz!3".+,!-*,A x+ 65 1O
y"D.+Y"D
5#Ns!A *,";6Bg9 1B!B
1*kz7H yB"D+"A !> +
" 67*C!DT
+sz! *CB+
069-+"; sz 1B
";BY>"D" x1+sz!(*(!INs!3 *( s
*(!>"xxD6zBF+*( +,!"D"k6
"zBF+C! 1ONs7";BIB!-+I10!":!OVx"! *
,:
)
K
A
zO7!
B"5x;!! 1ORO7!9B6u)BF+,"D+6+,"7 7L1MN!/u7+C 1"D5+,s!ox"! *39"D+T";B51uA.";6zBK+s) +T+s!& +*,"]
,: "B
sz 1O5B";O!o10)+sz! *(BF+dx"!N" T67xD+"Dx;!
03+sz!
*(B+gNs7";BKBs
21 795 3
uy!
internal minimum:
8 #
"
5
8 5 OD " 8 5 I 1OD upper bound minimum: " lower bound degeneracy: " 8 5 upper bound degeneracy: lower bound minimum:
"
Q
5%Q
(V
sz!- +s";BK 1z xE:B";Bq";Bg 7zxD";!O+,
+">"t +";
!
& C
& K
"
:
, ";Bg*C! *(!$x"Dz! *CxEO7!- !uy!-+FM4!!-9+,sz +> &y".+
1O
Kq
7+,"D>"t +";| O+sz!$>" !O95zx!>" 7Ld!
+sz!#|«2Nm"Bz +,6*- x/"V+,s!
9 *C!O6z9
B69CsR 7*(1ux;!z6*(!T«Y2NVuEy!-*0{*,>"7 | 8:9Cs:67*g9
9zB";B+"7 o107967x;!"
+,
+/
U- 1 / 1 " 5N7
6O!(*(BF+( O
7zO
>"!"!
==L
: :N *C!31*(!IO7">9+C -/ 0/ / 3 5L J Mos7";9Cs 9
«2N
sz 1BNT67xD+"Dzx!qBx6+";A !9(+C1*(B )
Ns!g"D.+C!(*(!
7zO
0/+sz!
| !"z1*(B4 *(!q+sz!
"&
1*+s7";Bx 1B+&";O7!
.
Definition 16.5 (The cone of matrix $1
1 7 S 5 31 L2/ +/ 3
, 1
5 3 1 '
:
3";B+,s!Ny, B"D+"A=!oBF 10 +,s!/9x67>zB10y +*," y B , , - q"A !
625 21MLA1
A *C"; ux;!/"D0
:
10 ' !-0F L:BxDA."7 5«2N
T6B+NB,xA !q+,s!$x"Dz! *oBE.B+,!u) 1B";9
,3";BN7H u 1B"9L= zOA=";9!qA=!(*(B, :
r/BK! . x "!O"
+,s!
7*(!-A.";6BqB!9(+,"L+sz!
z1+";103y B"D+"A."D+FEx E.BK >9!-.+*- x
*(x;!4"D95zx!
1 N/ 5 J
Definition 16.7.
'
8/ D1 N-/
I 21 1 N/ ' +/
(
1 /
S5
31
3 3
I 21 1 N/ -0/ 1 / / / 1 U-/ 1 / I 21 1 N
:
r/B";B|M4!!(+*C";9 +*C"9!!(+*,";9L1yB" +,"A !4B!" H O7!77"D+C!d +*,";9!
=
-matrices:
L +5 3«2N 5 3 1ML2/ L +5 «2N
L +5
31 L2/
S5
-matrices:
Copositive matrices:
7
Copositive plus matrices:
5" J
:*
+5 5 K7 1 " 3D5N7 7 W +5 !5 N7 1 " 3 5N7 7 P
31 L2/
S5
¤
}
s!!(+*C";9L+s!T*Cy!(*,+FE
m";Bg!=67"A
s) 1B3
roz1+sz!(*dO7"B+"z9(+";T0 ¤ +*C";9
+*C"9!
1:E
TL7 :E7*C"Dz9-"D) 1x>"*o"BoB+*C";9(+,xDE B"D+,"DA=!3ro:E7*C"9-"D xzB6uz +*C" S L"D+/";BNyB,B" uzx;!/+,sz +! C 0 L7"Bo+s:6BN":A=!(*,+"Dux;! } s!
L.+s"B4*(!
L 1B
> "
WQ 1
¤ P" P" : 4+s:6zBoB, +";B z! zO
+ 5 1BoO7!B"D*(!
uzB,!(*CA !> x;B+s) +
K": q";B 8/
! 17"7 4+s) +Y"D+Isz 1BF675kO7";B9.+":6" +,";!10
/#L2/ I 1 " 5N785 1 7 1M/ !/879/ S/ k O6z!/+C>
BC 19L$ x 1Bs7"D¬= A)L zOd67x A:BF¬:E D
1 +/.1
(m #,L +5
Q
/ 3 I 3 -"L
"D+sT+s"BO! 7 " +,";L /+sz!
