https://www.nicovideo.jp/watch/sm37654902 https://www.youtube.com/watch?v=8OD1uVGfo9c&t=1s

1. 2

2. 1.
2.

3. 1.
(a)
(b)
2.
(a)
(b)

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

6.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

7.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

8.
max c1x1 + c2x2 + ⋯ + cnxn
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
n x1, x2, ⋯, xn m y1, y2, ⋯, ym
min b1y1 + b2y2 + ⋯ + bmym
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

9.
max c1x1 + c2x2 + ⋯ + cnxn
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
m n
min b1y1 + b2y2 + ⋯ + bmym
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

10.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

11.
max c1x1 + c2x2 + ⋯ + cnxn
s . t .
a11x1 + a12x2 + ⋯ + a1nxn ≤ b1
a21x1 + a22x2 + ⋯ + a2nxn ≤ b2
⋮
am1x1 + am2x2 + ⋯ + amnxn ≤ bm
x1 ≥ 0, x1 ≥ 0, ⋯, xn ≥ 0
aij → aji
min b1y1 + b2y2 + ⋯ + bmym
s . t .
a11y1 + a21y2 + ⋯ + am1ym ≥ c1
a12y1 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

12.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

13.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

14.
xA xB
max 10000xA + 20000xB
s . t .
3xA + 2xB ≤ 18
2xA + 8xB ≤ 52
xA ≥ 0, xB ≥ 0
min 18y1 + 52y2
s . t .
3y1 + 2y2 ≥ 10000
2y1 + 8y2 ≥ 20000
y1 ≥ 0, y2 ≥ 0

15.
max 10000xA + 20000xB
s . t .
3xA + 2xB ≤ 18
2xA + 8xB ≤ 52
xA ≥ 0, xB ≥ 0
xA
xB
9
6
k = 10000xA + 20000xB
26
(2, 6)
k′ =
k
20000
min 18y1 + 52y2
s . t .
3y1 + 2y2 ≥ 10000
2y1 + 8y2 ≥ 20000
y1 ≥ 0, y2 ≥ 0
y2
5000
10000
3
k
52
2500
k = 18y1 + 52y2
(2000, 2000)

16.
max 10000xA + 20000xB
s . t .
3xA + 2xB ≤ 18
2xA + 8xB ≤ 52
xA ≥ 0, xB ≥ 0
xA
xB
9
6
k = 10000xA + 20000xB
26
(2, 6)
k′ =
k
20000
min 18y1 + 52y2
s . t .
3y1 + 2y2 ≥ 10000
2y1 + 8y2 ≥ 20000
y1 ≥ 0, y2 ≥ 0
y2
5000
10000
3
k
52
2500
k = 18y1 + 52y2
(2000, 2000)
(2, 6) 140,000
(2000, 2000) 140,000

18.
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 + a22x2 + ⋯ + am2ym ≥ c2
⋮
a1ny1 + a2ny2 + ⋯ + amnym ≥ cn
y1 ≥ 0, y1 ≥ 0, ⋯, ym ≥ 0

19.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m
x* y*

20.
x* = (x*1
, ⋯, x*n ) y* = (y*1
, ⋯, y*m)
c1x*1
+ c2x*2
+ ⋯ + cnx*n = b1y*1
+ b2y*2
+ ⋯ + bmy*m
x* y*

21.
(N, v)
min
∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊊ N
z * z * ≤ v(N)

22.
min
∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊊ N
⇔
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N

23.
min
∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊊ N
⇔
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N

24.
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N
x′1 − x′′1 + x′2 − x′′2 + x′3 − x′′3
x′1 − x′′1 ≥ v({1})
x′2 − x′′2 ≥ v({2})
x′3 − x′′3 ≥ v({3})
x′1 − x′′1 + x′2 − x′′2 ≥ v({1,2})
x′1 − x′′1 + x′3 − x′′3 ≥ v({1,3})
x′2 − x′′2 + x′3 − x′′3 ≥ v({2,3})
∑
i∈S
(x′i − x′′i ) ≥ v(S) yS
y{1}
y{2,3}

25.
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N
x′1 − x′′1 + x′2 − x′′2 + x′3 − x′′3
x′1 − x′′1 ≥ v({1})
x′2 − x′′2 ≥ v({2})
x′3 − x′′3 ≥ v({3})
x′1 − x′′1 + x′2 − x′′2 ≥ v({1,2})
x′1 − x′′1 + x′3 − x′′3 ≥ v({1,3})
x′2 − x′′2 + x′3 − x′′3 ≥ v({2,3})
x′1
1 ⋅ y{1} + 1 ⋅ y{1,2} + 1 ⋅ y{1,3} ≤ 1
x′′1
(−1) ⋅ y{1} + (−1) ⋅ y{1,2} + (−1) ⋅ y{1,3} ≤ − 1
y{1} + y{1,2} + y{1,3} = 1

26.
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N
x′1 − x′′1 + x′2 − x′′2 + x′3 − x′′3
x′1 − x′′1 ≥ v({1})
x′2 − x′′2 ≥ v({2})
x′3 − x′′3 ≥ v({3})
x′1 − x′′1 + x′2 − x′′2 ≥ v({1,2})
x′1 − x′′1 + x′3 − x′′3 ≥ v({1,3})
x′2 − x′′2 + x′3 − x′′3 ≥ v({2,3})
y{1} + y{1,2} + y{1,3} = 1
y{2} + y{1,2} + y{2,3} = 1
y{3} + y{1,3} + y{2,3} = 1
y{1}, y{2}, y{3}, y{1,2}, y{1,3}, y{2,3} ≥ 0

27.
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N
x′1 − x′′1 + x′2 − x′′2 + x′3 − x′′3
y{1} + y{1,2} + y{1,3} = 1
y{2} + y{1,2} + y{2,3} = 1
y{3} + y{1,3} + y{2,3} = 1
y{1}, y{2}, y{3}, y{1,2}, y{1,3}, y{2,3} ≥ 0
v({1})y{1} + v({2})y{2} + v({3})y{3}
+v({1,2})y{1,2} + v({1,3})y{1,3} + v({2,3})y{2,3}

28.
min
∑
i∈N
(x′i − x′′i )
s . t .
∑
i∈S
(x′i − x′′i ) ≥ v(S), ∀S ⊊ N
x′i ≥ 0, x′′i ≥ 0, ∀i ∈ N
x′1 − x′′1 + x′2 − x′′2 + x′3 − x′′3
y{1} + y{1,2} + y{1,3} = 1
y{2} + y{1,2} + y{2,3} = 1
y{3} + y{1,3} + y{2,3} = 1
y{1}, y{2}, y{3}, y{1,2}, y{1,3}, y{2,3} ≥ 0
v({1})y{1} + v({2})y{2} + v({3})y{3}
+v({1,2})y{1,2} + v({1,3})y{1,3} + v({2,3})y{2,3}
∑
S⊊N
v(S)yS
yS ≥ 0, ∀S ⊊ N
∑
S:i∈S⊊N
yS = 1, ∀i ∈ N

29.
min
∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊊ N
max
∑
S⊊N
v(S)yS
s . t .
∑
S:i∈S⊊N
yS = 1, ∀i ∈ N
yS ≥ 0, ∀S ⊊ N

30.
(N, v)
min
∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊊ N
z * z * ≤ v(N)
z* = min
∑
i∈N
xi w* = max
∑
S⊊N
v(S)yS
z* = w* w * ≤ v(N)

32. See you next time
2