[PDF] Manuel de Cours Optimisation tiellement au étudiants de Master

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??? ?????? ?? ?? ??????? ????? ??? ?? ??????? ???????x?? ? ? ? ? ? ??

X=

x 1 x 2 x n

X=x1x2... xn

X ′Y=Y′X=nX i=1x iyi????? X ′Y= 0????? ?? ????? ???? ? ??????X??? ? ′X????? Y?

A=

a

11a12... a1n

a

21a22... a2n

a m1am2... amn

A=

a

11a21... am1

a

12a22... am2

a

1na2n... amn

?? ??????? ?? ???? ???????A?m×p? ??B?p×n? ??? ??? ???????C c ij=pX k=1a ikbkji= 1,...,m j= 1,...,n????? X ′X=x1x2... xn x 1 x 2 x n =x21+x22+...+x2n XX x 1 x 2 x n x1x2... xn x

21x1x2... x1xn

x

2x1x22... x2xn

x nx1xnx2... x2n

I=

1 0...0

0 1...0

0 0...1

??AB′? ??A′B? ??AB′B ??? ???V??? ??? ?

AV=λV??????

?? ???????V??? ?????? ??????? ?????? ?? ?? ???????A??????? ? ?? ?????? ??????λ?????A= [3 2 ; 3-2],V= [2 1]′,λ= 4?? |A-λIn|= 0?????? ??????? ??????? ????? ????{λi}ni=1? |A|=nY i=1λ i?????? ?????? ??A? ?? ??????? ??????? ??? ?????A-1? ?? ? ????? ? AA -1=In??????

Tr(A) =nX

i=1a ii??????

Tr(A) =Pλi?

A=A′??????

???? ?? ??????? ??????A???? ??? ??????? ??????? ???? ?λ1= 1,λ2= 2?? 3= 5? ?? ??????? ? A=2 1 1 2 A=2 1 1 2 B=2-1 -1 2 ∀u,v∈Ωet∀α∈[0,1], on a x=αu+ (1-α)v∈Ω?????? ∇f=∂f∂X=h∂f∂x

1∂f∂x

2...∂f∂x

ni f(x,y) =x2+xy+y2 ∂f∂x = 2x+y, ∂f∂y =x+ 2y, ∂f∂X= [2x+y , x+ 2y] H f=∂2f(X)(∂X)2=

2f(X)(∂x1)2∂2f(X)∂x

1∂x2...∂2f(X)∂x

1∂xn

2f(X)∂x

2∂x1∂

2f(X)(∂x2)2...∂2f(X)∂x

2∂xn... ... ... ...

2f(X)∂x

n∂x1∂

2f(X)∂x

n∂x2...∂2f(X)(∂xn)2 f(x,y) =x2+xy+y2 ∂f∂x = 2x+y,-→∂2f(∂x)2= 2 ∂2f∂x∂y = 1 ∂f∂y =x+ 2y,-→∂2f(∂y)2= 2 ∂2f∂y∂x = 1

Hf=2 1

1 2 (2-λ)2-1 = 0 (2-λ-1)(2-λ+ 1) = 0 (1-λ)(3-λ) = 0

X∈ ℜn?

f(x,y) =x-y-x2y+xy2 ??fy=∂f∂y ???? ?? ????? ????? n?????? ???? ?? ????{Xi= [xi,yi]′}ni=1? ??? ?????? ???? f(a,b) =nX i=1(yi-(axi+b))2 f(x,y) =x-x2y+xy2 AB 1 2

3

[-1 0 2] -1 0 2 -2 0 4 -3 0 6 A ′B= [1 2 3] -1 0

2

= 5 (AB′)B= -1 0 2 -2 0 4 -3 0 6 -1 0

2

5 10

15

A(B′B) =

1 2

3

×5 =

5 10

15

?? ??????? ??????A??????? ???3??????? ???????λ1= 1,λ2= 2??λ3= 5? det(A-λI) = 0?????? ???? ????? ???? ?? ? ?? ????(λ-λ1)(λ-λ3)(λ-λ3) = 0 det(A) =3Y i=1λ i=λ1×λ2×λ3= 10?????? trace(A) =3X i=1λ i=λ1+λ2+λ3= 8?????? ? ????? ??????? A A=2 1 1 2 X ′AX= [x y]2 1 1 2 x y = 2x2+ 2xy+ 2y2 =x2+y2+ (x+y)2>0,∀(x,y)∈ ℜ∗2 ? ????? ??????? B B=2-1 -1 2 X ′BX= [x y]2-1 -1 2 x y = 2x2-2xy+ 2y2 =x2+y2+ (x-y)2>0∀(x,y)∈ ℜ∗2 ? ????? ??????? A det(A-λI2) = det2 1 1 2 -λ1 0 0 1 = det

2-λ1

1 2-λ

= (2-λ)2-1 ? ????? ??????? B det(B-λI2) = det2-1 -1 2 -λ1 0 0 1 = det

2-λ-1

-1 2-λ = (2-λ)2-1 ????Z=λX+ (1-λ)Y????λ∈[0,1]? ????? ? a ′Z=a′(λX+ (1-λ)Y) =λa′X+ (1-λ)a′Y ≥λβ+ (1-λ)β=β(on a λ≥0et(1-λ)≥0 ) a ′Z≥β⇒Z∈Ω f(x,y) =x-y-x2y+xy2 f x=∂f∂x = 1-2xy+y2 f y=∂f∂y =-1-x2+ 2xy y

2-x2= 0⇒y=±x

???? y=x????? ?? ? ?

1-2x2+x2= 0→1-x2= 0→x=±1

????? ?? ? ???? ?????? ?(1,1)??(-1,-1) ???? y=-x????? ?? ? ?

1 + 2x2+x2= 1 + 3x2= 0

2f(∂x)2=-2y

2f∂x∂y

=-2x+ 2y

2f(∂y)2= 2x

2f∂y∂x

=-2x+ 2y H f= -2y-2x+ 2y -2x+ 2y2x ? ????? ? ????(x,y) = (1,1) H f(1,1) = -2 0

0 2

-→ |Hf(1,1)|=-4<0 ? ????? ? ????(x,y) = (-1,-1) H f(-1,-1) = 2 0

0-2

-→ |Hf(-1,-1)|=-4<0 f(a,b) =nX i=1(yi-(axi+b))2 ∂f∂a =nX i=12(-yi+axi+b)xi = 2anX i=1x

2i+ 2bnX

i=1x i-2nX i=1y ixi ∂f∂b =nX i=12(-yi+axi+b) = 2anX i=1x i+ 2nb-2nX i=1y i n X i=1x

2ia+nX

i=1x ib=nX i=1y ixi n X i=1x ia+n b=nX i=1y i 11=nX i=1x 2i S

12=S21=nX

i=1x i S 22=n
Z 1=nX i=1y ixi Z 2=nX i=1y i

S11S12

S 21S22
a b =Z1 Z 2 a=S12Z2-S22Z1S

212-S11S22

b=S12Z1-S11Z2S

212-S11S22

f(x,y) =x-x2y+xy2 J f=∇f′= ∂f∂x ∂f∂y 1-2xy+y2 -x2+ 2xy H f=-2y-2x+ 2y -2x+ 2y2x det(H-λI) = 0 →(-2y-λ)(2x-λ)-(-2x+ 2y)2= 0

2+ 2(y-x)λ-(-2x+ 2y)2-4xy= 0

∆ = 4(y-x)2+ 16(y-x)2+ 16xy = 20(y-x)2+ 16xy 2 =x-y+p5(y-x)2+ 4xy 2 =x-y-p5(y-x)2+ 4xy y? det(Hf) =λ1×λ2 x-y+p5(y-x)2+ 4xy x-y-p5(y-x)2+ 4xy =-4(x2+y2-xy) J f=∇f′= ∂f∂x ∂f∂y 1-2xy+y2 -x2+ 2xy2 0

0

(1) (2) x

2-2xy= 0→x(x-2y) = 0

1-4y2+y2= 0→y=±r1

3 y=q1 3 →X1=h ,q1 3 i y=-q1 3quotesdbs_dbs6.pdfusesText_11

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