: L=*(!(7*(.O6z9
" 1 -/ 3P5 1 7962/ 3 I 50L2/ ' 5 L 7 /2') L " N/ L " / 5" - 5 L " I 79/ /87 J L " 1 " S5 I &1 */ 3 */ 32/ / 1 D5 N7 1 " 3 +5 5 / 0L2/ I 1 " 5N7 5" 1 7 J 8/ 79/ S /
Theorem 16.3 (Solvability of NCP).
+s";Bx E:B"xE :L79B";O7!-*w+,s!K! .) 1B"
IgB!!Ns M
24zB";O!(*
"
M
W
zO
+sz +
.Q
Q
LB,k+sz +
Ns:6zB
,
I 5 0/
!
16.5 Classical solution methods Ns67 s+,s!(*(!K";B 5zx!$x"D+,!-*( +,6*(!K 0{!-M
9-x 1B,B";9 x!-+sz=OB4 *C!q M
"+,s!$x"! *w*C *- >>"7 6BC =!
x;B,T+sz ++sz!
«2N
1y!-*( +,";BT":A xA !
A !9(+C1*$7*(=O769-+,BLz*$ +*,"|*- ¬.H(
6yO +,!
B+*C"9-+xE!=zx"9
10gB"Dt! © zOVB!|"O7!
24B";O7!-*«Y2N
+*,"
=*( '
";Bo9 xx!O g7*C"9O6z!+, $ ++C R&6*,+FE K zO ";B Nsz!|";O7! ";B5+s";B
"Bg+,z"9,¬#+sz!
W1 8/25 21 6879/S1 -0/ V
zO
"D0 1 1 * 1O7O#"D+$+C|B!-+ ¤"D0 1 1 M4!
O7!B9-*,"Duy!
7
repeat
8:xDA=!
8:!-+
.
if
0
, V
then
else
H O7"D!
C
S"¤ o M LA *C" ux;!
0!"R
r/B,B67!k" +
MN1*,¬B$0{1*
«2N
Mo"DxxxDE&uy!g9
";BdB,xA=!
x;! 1O7Bw+,T+FMNkyB,B"Du"x"D+";!10«2N
}
"D+s+sz!5 :"k6
+,s!KB,!(+ w o1+C! ! B
x;B
"O7!
+s) +oM4!K9 17+4!
F
" 09 1B,!
L." ! L:B,xA."7 K0{1*
N0IO"!"O7! B,!(+k0N+s!BxD67+"; T6B+k9
*C67x;!L +,s7";BK! 1B$+s) +
=
==
" +,sz + B+/
|
A
B,xA."7
«2N
"
10oB"Dt!
T+( ¬=!B
+gB+k
A
1y!-*( +,";BL +
0IB"Dt!/d
1y!(*- +,"BN *(!qz!
Ns!$H 9-E:9-x"7 k BB675+";>+,s!-*C!-0{1*(!$B6*CA."A !A *C" +";B
"D+k9 1u !
5 3
7 Mos"9Cs&";B*C!=67"D*C!B3*Cy!(*,+FE
T
16.5.2 Murty’s principal pivot method for MLCP Ns!
B"5x;!R+s) +
0
: YMo"D+s /0 19(+C1*C"t +,"
x *,"D+,s7 Ns!dxD"z! *y*Cux;!-B +,$BxDA=!3B+"xx=7xEK"D:A=xA !I+,s!d7*C"9-"D xB6u +*C"
&Nsz!T+*C"9,¬ s!-*C!T";BK+,Bx"D+K+sz!5"DzO7!#B!-+
z 1!-xEL
Mo"D+,s
7
O
";B4*C!7!O"D.+Cg+FMNk) *,+,BL:z 1!-xEL
+sz!$67y!-*4uy67OL O
".+,&+FMN| " *C6)B$ 1B
C
0
MN!-A=!(* M
u !-0{1*(!L Ly+s!5B,!(+
9
Q
" P" "
)
A
A
,
*(!$! . +,!*(! x;BwB
Mosz!(*(!/+,s!uy67zO>A !9(+C1*(B
uy!K"7"D+,!
Ns!$1*C" "z 1x)q!-xx;!(*o x 1*C"D+s +,s!
*C.9
+s) +o 1:E5* xx ,
, L:>" s.+
0{! B"Dux;! >A !
9 &u !q6B,!uE>+,s!$6y!(*ouy6O7B
) *+CB RB6z9Cs]+,sz +
! " ! "#%$'&)(+*-,*./ 1024335*,76829+:,;.52 3**-?@A3BC@AD EF*, GH(BIEF,7.J.=(+*%,=*K.J1024335*,76829 :,L.2M.=( *N.=BIO(P.Q?4@R3BC@AD EI*,HGH(BF05(SBF,J,6+EIBT. BI9P.52S.=( *M.BIO(P.
