日本時間 2020/11/03 9:00 公開 ニコニコ：https://www.nicovideo.jp/watch/sm37763386 Youtube：https://www.youtube.com/watch?v=njF6k1oIKm0&feature=youtu.be

1. 3

2. 1.

3. 1.
(a)
(b)
(c)

5.
max c1x1 + c2x2 + ⋯ + cnxn
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
min b1y1 + b2y2 + ⋯ + bmym
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22y2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

6.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*

7.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*
x = (x1, ⋯, xn) y = (y1, ⋯, ym)
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g
min b1y1 + b2y2 + ⋯ + bmym = g(y)
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22y2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0
max c1x1 + c2x2 + ⋯ + cnxn = f(x)
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0

8.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*
x = (x1, ⋯, xn) y = (y1, ⋯, ym)
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g
min b1y1 + b2y2 + ⋯ + bmym = g(y)
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22y2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0
max c1x1 + c2x2 + ⋯ + cnxn = f(x)
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
n = 3, m = 3
3
∑
j=1
cjxj ≤
3
∑
j=1
(
3
∑
i=1
aijyi)xj
c1x1 + c2x2 + c3x3
≤ (a11y1 + a21y2 + a31y3)x1
+(a12y1 + a22y2 + a32y3)x2
+(a13y1 + a23y3 + a33y3)x3
x1 ≥ 0, x2 ≥ 0,x3 ≥ 0

9.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*
x = (x1, ⋯, xn) y = (y1, ⋯, ym)
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g
min b1y1 + b2y2 + ⋯ + bmym = g(y)
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22y2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0
max c1x1 + c2x2 + ⋯ + cnxn = f(x)
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
n = 3, m = 3
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi
(a11y1 + a21y2 + a31y3)x1
+(a12y1 + a22y2 + a32y3)x2
+(a13y1 + a23y3 + a33y3)x3
= (a11x1 + a12x2 + a13x3)y1
+(a21x1 + a22x2 + a23x3)y2
+(a31x1 + a32x2 + a33x3)y3

10.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*
x = (x1, ⋯, xn) y = (y1, ⋯, ym)
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g
x = (x1, ⋯, xn) f ≤ g* f g*
g*
x*, y* x* f g*
x*

11.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
f* = c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m = g*
x* y*
x = (x1, ⋯, xn) y = (y1, ⋯, ym)
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g
y = (y1, ⋯, ym) f * ≤ g g f*
f*
x*, y* y* g f*
y*

12.
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g

13.
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g

14.
f =
n
∑
j=1
cjxj ≤
n
∑
j=1
(
m
∑
i=1
aijyi)xj =
m
∑
i=1
(
n
∑
j=1
aijxj)yi ≤
m
∑
i=1
biyi = g

16. See you next time
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