"
1 1
" x1, x2 > 0 x1x2 d" x1 + x2
2 2
"
1 1
" x1, x2 > 0 x1x2 = x1 + x2 Ô! x1 = x2.
2 2
x1
k
x2 . . . xk > 0 ´1 ´2 . . . ´k > 0 ´i = 1
i=1
k k
i
(xi)´ d" ´ixi, (A - G)
i=1 i=1
x1 = x2 = . . . = xk
f : (0, +") R
f(x) = - ln x,
1
C2(0, +") f (x) = x > 0 f
x2
(0, +")
k k k
" x1, x2, . . . , xk > 0 " ´1, ´2, . . . , ´k > 0, ´i = 1 f( ´ixi) d" ´if(xi),
i=1 i=1 i=1
x1 = x2 =
. . . = xk
k k k
" x1, . . . , xk > 0 " ´1, . . . , ´k > 0, ´i = 1 - ln( ´ixi) d" - ´i ln(xi)
i=1 i=1 i=1
k k k
" x1, . . . , xk > 0 " ´1, . . . , ´k > 0, ´i = 1 exp(ln( ´ixi)) e" exp( ´i ln(xi))
i=1 i=1 i=1
k k k
i
" x1, . . . , xk > 0 " ´1, . . . , ´k > 0, ´i = 1 ´ixi e" x´ .
i
i=1 i=1 i=1
 x1, x2, . . . , xk > 0
´1, ´2, . . . , ´k
x1, x2, . . . , xk > 0 ´1, ´2, . . . , ´k
Å„Å‚
2
ôÅ‚
ôÅ‚xy z3 max
ôÅ‚
òÅ‚
(Pa)
ôÅ‚x3 + y2 + z = 39
ôÅ‚
ôÅ‚
ół
x, y, z > 0.
(Pa)
Å„Å‚
ôÅ‚
òÅ‚h(x, y) := xy2(39 - x3 - y2)3 max
(Pa)
ôÅ‚
ółx, y > 0 '" x3 + y2 < 39.
h C2 R2 det Hh(1, 1) = -7759836988 (1, 1)
h
(Pa) x, z y, z
´1, ´2, ´3 > 0 ´1 + ´2 + ´3 = 1
2 2
3 3
1 2 3 1 2 3 1 2 3
39 = ´1(x )+´2(y )+´3(´z ) e"(A-G) (x )´ ·(y )´ ·(´z )´ = (´1)-´ (´2)-´ (´3)-´ ·x3´ y2´ z´
´1 ´2 3 ´1 ´2 3
Å„Å‚3´ 1 Å„Å‚
1
=
ôÅ‚2´ 2 ôÅ‚
ôÅ‚ ôÅ‚3´1 = ´2
2
ôÅ‚2´ 2 ôÅ‚
òÅ‚ òÅ‚3´ = ´3
2
=
2
1 3 9
´3 3
Ô! Ô! ´1 = , ´2 = , ´3 = .
13 13 13
ôÅ‚´1 + ´2 + ´3 = 1 ôÅ‚´1 + ´2 + ´3 = 1
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ół ół
´1, ´2, ´3 > 0 ´1, ´2, ´3 > 0
x, y, z > 0 x3 + y2 + z = 39
21 3 34
39 e" 13 · 3- 13
· (xy2z3)13 Ô! 313 e" xy2z3
y2
x3 z 1 1
= = Ô! x3 = z '" y2 = z.
´1 ´2 ´3 9 3
34
313 e" xy2z3 (Pa)
1
x = 33 , y = 3, z = 27
34
3
3
x1 4x2
f(x1, x2) = 4x1 + + min
x2 x1
2
x1, x2 > 0.
f
(0, +")2
1 4x2
4 + - = 0
x2 x2
1
2 1
"f(x1, x2) = [0, 0] Ô! Ô! (x1, x2) = (1, ).
4 2 2
1
-2x + = 0
x3 x1
2
f
8x2 2 4
-x -
3
x3 x2
1 2 1
Hf(x1, x2) =
2 4 6x1
-x -
3
x2 x4
2 1 2
Ô! "1(x1, x2) > 0, "2(x1, x2) = -x 4 (x4 - 8x2x3 + 4x6) ("2(1, 2) < 0).
4
x6 1 1 2 2
1 2
1
(x1, x2) = (1, ) f (0, +")2
2 2
f (0, +")2 Hf(1, 2)
1 1 1
"1(1, ) > 0 "2(1, ) > 0 (1, ) f
2 2 2 2 2 2
(A - G) x1, x2 > 0
4x2 4x2 3
1 1 1 1 1 2
f(x1, x2) = ´1(4x ) + ´2(xx ) + ´3(x ´3 ) e"(A-G) (4x )´ · (xx )´ · (x ´3 )´
2 2
´1 ´2 ´1 ´2 1
1
2 2
1 2 3 1 2
(´4 )´ (´2)-´ (´4 )-´ x´ +´2-´3x-2´ +´3
1 2
1 3
f
x1, x2 ´1, ´2, ´3
Å„Å‚
ôÅ‚
ôÅ‚´1, ´2, ´3 > 0
ôÅ‚
òÅ‚´ + ´2 + ´3 = 1
1
1 1
Ô! ´1 = ´2 = ´3 = .
4 2
ôÅ‚´1 + ´2 - ´3 = 0
ôÅ‚
ôÅ‚
ół
-2´2 + ´3 = 0
" x1, x2 > 0 f(x1, x2) e" 8,
4x1 x1 4x2 1
1 2
= = Ô! 16x1 = 4x = 8x Ô! x1 = x2 = .
´1 x2´2 x1´3 x2 x1 2
2 2
1
" x1, x2 > 0 f(x1, x2) e" f(1, ).
2 2
m n
2
ci > 0 i = 1, . . . , n Ä…ij " R i = 1, . . . , n j = 1, . . . , m Ä…ij > 0
j=1 i=1
Å„Å‚
n m
ôÅ‚
òÅ‚g(t1, . . . , tm) = ci (tj)Ä…ij
min
(GP )
i=1 j=1
ôÅ‚
ół
t1, . . . , tm > 0
(GP ) t " (0, +")m g
(0, +")m
(GP )
´i
n n
ij ij
ci m (tj)Ä… ci m (tj)Ä…
j=1
g(t1, . . . , tm) = ´i j=1 e"(A-G) =
´i ´i
i=1 i=1
´i n
n
n
ci ´i n m ci ´i m
ij
i=1
= tÄ… = tj Ä…ij´i
´i i=1 j=1 j ´i j=1
i=1 i=1
´1, . . . , ´n > 0
n
´i = 1
i=1
n
" j = 1, . . . , m Ä…ij´i = 0 .
i=1
´i i = 1, . . . , n
n
ci ´i
g(t1, . . . , tm) e" .
´i
i=1
n
ci ´i
v(´1, . . . , ´n) := .
´i
i=1
g(t1, . . . , tm) e" v(´1, . . . , ´n),
tj > 0 j = 1, . . . , m ´i i = 1, . . . , n
Å„Å‚
n
ôÅ‚
ôÅ‚v(´1, . . . , ´n) = ci ´i max
ôÅ‚
ôÅ‚
ôÅ‚
´i
ôÅ‚
ôÅ‚ i=1
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
òÅ‚´ , . . . , ´n > 0
1
(DGP )
n
ôÅ‚
ôÅ‚
ôÅ‚
´i = 1
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
i=1
ôÅ‚
ôÅ‚ n
ôÅ‚
ôÅ‚
ôÅ‚" j = 1, . . . , m Ä…ij´i = 0
ôÅ‚
ół
i=1
(GP )
(´1, . . . , ´n) " Rn
(DGP )
(DGP ) (DGP )
(DGP ) v
(DGP )
t (GP ) ´
(DGP )
g(t ) e" v(´ ) .
t = (t 1, . . . , t m)
(GP ) (DGP )
´ = (´1, . . . , ´n)
ui(t )
´i = , i = 1, . . . , n,
g(t )
n
ij
ui(t ) = ci m (t j)Ä… i g(t ) = ui(t )
j=1 i=1
(DGP )
g(t ) = v(´ )
m n
2
ci > 0 i = 1, . . . , n Ä…ij " R i = 1, . . . , n j = 1, . . . , m Ä…ij > 0
j=1 i=1
Å„Å‚
n m
ôÅ‚
òÅ‚g(t1, . . . , tm) = ci (tj)Ä…ij
min
(GP )
i=1 j=1
ôÅ‚
ół
t1, . . . , tm > 0,
ij
ui(t) = ci m (tj)Ä… i = 1, . . . , n (GP )
j=1
F (DGP )
´ " Rn
Å„Å‚
ôÅ‚
1
ôÅ‚´ , . . . , ´n > 0
ôÅ‚
ôÅ‚ n
ôÅ‚
ôÅ‚
òÅ‚
´i = 1
i=1
ôÅ‚
n
ôÅ‚
ôÅ‚
ôÅ‚" j = 1, . . . , m Ä…ij´i = 0
ôÅ‚
ôÅ‚
ół
i=1
F
(GP )
{´ } ´ (DGP )
´ (DGP )
´ t (GP )
ui(t )
´i = , i = 1, . . . , n.
v(´ )
g(t ) = v(´ )
(DGP ) F
´ F
(DGP )
(t 1, . . . , t m)
i1 im
ui(t ) = ci(t 1)Ä… . . . (t m)Ä…
 (t 1, . . . , t m)
(ln t 1, . . . , ln t m)
Ä…i1 ln t 1 + . . . + Ä…im ln t m = ln ´1 - ln ci + ln v(´ ), i = 1, . . . , n.
(DGP ) (GP ) (DGP )
i ´i ´ (DGP ) i
ui(t ) g(t ) (GP )
1000
g(t1, t2) = + 2t1 + 2t2 + t1t2 min
t1t2
(GP )1
t1, t2 > 0
"g(t1, t2) = [-1000 + 2 + t2, -1000 + 2 + t1]
t2t2 t1t2
1 2
"g(t1, t2) = [0, 0] Ô! t1 = t2 '" t4 + 2t3 - 1000 = 0
1 1
2000
1
t3t2
1
Hg(t1, t2) =
2000
1
t1t3
2
g
(0, +")2 g (0, +")2
g
(DGP )1 (GP )1
Å„Å‚
-´4
1 2 3
ôÅ‚
ôÅ‚v(´1, ´2, ´3, ´4) = (1000)´ · (´2 )´ · (´2 )´ · ´4 max
´1
2 3
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
òÅ‚´ , ´2, ´3, ´4 > 0
1
(DGP )1
ôÅ‚´1 + ´2 + ´3 + ´4 = 1
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚-´1 + ´2 + ´4 = 0
ôÅ‚
ôÅ‚
ôÅ‚
ół
-´1 + ´3 + ´4 = 0
(DGP )1
(DGP )1
1 1 3 1
F = {(1 - ´2, ´2, ´2, - ´2) : ´2 " (0, )}.
2 2 2 2 3
(DGP )1
1 1
1 3
- x
1 1 3 1000 2 3 1
2 2
f(x) := v(1 - x, x, x, - x) = · (x)2x · (1 - x)- 2 + 2 x, x " (0, ).
1 1
2 2 2 2 2 2 3
- x
2 2
 f
h(x) := ln(f(x)) (0, +") x ln x f h
1
h(x) = 3(1-1x) ln 10-(1-1x) ln(1-1x)+2x ln 2-2x ln x-(1-3x) ln(1-3x), x " (0, ).
2 2 2 2 2 2 2 2 2 2 3
1 1 3 3
h (x) = -3 ln 10 + ln(1 - x) + 2 ln 2 - 2 ln x + ln(1 - x)
2 2 2 2 2 2 2
1 2 9 1
h (x) = - - - , h (x) < 0, x " (0, )
1 1 1 3
x 3
4( - x) 4( - x)
2 2 2 2
1
h (0, )
3
h
1 3
4(1 - x)(1 - x)3 = 1000x4
2 2 2 2
´2 H" 0.09527312707
(GP )1
ui(t )
´i = , i = 1, 2, 3, 4 Ô!
v(´ )
2t 1 2t 2 1 3 t 1t 2
1 1 1000
- ´2 = , ´2 = , ´2 = , - ´2 = Ô!
2 2 t 1t 2v(´ ) v(´ ) v(´ ) 2 2 v(´ )
´2 v(´ )
t 1 = t 2 = .
2