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ゲーム理論NEXT 線形計画問題第5回 -双対定理証明-
1.
5
2.
1.
3.
1. (a) (b) 2. (a)
4.
5.
ℜn C ∀x ∈ C,
∀α ≥ 0, αx ∈ C C ℜn C ∀x ∈ C, ∀y ∈ C, ∀α ∈ [0,1], αx + (1 − α)y ∈ C C C
6.
C Rn clC y ∈
Rn clC α ∈ Rn αt y < inf x∈C αt x
7.
8.
w* C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (0, 0, 0) ∈ C t = x1 = x2 = x3 = x4 = 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
9.
w* C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t = 0 1 = c1x1 + c2x2 0 = a11x1 + a12x2 + x3 0 = a21x1 + a22x2 + x4 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
10.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t = 0 1 = c1x0 1 + c2x0 2 0 = a11x0 1 + a12x0 2 + x0 3 0 = a21x0 1 + a22x0 2 + x0 4 x1 = (x1 1, x1 2, x1 3, x1 4) x2 = x1 + kx0 , ∀k > 0 a11x2 1 + a12x2 2 + x2 3 = a11(x1 1 + kx0 1) + a12(x1 2 + kx0 2) + (x1 3 + kx0 3) = a11x1 1 + a12x1 2 + x1 3 + k(a11x0 1 + a12x0 2 + x0 3) = a11x1 1 + a12x1 2 + x1 3 = b1 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
11.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t = 0 1 = c1x0 1 + c2x0 2 0 = a11x0 1 + a12x0 2 + x0 3 0 = a21x0 1 + a22x0 2 + x0 4 x1 = (x1 1, x1 2, x1 3, x1 4) x2 = x1 + kx0 , ∀k > 0 c1x2 1 + c2x2 2 = c1x1 1 + c2x1 2 + k(c1x2 0 + c2x2 0) = c1x1 1 + c2x1 2 + k k > 0 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
12.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t > 0 t x3 i = xi t max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
13.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t > 0 t x3 i = xi t 1 t = c1x3 1 + c2x3 2 − w* 0 = a11x3 1 + a12x3 2 + x3 3 − b1 0 = a21x3 1 + a22x3 2 + x3 4 − b2 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
14.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} (1, 0, 0) ∈ C 1 = c1x1 + c2x2 − tw* 0 = a11x1 + a12x2 + x3 − tb1 0 = a21x1 + a22x2 + x4 − tb2 t > 0 t x3 i = xi t 1 t = c1x3 1 + c2x3 2 − w* 0 = a11x3 1 + a12x3 2 + x3 3 − b1 0 = a21x3 1 + a22x3 2 + x3 4 − b2 x3 = (x3 1, x3 2, x3 3, x3 4) c1x3 1 + c2x3 2 = w* + 1 t > w* w* max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
15.
(1, 0, 0)
∉ C C (α, d1, d2) ∈ R3 (α, d1, d2) ( 1 0 0) < inf{(α, d1, d2) ( r u1 u2 ) ( r u1 u2 ) ∈ C} ( r u1 u2 ) ∈ C αr + d1u1 + d2u2 < 0 C k > 0 kr ku1 ku2 ∈ C α(kr) + d1(ku1) + d2(ku2) < 0 k(αr + d1u1 + d2u2) < α max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
16.
(1, 0, 0)
∉ C C (α, d1, d2) ∈ R3 (α, d1, d2) ( 1 0 0) < inf{(α, d1, d2) ( r u1 u2 ) ( r u1 u2 ) ∈ C} (0, 0, 0) ∈ C = 0 (α, d1, d2) ( 1 0 0) < 0 ⇒ α < 0 α = − 1 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
17.
(1, 0, 0)
∉ C C (α, d1, d2) ∈ R3 (α, d1, d2) ( 1 0 0) < inf{(α, d1, d2) ( r u1 u2 ) ( r u1 u2 ) ∈ C} (0, 0, 0) ∈ C = 0 (α, d1, d2) ( 1 0 0) < 0 ⇒ α < 0 α = − 1 (r, u1, u2) ∈ C inf{(−1, d1, d2) ( r u1 u2 ) } = 0 ≤ (−1, d1, d2) ( r u1 u2 ) ⇒ 0 ≤ (−1)r + d1u1 + d2u2 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
18.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} 0 ≤ (−1)r + d1u1 + d2u2 (r, u1, u2) ∈ C t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 ⇒ 0 ≤ (−1)(c1x1 + c2x2 − tw*) + d1(a11x1 + a12x2 + x3 − tb1) + d2(a21x1 + a22x2 + x4 − tb2) ⇒ 0 ≤ t(w* − b1d1 − b2d2) +x1(−c1 + d1a11 + d2a21) +x2(−c2 + d2a12 + d2a22) +x3d1 +x4d2 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
19.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} 0 ≤ (−1)r + d1u1 + d2u2 (r, u1, u2) ∈ C t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 ⇒ 0 ≤ (−1)(c1x1 + c2x2 − tw*) + d1(a11x1 + a12x2 + x3 − tb1) + d2(a21x1 + a22x2 + x4 − tb2) ⇒ 0 ≤ t(w* − b1d1 − b2d2) +x1(−c1 + d1a11 + d2a21) +x2(−c2 + d2a12 + d2a22) +x3d1 +x4d2 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 w* − b1d1 − b2d2 ≥ 0 −c1 + d1a11 + d2a21 ≥ 0 −c2 + d2a12 + d2a22 ≥ 0 d1 ≥ 0 d2 ≥ 0 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
20.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} 0 ≤ (−1)r + d1u1 + d2u2 (r, u1, u2) ∈ C t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 ⇒ 0 ≤ (−1)(c1x1 + c2x2 − tw*) + d1(a11x1 + a12x2 + x3 − tb1) + d2(a21x1 + a22x2 + x4 − tb2) ⇒ 0 ≤ t(w* − b1d1 − b2d2) +x1(−c1 + d1a11 + d2a21) +x2(−c2 + d2a12 + d2a22) +x3d1 +x4d2 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 w* ≥ b1d1 + b2d2 d1a11 + d2a21 ≥ c1 d1a12 + d2a22 ≥ c2 d1 ≥ 0 d2 ≥ 0 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
21.
C = {(r,
u1, u2)|r = c1x1 + c2x2 − tw* u1 = a11x1 + a12x2 + x3 − tb1 u2 = a21x1 + a22x2 + x4 − tb2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0} 0 ≤ (−1)r + d1u1 + d2u2 (r, u1, u2) ∈ C t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 ⇒ 0 ≤ (−1)(c1x1 + c2x2 − tw*) + d1(a11x1 + a12x2 + x3 − tb1) + d2(a21x1 + a22x2 + x4 − tb2) ⇒ 0 ≤ t(w* − b1d1 − b2d2) +x1(−c1 + d1a11 + d2a21) +x2(−c2 + d2a12 + d2a22) +x3d1 +x4d2 t ≥ 0, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 w* ≥ b1d1 + b2d2 d1a11 + d2a21 ≥ c1 d1a12 + d2a22 ≥ c2 d1 ≥ 0 d2 ≥ 0 max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 (d1, d2) w* min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
22.
w* ≥ b1d1
+ b2d2 d1a11 + d2a21 ≥ c1 d1a12 + d2a22 ≥ c2 d1 ≥ 0 d2 ≥ 0 (d1, d2) w* x = (x1, ⋯, xn) y = (y1, ⋯, ym) f = n ∑ j=1 cjxj ≤ m ∑ i=1 biyi = g w* x = (x*1 , x*2 ) w* ≤ b1d1 + b2d2 = g max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
23.
w* ≥ b1d1
+ b2d2 d1a11 + d2a21 ≥ c1 d1a12 + d2a22 ≥ c2 d1 ≥ 0 d2 ≥ 0 (d1, d2) w* w* ≤ b1d1 + b2d2 w* = b1d1 + b2d2 (d1, d2) w* w* max c1x1 + c2x2 s . t . { a11x1 + a12x2 + x3 = b1 a21x1 + a22x2 + x4 = b2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0 min b1y1 + b2y2 s . t . { a11y1 + a21y2 ≥ c1 a12y1 + a22y2 ≥ c2 y1 ≥ 0, y2 ≥ 0
24.
25.
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