Problèmes généraux : dualité et solvabilité (résumé de cours) 1 Le

n
X
j=1
cjxj
Pn
j=1 aij xj≤bi, i ∈I
Pn
j=1 aij xj=bi, i ∈E
xj≥0, j ∈R.
n m j ∈R F := {1, . . . , n} \ R j ∈F
n
X
j=1 m
X
i=1
aij yixj≤
m
X
i=1
biyi,
yi≥0i∈Ix
y=y1· · · ym
y
m
X
i=1
aij yi≥cj, j ∈R
m
X
i=1
aij yi=cj, j ∈F
cjxj≤m
X
i=1
aij yixj, j = 1, . . . , n
j∈F
n
X
j=1
cjxj≤
n
X
j=1 m
X
i=1
aij yixj.
x
n
X
j=1
cjxj≤
m
X
i=1
biyi.

m
X
i=1
biyi
Pm
i=1 aij yi≥cj, j ∈R
Pm
i=1 aij yi=cj, j ∈F
yi≥0, i ∈I.
Pn
j=1 aij xj≤bi, i ∈I
Pn
j=1 aij xj=bi, i ∈E,
x∈Rnx
x1+ 3x2+ 2x3+ 4x4≤5
3x1+x2+ 2x3+x4≤4
5x1+ 3x2+ 3x3+ 3x4= 9
−x3≤0
−x4≤0
y=1 3 −2210≤ −1
y∈Rm
Pm
i=1 aij yi= 0, j = 1, . . . , n
Pm
i=1 biyi<0
yi≥0, i ∈I.
m
X
i=1
xn+i
Pn
j=1 aij xj+wixn+i≤bi, i ∈I
Pn
j=1 aij xj+wixn+i=bi, i ∈E
xn+i≥0, i = 1, . . . , m

wi= 1 bi≥0wi=−1bi<0
<0
m
X
i=1
biyi
Pm
i=1 aij yi= 0, j = 1, . . . , n
wiyi≥ −1, i = 1, . . . , m
yi≥0, i ∈I
y∗<0y∗
Ax ≤bn
n+ 1 yi>0
m≥0
m > 0
∃y∗≥0
y∗A=0 y∗b=: c < 0
yA =0
yb =c
n+1 y∗≥0
y≥0n+ 1 >0
(1) −x1−4x2≤ −9
(2) −2x1−x2≤ −4
(3) x1−2x2≤0
(4) x1≤4
(5) 2x1+x2≤11
(6) −2x1+ 6x2≤17
(7) −6x1−x2≤ −6
x1x2x2
x2
(6) −2x1+ 6x2≤17 ⇔x2≤17
6+1 3x1
(1) −x1−4x2≤ −9⇔x2≥9
4−1 4x1.

x1≤
9
4−1 4x1≤17 6+1 3x1⇔x1≥ −1.
x2
x2
x2
Ax ≤bxki, j aik >0
ajk <0
aix≤bi⇔aikxk≤aikxk−aix+bi
ajx≤bj⇔(−ajk)xk≥ −ajkxk+ajx−bj.
(−ajk)aik
aikajx−aikbj≤ −(−ajk )aix+ (−ajk)bi
(aikaj+ (−ajk)ai)x≤aik bj+ (−ajk)bi
Pn
j=1 aij xj≤bi, i ∈I
Pn
j=1 aij xj=bi, i ∈E.
∃y∈Rmyi≥0∀i∈I
m
X
i=1
aij yi= 0,∀j= 1, . . . , n
m
X
i=1
biyi<0.
Ax ≤b x ∈Rnb∈Rma0x≤b0
a0x≤b0∀x
∃y≥0 yA =a0yb ≤b0
∃y≥0 yA =0Tyb <0
Ax ≤bP
P P
Am×nb∈Rm
∃x∈RnAx =b x ≥0
∃y∈RmyA ≥0Tyb <0
A=
a1· · · an
b a1, . . . , an
h0
h={x∈Rm|yx = 0}
y a1, . . . , anb

m
X
i=1
xn+i
Pn
j=1 aij xj+wixn+i≤bi, i ∈I
Pn
j=1 aij xj+wixn+i=bi, i ∈E
xn+i≥0, i = 1, . . . , m
wi= 1 bi≥0wi=−1bi<0
y