feuilles - Ceremade - Université Paris

A N ×Nsp(A)
A A ρ(A)
A A
ρ(AT) = ρ(A)
A AT=Asp(A)
ρ(A2) = ρ(A)2
A2−λ2I= (A−λ I)(A+λ I)
A λ ∈sp(A−1)⇔λ−1∈sp(A)
kA Bk ≤ kAkkBkρ(A)≤ kAk
k · kkRN
k·kk
kAkk= sup
kxkk≤1kA xkk,∀A∈RN×N.
ρ(A)≤ kAkk
kA xk22kxk22=PN
i=1 x2
i≤1
kAk2=ρ(ATA)1/2
kAxk22=x, ATAx
ρ(A)kAk2A=1 2
−1 1 
kAk∞= max
i∈{1,...,N}PN
j=1 |Aij | kAk1= max
j∈{1,...,N}PN
i=1 |Aij |
kxk1= sup
kyk∞≤1hx, yi kxk∞= sup kyk1≤1hx, yi
A N ×N γk(A) =
kAkkkA−1kkAk·kkRN
A x =b A(x+ ∆x) = b+ ∆b A ∈RN×NbRN
k∆xkk
kxkk≤γk(A)k∆bkk kbkk
.

A=
10 7 8 7
7 5 6 5
8 6 10 9
75910
x=
1
1
1
1
∆x=
82
−136
35
−21
b=A x =
32
23
33
31
∆b=A(∆x) = 
1
−1
1
−1
k∆xk2'164
γ2(A)
kAk= (ATA)1/2=PN
i,j=1 |Aij |21/2
RN×N
P N ×N N ×N
PTP=IkP Ak=kA P k=kAk kPTA P k=kAk
k·k RN
kIk=√N
qRN
q(x) = hA x, xi,∀x∈RN,
A N ×N
∇q(x)q(x)
kxk22= 1
λminkxk22≤q(x)≤λmaxkxk22λminI≤A≤λmaxI
A N ×N
q(x) = hA x, xi
q(x) = 0 ⇒x= 0
Asp(A)⊂[0,+∞[
sp(A)⊂]0,+∞[
A N ×N v RNa
w=A−1v B (N+ 1) ×(N+ 1)
B=A v
vTadet(B)>0
det(B) = 
A v
vTa
=
A v −A w
vTa−vTw
= (a− hw, A wi) det(A).

(uTα)A v
vTa u
α≥ hu+α w, A(u+α w)iu∈
RNα∈R
B
B=A v
vTadet(B)>0
A a > 0
A=
2−1 0 0
−1 2 −1 0
0−1 2 −1
0 0 −1 2
FRN
F(x) = 1
2hA x, xi−hb, xi+c, ∀x∈RN,
A N ×N b RNc∈RA
M×N b RMF(x) = kA x −bk2
RN
∇F(x)∇2F(x)
A N F
x?ATAx =ATb
A b x?F(x1, x2)=(x1+x2−2)2+(2 x1+x2−3)2+(3 x1+x2−5)2
FC2RN
c F
v, ∇2F(x)v≥ckvk2x v RN
c F F (x)≥F(x0)+h∇F(x0), x −x0i+c
2kx−x0k2
x0xRN
lim
kxk→+∞F(x) = +∞
F(x) = 1
2hA x, xi−hb, xi+cRN
A N
A M ×N b RMF(x) = kA x−bk2
RN(A) = N
F(x)=(xN−1)2+x2
0+PN−1
i=0 (xi+1 −xi)2
RN+1
F G R2
F(x, y) = x2+ 2 y2+ 3 x y G(x, y) = 2 x2+ 3 y2+ 4 x y,
inf Ginf F=−∞
x?A x =b

min
p+q+r=1
3p+6q+9r=5.25
p,q,r≥0
(pln p+qln q+rln r)
x1x2x30≤xi≤42 i= 1,2,3x1+2 x2+2 x3≤72
max
0≤xi≤42, i=1,2,3,
x1+2 x2+2 x3≤72.
x1x2x3.
min
0<xi≤42, i=1,2,3,
x1+2 x2+2 x3≤72.
(−ln x1−ln x2−ln x3),
( ˆx1,ˆx2,ˆx3)
x1+ 2 x2+
2x3≤72 ( ˆx1,ˆx2,ˆx3) ˆx2= ˆx3
x y z
V > 0
min
x y z=V
x,y,z>0
(x y + 2 x z + 2 y z).
u= ln x v = ln y w = ln z

(P) max
x≥0
Pn
i=1 xi=1
n
X
i=1
ln(xi+ai),
a= (a1, . . . , an)Rn
x= (x1, . . . , xn)
ˆx(P)
(P)
ξˆxi= max(0, ξ −ai)i= 1, . . . , n
n= 4 a= (0.25,0.5,0.75,1) ˆx
(P) min
A x≤b
1
2kx−ak2,
a∈RNb∈RMA∈RM×N
(P)
M= (µ1, . . . , µM)
(P)
(P?) min
M≤0
1
2
ATM
2+hM, A a −bi.
(P?)A M
(P?)a= ( 1 2.431.7 0.9)TA= ( 11111)
b= 1 (P)
min
x∈R2(kx−ak+kx−bk+kx−ck),
a= (1,4) b= (−3,−4) c= (5,1)
x0
x0
R2F(x, y) = x2+ 2 y4−3x y2
(0,0) F
(u, v)R2ϕ(t) = F(t u, t v)
t > 0F
(0,0)
ABC
M M A +MB +M C
ABC