1. The document presents an optimization problem for finding a minimum value M given a set function v defined on subsets of a ground set N. The objective is to minimize M subject to several inequality constraints involving M and a vector x. 2. An example is given with N={1,2,3}, v defined on the power set of N, and the goal is to find a minimum value of M and a corresponding vector x = (x1, x2, x3) that satisfies the constraints. 3. The constraints define upper and lower bounds on x1, x2, x3 involving M, and their sum must equal v(N) while remaining non-negative. Analysis shows the minimum

1. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿਔ

2. ਔͷఆٛ
ਔΛ‫ٻ‬ΊΔ

3. ਔͷఆٛͷ֬ೝ
ఆٛɿఏ‫ܞ‬ ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ഑෼ ͱఏ‫ܞ‬ ΛͱΔͱ͖
Λ഑෼ ʹର͢Δఏ‫ܞ‬ ͷෆຬͱ͍͏
ʹର֤ͯ͠ఏ‫ܞ‬ ͷෆຬ Λ
େ͖ͳ΋ͷ͔Βॱʹฒ΂ͨϕΫτϧΛ ͱද͢
ͨͩ͠ ͢΂ͯͷ഑෼ʹରͯ͠ ͱ ʹؔͯ͠͸ෆຬ͸ͳͷͰআ֎ͯ͠ߟ͑Δ
Ώ͑ʹ ͸ ‫ݸ‬ͷෆຬΛฒ΂ͨϕΫτϧʹͳΔ
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
x ∈
𝒜
(v) S ⊆ N e(S, x)
θ(x)
N ∅
θ(x) 2n
− 2
#4*$
·ͨ͸ԋशࢀর

4. ਔͷఆٛͷ֬ೝ
ఆٛɿड༰త
̎ͭͷ഑෼ ʹରͯ͠ ͋Δ ͕ଘࡏͯ͠
ͱͳΔͱ͖ ͸ ΑΓ΋ड༰తͰ͋Δͱ͍͍ ͱද͢
ͷҙຯɿ ͩͱ ͱͳΔͨΊ
Ұఆ
x, y ∈
𝒜
(v) k, 1 ≤ k ≤ 2n
− 4
θl(x) = θl(y) ∀l = 1,⋯, k, θk+1(x) θk+1(y)
x y x ≫ y
2n
− 4 θl(x) = θl(y) ∀l = 1,⋯,2n
− 3 θ2n−2(x) = θ2n−2(y)
n−2
∑
l=1
θl(x) =
∑
S⊂N,S≠N,∅
(
v(S) −
∑
i∈S
xi)
=
∑
S⊂N,S≠N,∅
v(S) −
∑
S⊂N,S≠N,∅
∑
i∈S
xi
=
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1 ∑
i∈N
xi =
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1v(N)
#4*$
·ͨ͸ԋशࢀর

5. ਔͷఆٛͷ֬ೝ
ఆٛɿਔ
ͦΕΑΓ΋ड༰తͳ഑෼ͷଘࡏ͠ͳ͍഑෼ͷશମΛਔͱ͍͏
ఏ‫ؒܞ‬ͷෆຬΛͰ͖Δ͚ͩ‫ۉ‬౳Խ͢Δͱ͍͏ߟ͑ʹ‫ͮ͘ج‬ղ
ਔͷϙΠϯτ
ఆཧɿΧʔωϧʹඞͣ‫·ؚ‬ΕΔਔ Χʔωϧ ަবू߹
ఆཧɿਔ͸ඞͣଘࡏ͢Δ
ఆཧɿਔ͸Ұ఺ͷΈ‫ؚ‬Ήू߹ͱͳΔ
⊆ ⊆
#4*$
·ͨ͸ԋशࢀর

6. ਓήʔϜʹ͓͍ͯ ઓུ‫ܗ‬ήʔϜ͔Β‫ٻ‬Ίͨಛੑؔ਺‫ܗ‬ήʔϜʹ͓͍ͯ ਔΛ‫ٻ‬ΊΑ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
ਔΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

7. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
ਔΛ‫ٻ‬ΊΔ

8. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢Δ഑෼Λߟ͑Δ ͢ͳΘͪҎԼͷઢ‫ܭܗ‬ը໰୊Λղ͘ɿ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . v({1,2}) − (x1 + x2) ≤ M
v({1,3}) − (x1 + x3) ≤ M
v({2,3}) − (x2 + x3) ≤ M
v({1}) − x1 ≤ M v({2}) − x2 ≤ M v({3}) − x3 ≤ M
x1 + x2 + x3 = 6, x1, x2, x3 ≥ 0
ਔΛ‫ٻ‬ΊΔ
ఏ‫ܞ‬ ͷෆຬ
{1,2}

9. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢Δ഑෼Λߟ͑Δ ͢ͳΘͪҎԼͷઢ‫ܭܗ‬ը໰୊Λղ͘ɿ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . 1 − (x1 + x2) ≤ M
4 − (x1 + x3) ≤ M
2 − (x2 + x3) ≤ M
−x1 ≤ M −x2 ≤ M −x3 ≤ M
x1 + x2 + x3 = 6, x1, x2, x3 ≥ 0
ਔΛ‫ٻ‬ΊΔ
min M
s . t . x3 ≤ M + 5 x2 ≤ M + 2 x1 ≤ M + 4
−x1 ≤ M −x2 ≤ M −x3 ≤ M
−3M ≤ x1 + x2 + x3 = 6 ≤ 11 + 3M, x1, x2, x3 ≥ 0

10. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢Δ഑෼Λߟ͑Δ ͢ͳΘͪҎԼͷઢ‫ܭܗ‬ը໰୊Λղ͘ɿ
͜ͷ੍໿৚݅ʹ͓͍ͯ ͕ଘࡏ͢ΔͨΊͷ৚݅͸
Ҏ্ͷ৚݅Λຬͨ͢தͰ ࠷খͱͳΔ ͸ ͜ͷͱ͖
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . −M ≤ x1 ≤ M + 4 −M ≤ x2 ≤ M + 2 −M ≤ x3 ≤ M + 5 −3M ≤ x1 + x2 + x3 = 6 ≤ 11 + 3M, x1, x2, x3 ≥ 0
x = (x1, x2, x3)
−M ≤ M + 4 ⇔ − 2 ≤ M −M ≤ M + 2 ⇔ − 1 ≤ M −M ≤ M + 5 ⇔ −
5
2
= −
15
6
≤ M
−3M ≤ 6 ⇔ − 2 ≤ M 6 ≤ 11 + 3M ⇔ −
5
3
= −
10
6
≤ M
M −1 x2 = 1
ਔΛ‫ٻ‬ΊΔ
−2
−
5
2
−1
M
−
5
3

11. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
࠷େෆຬ ͳͷͰ ֤ఏ‫ܞ‬ͷෆຬ͸
ෆຬΛฒ΂Δͱ
Ώ͑ʹ൪໨ʹେ͖͍ෆຬ Λߟ͑ ࠷খԽ໰୊Λߟ͑Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M = − 1 x2 = 1
1 − (x1 + x2) = − x1 = ?
4 − (x1 + x3) = − 2 + x2 = − 1
2 − (x2 + x3) = 1 − x3 = ?
−x1 = ? −x2 = − 1 −x3 = ?
(−1, − 1, θ3(x), θ4(x), θ5(x), θ6(x))
M′

ਔΛ‫ٻ‬ΊΔ
−2
−
5
2
−1
M
−
5
3
ఏ‫ܞ‬ ͷෆຬ͸֬ఆ
{1,3}
ఏ‫ܞ‬ ͷෆຬ͸֬ఆ
{2}

12. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
Λલఏͱ͠ ൪໨ʹେ͖͍ෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢Δ഑෼Λߟ͑Δ
͢ͳΘͪҎԼͷઢ‫ܭܗ‬ը໰୊Λղ͘ɿ
৚݅Λ੔ཧ͢Δͱ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
x2 = 1 M′

( − 1)
min M′

s . t . 1 − (x1 + x2) = − x1 ≤ M′

⇔ x3 − 5 ≤ M′

2 − (x2 + x3) = 1 − x3 ≤ M′

⇔ x1 − 4 ≤ M′

−x1 ≤ M′

−x3 ≤ M′

x1 + 1 + x3 = 6, x1, x3 ≥ 0
−M′

≤ x1 ≤ M′

+ 4 1 − M′

≤ x3 ≤ M′

+ 5 1 − 2M′

≤ x1 + x3 = 5 ≤ 2M′

+ 9
ਔΛ‫ٻ‬ΊΔ

13. ഑෼Λ ͱ͢Δ഑෼ͳͷͰ
Λલఏͱ͠ ൪໨ʹେ͖͍ෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢Δ഑෼Λߟ͑Δ
͜ΕΒ৚݅ʹ͓͍ͯ ͕ଘࡏ͢ΔͨΊʹ͸
Ҏ্ͷ৚݅Λຬͨ͢தͰ ࠷খͱͳΔ ͸ ͜ͷͱ͖ ͱͳΔ
Αͬͯ ਔ͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
x2 = 1 M′

( − 1)
−M′

≤ x1 ≤ M′

+ 4 1 − M′

≤ x3 ≤ M′

+ 5 1 − 2M′

≤ x1 + x3 = 5 ≤ 2M′

+ 9
x1, x3
−M′

≤ M′

+ 4 ⇔ − 2 ≤ M′

1 − M′

≤ M′

+ 5 ⇔ − 2 ≤ M′

1 − 2M′

≤ 5 ⇔ − 2 ≤ M′

5 ≤ 2M′

+ 9 ⇔ − 2 ≤ M′

M′

−2 x1 = 2 x3 = 3
(x1, x2, x3) = (2, 1, 3)
ਔΛ‫ٻ‬ΊΔ

14. ίΞͱਔͷؔ܎͸ਤͷΑ͏ʹͳΔ
ਔΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
1
2 3
x1 ≤ 4
x2 ≤ 2
x3 ≤ 5
(2,1,3)
ਔ
ίΞ

15. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿਔ