20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1.................................... 7 1.2.2............................... 8 1.3..................................... 8 1.3.1....................................... 9 1.3.2...................................... 11 1.3.3....................................... 11 1.3.4................................... 13 1.4................................... 14 1.4.1.......................... 14 1.4.2................................... 14 1.4.3.................................. 16 1.4.4.................................... 17 1.4.5.................................. 18 1.5....................................... 20 1.5.1 1............................ 20 1.5.2............................ 21 2 22 2.1............................................ 22 2.1.1.................................. 22 2.1.2 1.............................. 24 2.1.3 1.............................. 24 2.1.4............................... 26 2.1.5...................................... 32 2.2........................................... 34 2.2.1............................... 34 2.2.2................................... 35 2.2.3 3................................. 36 1

2 2.2.4............................. 36 2.3.................................. 38 2.3.1.................................. 38 2.3.2......................... 41 2.4....................................... 41 2.4.1.............................. 41 2.4.2............................... 44 2.5 2................................... 46 2.5.1 2...................................... 46 2.5.2 2............................. 48 3 53 3.1..................................... 53 3.1.1............................. 53 3.1.2.................................... 53 3.1.3............................... 55 3.2................. 57 3.2.1....................... 57 3.2.2.................................. 62 3.3................................... 63 3.3.1.............................. 64 3.3.2.................................... 68 3.3.3.......................... 70 4 74 4.1.................................. 74 4.2........................................ 75 4.2.1............................... 75 4.2.2.................................. 76 4.2.3.................................... 78 4.2.4.................................. 82 4.3........................ 88 4.4................................. 88 5 91 5.1.......................................... 92 5.1.1 1................................. 92 5.1.2................................. 93 5.1.3.................................... 96 5.2........................................ 97 6 99 6.1.......................................... 99 6.1.1............................... 99 6.1.2.................................. 100 6.2................................. 102

3 6.2.1....................................... 102 6.2.2................................... 106 6.2.3............................... 109 6.2.4...................................... 116 6.3..................................... 119 6.4..................................... 120 7 123 7.1....................................... 123 7.1.1............................. 123 7.1.2............................ 124 7.1.3............................ 125 7.2....................................... 126 7.2.1....................................... 126 7.2.2........................ 127 7.2.3................................... 129 7.2.4.................................. 131 7.3................................... 132 7.3.1 1.................................. 132 7.3.2 2............................. 134 7.4.......................... 139 7.4.1............................... 139 7.4.2 1 2............ 141 7.4.3.................... 143 7.5....................................... 145 7.5.1................................... 145 7.5.2....................................... 146

4 1 1.1 1 1.1.1 R x 1, x 2, x 3,... ( x n ) (n = 1, 2, 3,... ) 1.1 ( ). R ( x n ) x R ɛ N N n x n x < ɛ ( x n ) x ɛ > 0, N N, n > N : x n x < ɛ ( x n ) x n xn x lim x n = x x n x (n ) n x ( x n ) 1.1. x n = 1/n ( x n ) 0 ɛ N 1/ɛ N N n > N 0 < x n = 1 n < 1 N ɛ n > N : x n 0 < ɛ ( x n ) 0 1.2. ( x n ) 1 x n = n n 1 n R ( x n ) 0 1.1 1.1 ɛ > 0, N N, n > N : x n x < ɛ ɛ > 0, N N, n > N : x n x ɛ (1.1) ɛ N N n x n x ɛ ɛ = 1/2 n x n 0 = 1 0 = 1 > ɛ (1.2)

5 ɛ 1/2 N n (1.1) ( ) x n 0 ( ) x n 1 n 1.1. 1.2 ( ) x n R 1.2. 1.1.2 1 f : R R ( ) x n f(x1 ), f(x 2 ), f(x 3 ),... 1.3. f : R R f(x) = x 3 + 1 (1.3) ( x n ) 1.1 xn = 1/n f(x n ) = x 3 n + 1 = 1 n 3 + 1 f(x n ) 1 (n ) f(x n ) 1 x n = ( 1) n /n f(x n ) = x 3 n + 1 = ( 1)n n 3 + 1 f(x n ) 1 (n ) 1 1.3 ( x n ) lim n x n = 0 0 (1.3) f 0 ( x n ) f(x n ) 1 (n ) 1.2 ( ). x R f R x n x (n ) ( x n ) f(x n ) f(x) (n ) lim n f(x n) = f(x) f x f(y) f(x) (y x) lim y x f(y) = f(x)

6, (x n x) ( ) (f(x)) (f(x n )) lim f(x n) n lim n f(x n ) = f (lim n x n ) 1.3 f 0 1.2 ɛ > 0, δ > 0, y R : x y < δ f(x) f(y) < ɛ 1.3. 1.2 1.4. R f 1 x 0 f(x) = 0 x < 0 x = 0 0 x n = 1/n x n = 1/n f n, x n = 1 n > 0 n, f(x n ) = 1 f(x n ) 1 (n ) f f(0) = 1 f(x n ) f(0) (n ) 0 1 f n, x n = 1 n < 0 n, f(x n ) = 0 f(x n ) 0 (n ) 0 f(x n ) f(0) (n ) 0 2 f 0 0 3 0 1 2 3

7 1.2 1.2.1 L R L x 1, x 2, x 3,... ( x n ) (n = 1, 2, 3,... ) Eucliden Norm 1.3 ( ). R L x = ( x 1, x 2, x 3,..., x L) x x R L x = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 + + (x L ) 2 (1.4) L = 2, 3 1.4. R L ( x n ) x R L lim x n x = 0 x n x 0 (n ) n ( x n ) x x ( x n ) lim x n = x x n x (n ) (1.5) n ( x n ) x ( xn ) x x n = ( x 1 n, x 2 n, x 3 n,..., x L n) 1.4 l = 1, 2, 3,..., L : x l n x l (n ) (1.6) 1.4. 1.4 (1.6) 1.5. R 2 0 = (0, 0) ( ) 1 x n = n, 0 x n = x n = x n = ( 0, 1 ) n ( 1 n, 1 ) n ( 1 n cos n, 1 n sin n ) (1.7) (1.8) (1.9) (1.10) xn 0 (n ) 0 R 2 0

8 1.2.2 L F : R L R M 1 R L R M F : R L R M F (x) = F (x 1, x 2,..., x L ) x R L F 1 (x 1, x 2,..., x L ) F 2 (x 1, x 2,..., x L ) F (x) =. F M (x 1, x 2,..., x L ) 1.5. x R L F R L x n x (n ) ( x n ) F (x n ) F (x) (n ) x n x 0 (n ) ( x n ) F (x n ) F (x) 0 (n ) F x lim F (y) = F (x) F (y) F (x) (y x) y x F 0 F (0) 1 ɛ - δ 1.5 ɛ > 0, δ > 0, x R L : x u < δ f(x) f(u) < ɛ 1 1 1 1.3 (x 1, x 2,..., x L ) F

9 1.3.1 L = 2 M = 1 F (x 1, x 2 ) = F (x, y) 1.6 (1 ). 1 f : R R x R f(x + h) f(x) lim h 0 h f x f(x + h) f(x) lim = df h 0 h dx (x, y) = f (x) f x f x R f R f R R x f (x) f f : R R f x f x x f R 1.5. n f : R R ( ) 1 x n sin (x 0 ) f(x) = x 0 (x = 0 ) 1. n = 0 x 0 f(x) = sin (1/x) f x = 0 2. n = 1 f x = 0 3. n = 2 f x = 0 1.7 ( ). F : R 2 R (x, y) R 2 g(h) = F (x + h, y) g : R R g h = 0 g(h) r(0) F (x + h, y) F (x, y) lim = lim h 0 h h 0 h F (x, y) x F (x, y) x F x (x, y) R 2 F x y F (x, y + h) F (x, y) lim h 0 h F (x, y) y F (x, y) y F y (x, y) R 2 F y

10 1.7 1 F 1.6. (K, L) (K +dk, L+dL) Q = F (K, L) F (K, L) K K + dk Q F K (K, L)dK L L + dl Q F L (K, L)dL Q dq 4 dq = F F (K, L)dK + (K, L)dL K L 1.6 1 F F 1.7. 2 F (x, y) 0 xy = 0 F (x, y) = x + y xy 0 2 h (x, y) (0, 0) (h, h) F F (x, y) (0, 0) F F (0 + h, 0) F (0, 0) (0, 0) = lim = 0 x h 0 h F F (0, 0 + h) F (0, 0) (0, 0) = lim = 0 y h 0 h F df df = F F (0, 0)h + (0, 0)h = 0 (1.11) x y F (1.11) df = F (0 + h, 0 + h) F (0, 0) = h 0 = h F (0 + h, 0 + h) F (0, 0) lim = 1 h 0 h F 4 Q dq = F (K + dk, L + dl) F (K, L)

11 1.3.2 1.7 1 F 1.8 ( ). F : R 2 R (x, y) R 2 (u, v) R 2 g : R R g(ɛ) = F (x + ɛu, y + ɛv) g ɛ = 0 g(ɛ) g(0) F (x + ɛu, y + ɛv) F (x, y) lim = lim ɛ 0 ɛ ɛ 0 ɛ (1.12) F (x, y) (u, v) (1.12) F (x, y) 2 (u, v) F 1.8 (u, v) R 2 (u, v) = (1, 0) (u, v) = (0, 1) x y 1.7 F 2 (0, 0) (ɛ, ɛ) F (u, v) = (1, 1) F (0 + ɛ, 0 + ɛ) F (0, 0) ɛ = ɛ 0 ɛ (0, 0) (u, v) = (1, 1) F 1 = 1 1.3.3 (1.10) (u, v) 1 1 1 f : R R x R f(x + h) f(x) lim h 0 h f x f(x + h) f(x) lim = df h 0 h dx (x, y) = f (x) f x R f

12 1 1.6 1. f x 2. c R f(x + h) ( f(x) + ch ) lim = 0 (1.13) h 0 h (1.13) c c = f (x) f (1.13) 1 (1.13) 1.9 ( ). F : R 2 R (x, y) R 2 (c, d) R 2 F (x + h, y + k) ( F (x, y) + ch + dk ) lim = 0 (1.14) (h,k) (0,0) (h, k) F (x, y) F (x, y) (c, d) ( ) F F (c, d) = (x, y), (x, y) x y F (x, y) F (x, y) F F (1.14) lim (h n, k n ) = (0, 0) n {(h n, k n )} 1.5 (1.7) (1.8) (1.9) (1.10) (1.14) 1.6. 2 F : R 2 R F (x, y) = x y. 1. (x, y) = (0, 0) F 2. (x, y) = (0, 0) F 3. (x, y) = (0, 0) F

13 1.3.4 1.9 (c, d) (x, y) (grdient vector) ( ) F F F (x, y) = (x, y), (x, y) x y mx F (x + h, y + k) F (x, y) (h,k) subject to (h, k) ɛ (x, y) (x + h, y + k) F (x, y) (x, y) ɛ F F 0 ɛ (h ɛ, k ɛ ) = F (x + h ɛ, y + k ɛ ) F (x + h ɛ, y + k ɛ ) (1.15) lim h ɛ = 0, lim k ɛ = 0 ɛ 0 ɛ 0 (1.15) ɛ ɛ 0 F 1 lim ɛ 0 ɛ (h 1 ɛ, k ɛ ) = F (x, y) (1.16) F (x, y) (1.16) (x, y) F F (x, y) (1.16) F F (x, y) F (x, y) lim (ɛh,ɛk) (0,0) F (x + ɛh, y + ɛk) ( F (x, y) + F x (ɛh, ɛk) F (x + ɛh, y + ɛk) F (x, y) lim = ɛ 0 ɛ (h, k) F x F (x, y)ɛh + y (x, y)ɛk) F (x, y)h + y (x, y)k (h, k) (h, k) ( ) 1 (h, k) = F (x, y) F (x, y) = 1 F F (x, y), (x, y) F (x, y) x y (h, k) = 1 = 0 (1.17) F (x + ɛh, y + ɛk) F (x, y) lim = F (x, y) (1.18) ɛ 0 ɛ (1.18) (h, k) (1.17) (h, k) 1 (1.18) F

14 1.4 1.4.1 1.10. F : R L R U R L k F U k C k 1.1. F : R 2 R R 2 F (x, y) 1 F (x, y) 1.4.2 F : R 2 R (x, y) x (x, y) (x, y) F (x, t) x F x : R2 R (1.19) (x, y) F 2 y 2 F 2 (x, y) x 2 F (x, y) y x F y : R2 R (1.20) (x, y) (1.20) 2 F (x, y) x y 2 F 2 (x, y) y 2 F/ y x 2 F/ x y x y y x 1.2 ( ). F : R 2 R (x, y) x y F x : R2 R F y : R2 R

15, (, b) 1.2. h, k R {0} 2 F x y (, b) = 2 F (, b) y x = F ( + h, b + k) F ( + h, b) F (, b + k) + F (, b) φ (x) = F (x, b + k) F (x, b) = φ ( + h) φ () F (, b) x φ (x) = F F (x, b + k) (x, b) x x (, + h) φ (x) θ (0, 1) φ ( + h) φ () h = φ ( + θh) h = F F ( + θh, b + k) ( + θh, b) x x h = k F/ x F F ( + θh, b + k) = x x (, b) + F θh 2 x 2 (, b) + h 2 F (, b) + o (h) y x F F ( + θh, b) = x x (, b) + F θh 2 (, b) + o (h) x2 o ( ) lim h 0 o (h) /h = 0 h 2 = 2 F o (h) (, b) + y x h lim h 0 h 2 = 2 F (, b) y x x y lim h 0 h 2 = 2 F (, b) x y 1.3 ( ). F : R 2 R (x, y) x y 2 2 F x y : R2 R 2 F y x : R2 R

16 (x, y) (, b) 2 F x y (, b) = 2 F (, b) y x 1.3. h = F F ( + θh, b + k) ( + θh, b) x x 2 F/ x y y 1 θ (0, 1) 2 F/ x y (, b) hk = 2 F y x ( + θh, b + θ k) lim (h,k) (0,0) hk = 2 F (, b) y x x y lim (h,k) (0,0) hk = 2 F (, b) x y 2 F x y (, b) = 2 F (, b) y x (, b) 2 (x, y) (, b) 1.4.3 1 1.4 ( 1). f : R R g : R R f g : R R F 1 2 F/ t 1, F/ t 2 f g f g df ( g(x) ) dx = df(y) dg(x) dy dx

17 1.5 ( 2). F : R 2 R G : R R 2 F G : R R F G F G df ( G 1 (x), G 2 (x) ) = F (G 1 (x), G 2 (x)) dg 1 (x) + F (G 1 (x), G 2 (x)) dg 2 (x) dx t 1 dx t 2 dx 1.6 ( 3). F : R 2 R G : R 2 R 2 F G : R 2 R F G F G F ( G 1 (x 1, x 2 ), G 2 (x 1, x 2 ) ) = F (G 1(x 1, x 2 )) G 1 (x 1, x 2 ), + F (G 1(x 1, x 2 )) G 2 (x 1, x 2 ) x 1 t 1 x 1 t 2 x 1 F ( G 1 (x 1, x 2 ), G 2 (x 1, x 2 ) ) = F (G 1(x 1, x 2 )) G 1 (x 1, x 2 ), + F (G 1(x 1, x 2 )) G 2 (x 1, x 2 ) x 2 t 1 x 2 t 2 x 2 1.4.4 1.7 ( ). F : R 2 R c R (, b) R 2 F (, b) = c F (, b) 0 y I R b J R x I y J : F (x, y) = c y = g(x) g : I J x I y J : ( ) F dg dx (x) = x x, g(x) ( ) x, g(x) F y F (x, y) = c (x, y) (, b) x 1.8. F (K, L) c K L F (K, L) = Q (1.21) (K, L) Q (K, L) (1.21) F (K, L) = Q L = g(k) (1.22) L g (1.22)

18 g g ( ) F dg dk (K) = K K, g(k) ( ) K, g(k) F L (K, L) 1.7. 1 F Q = F (K, L) = K 2 + K 5 + L 7 + L 9 K L Q 1. F (1, 1) 2. 1 g(k) F (K, g(k)) = F (1, 1) g (1) 3. 1 h(l) F (h(l), L) = 38 h (1) 1.4.5 1.11 ( ). F : R L ++ R x R L ++, t > 0 : F (tx) = t m F (x) F m 5 1.9. U(x, y) = x 1/2 y 1/2 t > 0 U(tx, ty) = t 1/2 x 1/2 t 1/2 y 1/2 = tx 1/2 y 1/2 = tu(x, y) 1 1 1.10. 1.9 U V V (x, y) = log U(x, y) = 1 2 log x + 1 2 log y V U V (tx, ty) = 1 2 log tx + 1 log ty 2 = 1 2 log t + 1 2 log x + 1 2 log t + 1 2 log y = log t + V (x, y) m t m V (x, y) x = y = 1 V 1 5 x t x R++ n t > 0 F (tx) = tm F (x) F m

19 1.8. α β x y 2 F 1. F (x, y) = x α y β. 2. F (x, y) = x α + y β. 3. G(x, y) F (x, y) = (G(x, y)) α. 1.8 ( ). F : R++ n R m x R n ++ : x F (x) = mf (x) x = x 1 x 2. Rn ++ : x 1 F x 1 (x) + x 2 F x 2 (x) + + x n F x n (x) = mf (x 1, x 2,..., x n ) x n 6 1.8. F m t > 0 t t = 1 F (tx) = t m F (x) x 1 F x 1 (tx) + x 2 F x 2 (tx) + + x n F x n (tx) = mt m 1 F (x) x 1 F x 1 (x) + x 2 F x 2 (x) + + x n F x n (x) = mf (x 1, x 2,..., x n ) 1.11. Q = F (K, L) F 1 K F (K, L) + L F (K, L) = Q (1.23) K L (K, L) (K, L ) (Q ) (1.23) p 6 K F K (K, L ) + L F L (K, L ) = Q K p F K (K, L ) + L p F L (K, L ) = pq (1.24) F m x R n ++ : x F (x) = mf (x)

20 1 p F K (K, L ) p F L (K, L ) (1.24) 1.5 1.5.1 1 1 f : R R f(x) x x + h f f(x + h) 1.9 (( )). f : R R x f k + 1 θ [0, 1] f(x + h) = f(x) + f (x)h + 1 2! f 2 + + 1 k! f (k) (x)h k + R k+1 (h; x) (1.25) R k (h; x) = 1 (k + 1)! f (k+1) (x + θh)h k+1 f x k f (k) (x) f k x R k+1 (h; x) lim h 0 h k = 0 h x + h x f(x + h) f(x) + f (x)h + f 2 + + f (k) (x)h k f(x + h) x f h f(x + h) 1 k = 1 f(x + h) f(x) + f (x)h 2 k = 2 f(x + h) f(x) + f (x)h + 1 2 f 2 k = 0 f (x + h) = f (x) + f (x + θh) h f (x + h) f (x) h = f (x + θh)

21 1.5.2 n F : R n R F (x) x x + h F F (x + h) x F k + 1 F (x + h) = F (x) + n i=1 F (x) x i h i + 1 2! + 1 k! n n n i=1 j=1 n i 1 =1 i 2 =1 2 F (x) x i x j h i h j +... n i k =1 k F (x) x i1 x i2... x ik h i1 h i2... h ik + R k+1 (h; x) 1 θ [0, 1] R k (h; x) = 1 (k + 1)! n n i 1 =1 i 2 =1 n i k+1 =1 R k+1 (h; x) lim h 0 h k = 0 k+1 F (x + θh) x i1 x i2... x ik+1 h i1 h i2... h ik+1 F (x + h) 1 F (x + h) F (x) + F (x) h 2 F (x + h) F (x) + F (x) h + 1 2 h 2 F (x)h n n 3 1.9. 2 F 2 2 F (x, y) x 2, 2 F (x, y) x y, 2 F (x, y) y x, 2 F (x, y) y 2 (x, y) R 2 c. h 1/3 k 1/3. F (x, y) 1 F (x+h, y +k) c

22 2 2.1 2.1.1 m n A R m n 11 12... 1n.... 21 A =..... m1........ mn 1j 2j A = ( 1 2... n ), j =. Rm, j = 1, 2, 3,..., n m1 A n x = x 1 x 2. Rn, x n 11 x 1 + 12 x 2 + 13 x 3 + + 1n x n 21 x 1 + 22 x 2 + 23 x 3 + + 2n x n Ax =. = x 1 1 + x 2 2 + + x n n R m m1 x 1 + m2 x 2 + m3 x 3 + + mn x n R n x R m Ax x Ax 2.1 ( ). F : R n R m x R n, y R n : F (x + y) = F (x) + F (y) x R n, t R : F (tx) = tf (x) F 2.1. A R m n F : R n R m F (x) = Ax F

23 F : R n R m x R n, F (x) = Ax A R m n 2.1. j 1 0 R n e j (j = 1, 2, 3,..., n) F x R n x = x 1 x 2. x n 1 0 0 = x 0 1. + x 1 2. + + x 0 n. = x 1 e 1 + x 2 e 2 + + x n e n 0 e 1, e 2, e 3,..., e n e 1, e 2, e 3,..., e n F F F (e j ) = F (x) = F (x 1 e 1 + x 2 e 2 + + x n e n ) 1j 2j. m1 = x 1 F (e 1 ) + x 2 F (e 2 ) + + x n F (e n ) 0 = j j = 1, 2, 3,..., n, A = ( 1 2... n ) 1 F (x) = x 1 1 + x 2 2 + + x n n = ( 1 2... n ) x 1 x 2. = Ax x n F A R m n F (x) = Ax 2.1. A F F (x) = Bx B R m n A A = B 2.1 2.1

24 2.1.2 1 1 11 x 1 + 12 x 2 + 13 x 3 + + 1n x n = b 1 21 x 1 + 22 x 2 + 23 x 3 + + 2n x n = b 2. m1 x 1 + m2 x 2 + m3 x 3 + + mn x n = b m 11 12... 1n.... 21 A =..... m1........ mn Ax = b x = x 1 x 2. b 1 b 2 Rn, b =. Rm x n b m, x 1 1 + x 2 2 + + x n n = b. b A 1 1 1. 1 Ax = b x 2. b A 1 2.1.3 1 2.2 (1 ). n m 1, 2,..., n x = x 1 x 2. x n Rn : x 1 1 + x 2 2 + + x n n = 0 x = 0 1, 2,..., n 1, 1,, x = x 1 x 2. x n Rn \ {0} : x 1 1 + x 2 2 + + x n n = 0 1, 2,..., n 1

25 1, 2,..., n 1 1 2.2 x R n, k, x k 0 : x 1 1 + x 2 2 + + x k k +... x n n = 0 k = x 1 x k 1 x 2 x k 2 x k 1 x k k 1 x k+1 x k k+1 x n x k n 2.1. 1 1, 2,..., n 1 j : j 0 j, k : j k j k 1 j : j = 0 j, k, j k : j = k 1, 2,..., n 1 2.2. 1 1 1, 2,..., n 1 i1, i2,..., ik (k n) 1 1 1 1, 2,..., n 1 k 1, 2,..., n, n+1,..., n+k 1 2.2. n α F : R n ++ R F (x) = (min{x 1, x 2,..., x n }) α 1. F 2. x R++ n x x 3. x R n ++ x 1 = x 2 = = x n 2.3. n m n > m n x 1,..., x n m 1 11 x 1 + + 1n x n = 0,. m1 x 1 + + mn x n = 0 x j 0 j (x 1,..., x n ) m

26 2.3. 1, 2,..., n R m 1 n m 2.3. n > m. 1,..., n 1,, α 1 = = α n = 0., α 1 1 + + α n n = 0 11 α 1 + + 1n α n = 0,. m1 α 1 + + mn α n = 0., 1,..., n n., 1,..., n 1., n m. 2.1.4 2.3 ( ). R m V v V, w V : v V, t R : v + w V tv V V R m : R 2 R 3 2.4. n m 1, 2,..., n 1 1, 2,..., n R m 1, 2,..., n R m 2.4 ( ). R m V, V n 1, 2,..., n 1. 1, 2,..., n V 1, 2,..., n = V 2. 1, 2,..., n 1 1, 2,..., n V 1 2.1. V = R 2 V ( ) ( ) 1 0 1 =, 2 = 0 1

27 1, 2 1 ( ) x = x 1 x 2 V : x = x 1 1 + x 2 2 1, 2 = V 1, 2 V ( ) ( ) 1 1 1 =, 2 = 0 1 1, 2 V 1, 2 1 ( ) x = x 1 x 2 1, 2 = V V : x = (x 1 x 2 ) 1 + x 2 2 2.5. 1,..., k b 1,..., b l V k = l 2.5. k > l. 1,..., k b 1,..., b l V, 1,..., k b 1,..., b l., 1 = β1b 1 1 + + βl 1b l,. k = β1 k b 1 + + βl kb l. 1,..., k 1, ( ) α 1 1 + + α k k = 0 ( ), α 1 = = α k = 0. ( ) ( ), b 1,..., b l 1, β 1α 1 1 + + β1 k α k = 0,. βl 1α 1 + + βl kα k = 0. ( ) k > l, ( )., 1,..., k 1., k l., k l., k = l. 2.5 ( ). V V dimv 2.6. F : R n R m A = ( 1 2... n ) R m n F (x) = Ax 1. F 1, 2,..., n = R n 2. F 1, 2,..., n 1 3. F 1, 2,..., n R n

28 2.6 ( ). m n A = ( 1 2... n ), j = 1j 2j. Rm, j = 1, 2, 3,..., n A R m A ColA 2.7 ( ). m n 1 2 A =. m m1 ColA = 1, 2,..., n, i = ( i1 i2... in ) R n, i = 1, 2, 3,..., m A A RowA RowA = 1, 2,..., m 2.8 ( ). m n 1 Ax = 0 x R n A KerA KerA R n. KerA = { x R n Ax = 0 } 2.7. m n A R m n dim (ColA) + dim (KerA) = n 2.7. v 1, v 2,..., v l KerA dim (KerA) = l, KerA R n v 1, v 2,..., v l = R n l = n x R n Ax = 0 A = 0 dim (ColA) = 0 dim (ColA) + dim (KerA) = 0 + n = n v 1, v 2,..., v l R n v 1, v 2,..., v l. v l+1, v 1,..., v l, v l+1 1

29. n l, (n l) v l+1,..., v n R n v1, v 2,..., v l, v l+1, v l+2,..., v n = R n Av l+1,..., Av n ColA. y ColA y A 1, 2,..., n x R n : y = Ax v 1, v 2,..., v l, v l+1, v l+2,..., v n R n x z 1 z n z 2 z =. Rn : x = z 1 v 1 + z 2 v 2 + + z l v l + z l+1 v l+1 + + z n v n y = A (z 1 v 1 + z 2 v 2 + + z l v l + z l+1 v l+1 + + z n v n ) = z 1 Av 1 + z 2 Av 2 + + z l Av l + z l+1 Av l+1 + + z n Av n v 1, v 2,..., v l KerA Av 1 = Av 2 = = Av l = 0 y = z l+1 Av l+1 + + z n Av n y (n l) Av l+1, Av l+2,..., Av n y ColA y Av l+1, Av l+2,..., Av n ColA Av l+1, Av l+2,, Av n y Av l+1, Av l+2,..., Av n z = c l+1 c l+2. c n Rn l : y = c l+1 Av l+1 + c l+2 Av l+2 + + c n Av n y = A(c l+1 v l+1 + c l+2 v l+2 + + c n v n ). y Av l+1, Av l+2,..., Av n y ColA, Avl+1, Av l+2,, Av n ColA.. Avl+1, Av l+2,..., Av n = ColA

30 z l+1 Av l+1 + z l+2 Av l+2 + + z n Av n = 0 A (z l+1 v l+1 + z l+2 v l+2 + + z n v n ) = 0 z l+1 v l+1 + z l+2 v l+2 + + z n v n KerA z 1 z l z 2 z =. Rl : z l+1 v l+1 + z l+2 v l+2 + + z n v n = z 1 v 1 + z 2 v 2 + + z l v l z 1 v 1 + z 2 v 2 + + z l v l z l+1 v l+1 z l+2 v l+2 z n v n = 0 v 1, v 2,..., v l, v l+1, v l+2,..., v n R n 1 z 1 = z 2 = = z l = z l+1 = z l+2 = = z n = 0 z l+1 Av l+1 + z l+2 Av l+2 + + z n Av n = 0 z l+1 = z l+2 = = z n = 0 Av l+1, Av l+2,..., Av n 1 (n l) Av l+1, Av l+2,..., Av n ColA dim (ColA) = n l dim (KerA) = l dim (ColA) + dim (KerA) = n l + l = n 2.9 ( ). R n V V n V = { x R n x z = 0, z V } V R n. 2.8. R n V dimv + dimv = n

31 2.8., V v 1,..., v r, i j, e i e i = 1, e i e j = 0 (i j) e 1,..., e r. (. 7 ) R n x, r x 1 = (x e i )e i i=1, x 1 V. x 2 = x x 1, x 2 e 1,..., e r, x 2 V., R n V V., V V = {0}, R n V V., dimr n = dimv + dimv. 2.9. m n A R m n dim (ColA) = dim (RowA) dim (ColA) = dim (ColA ), dim (RowA) = dim (RowA ) 2.9. x Ax = 0 x KerA 1 x = 2 x = = m x = 0 z RowA : z x = 0 x (RowA) KerA = (RowA) dim (KerA) = dim ( (RowA) ) 2.7 2.8 dim (KerA) = n dim (ColA) dim ( (RowA) ) = n dim (RowA) dim (ColA) = dim (RowA) 2.7, 2.8 2.9. 2.8 2.9 m n A R m n dim (KerA) = n dim (RowA) (2.1) 1 Ax = 0 dim (KerA) n dim (RowA) 8 (2.1) ( ) = ( ) ( ) 2.10 (rnk A). A dim (ColA) = dim (RowA) = rnka 7,,,,, p.121. 8, m = c 1 1 + + c m 1 m 1 1 x = = m 1 x = 0, m x = 0.

32 2.1.5,. 2.11 ( ). A A 2.12 ( ). A R n n B R n n AB = I BA = I B A 2.10. A R n n 1. A, A 1. 2.11. A B n AB = I BA = I, B A, AB = I BA = I. 2.4. 3 2 A 1 2 0 1 0 1 1. BA = [ 1 0 0 1 ] 2 3 B B 2. AB = 1 0 0 0 1 0 0 0 1 2 3 B B 2.5. 2 3 A [ 1 2 3 4 8 ] 1. BA = 1 0 0 0 1 0 0 0 1 3 2 B B 2. AB = [ 1 0 0 1 ] 3 2 B B

33 2.12. A R n n F : R n R n 3 1. F 2. F 3. F 2.12. 1 2: F, 1,..., n = R n., dimr n = dim(cola) = n, dim(kera) = 0., 0 = α 1 1 + + α n n, α 1 = = α n = 0., 1,..., n 1., F. 2 1: dimr n = n., F, 1,..., n n 1., R n x 1,..., n., 1,..., n = R n., F., 1,..., n R n, 3. 2.13. F : R n R n F 1 : R n R n F F 1 2.13 ( ). A R n n A, F : R n R n F (x) = Ax, F F 1. 2.14. A R n n F : R n R n A 2.14. A R n n F : R n R n F (x) = Ax F, F 1 : R n R n, 2.13, F 1 F 1 n B F 1 (x) = Bx F F 1 F F 1 : R n R n (F F 1 )(x) = F (Bx) = ABx (2.2) F F 1 F F 1 : R n R n (F F 1 )(x) = x (2.3) (2.2) (2.3) AB I A B n 2.11 BA I B A B = A 1

34 2.2 m n m n A R n n det : R n n R deta deta A 2.2.1 2 2.14 (2 ). 2 ( ) 11 12 A = R 2 2 21 22 det : R 2 2 R deta = 11 22 12 21 A 2.14 deta A A 2 2 2 ( ) 2 ( ) ( ) 1 0 e 1 =, e 2 = 0 1 A A e 1 Ae 1, e 2 Ae 2 Ae 1 Ae 2 e 1, e 2 A Ae 1, Ae 2 Ae 1 Ae 2 ( Ae 1 = 11 21 ), Ae 2 = A 1 2 1 2 e 1 e 2 ( 21 22 )

35 e 1 e 2 e 1 e 2 180 1 2 1 2 e 1 e 2 1 2 1 2.2.2 2.15 ( ). 1. ( ) ( ) ( ) 11 12 + 12 11 12 11 12 det = det + det 21 22 + 22 21 22 21 22 2. t R ( ) ( ) 11 t 12 11 12 det = tdet 21 t 22 21 22 ( ) ( ) t 11 12 11 12 det = tdet t 21 22 21 22 2.16 ( ). ( ) ( ) 11 12 12 11 det = det 21 22 22 21 2.15. det : R 2 2 R 1. 2. 3. 1 2.14 2.16. 1. 2.

36 3. 1 3 F : R 2 2 R. 2.15 2.16 1 2.2.3 3 3 2.16 n 2.17. 1. 2. 3. 1 3 F : R n n R,. 2.17 n 2 1 2.2.4,,. 2.17 ( ). n 1, 2,..., n M = { 1, 2, 3,..., n } π : M M π M 2.18 ( ). 2 k = 1, 2, 3,..., n i k = j σ(k) = j k = i k k i, j σ : M M 2.18. 2.18

37 2.19. π π = τ 1 τ 2 τ r π = π 1 π 2 π s 2 r s r s 2.19 2.19 ( ). sgn(π) 1 π sgn(π) = 1 π 2.20. M = { 1, 2, 3,..., n } Π n 11 12... 1n.... 21 A = R. n n.... n1........ nn (sgnπ) π(1)1 π(2)2... π(n)n = deta π Π (,, deti=1, ) 2.21. n A A deta = deta 2.21. deta = π Π(sgnπ) π(1)1 π(2)2... π(n)n. π Π, π π 1 Π, deta = π Π(sgnπ 1 ) π 1 (1)1 π 1 (2)2... π 1 (n)n. π 1 (1), π 1 (2),..., π 1 (n), π 1 (i) = k i = π(k), deta = π Π(sgnπ) 1π(1) 2π(2)... nπ(n)., deta.

38 2.20 ( ). n 11 12... 1n.... 21 A =..... R n n n1........ nn i j n 1 11... 1 j 1 1 j+1... 1n.......... A ij = i 1 1... i 1 j 1 i 1 j+1... i 1 n i+1 1... i+1 j 1 i+1 j+1... R (n 1) (n 1) i+1 n.......... n1... n j 1 n j+1... nn A ij det (A ij ) A n 1 2.22. n A n ij ( 1) i+j deta ij = deta j=1 n ij ( 1) i+j deta ij = deta i=1 i j 2.23 ( ). A n, b R n, Ax = b x. x j = det( 1,..., j 1, b, j+1..., n ) deta 2.3 2.3.1 2.2. 4 2 ( ) ( ) ( ) ( ) 4 0 4 0 8 0 5 3 A =, B =, C =, D = 0 4 0 4 0 2 3 5 deta = detb = detc = detd = 16

39 2 2 2 ( ) ( ) 1 0 e 1 =, e 2 = 0 1 A e 1 Ae 1, e 2 Ae 2 Ae 1 Ae 2 A e 1, e 2 A Ae 1, Ae 2 A B C A B C e 1 e 2 A 4 4 B 4 4 C 8 2 ( ) ( ) 4 0 Ae 1 = = 4e 1, Ae 2 = = 4e 2 0 4 ( ) ( ) 4 0 Be 1 = = 4e 1, Be 2 = = 4e 2 0 4 ( ) ( ) 8 0 Ce 1 = = 8e 1, Ce 2 = = 2e 2 0 2 D D e 1 e 2 ( ) ( ) 1 1 b 1 =, b 2 = 1 1 D Db 1 Db 2 ( ) ( ) 8 2 Db 1 = = 8b 1, Db 2 = = 2b 2 8 2 e 1 e 2 C 8 2 D b 1 b 2 8 2 D

40 2.21 ( ). n A Ax = λx, x R n \ {0}, λ R λ A x λ 2.2 A 4 e 1 e 2 B 4 e 1 e 2 C 8 2 e 1 e 2 D 8 2 b 1 b 2 2.3. 2 ( ) 0 1 A = 1 0 Ax = λx, x R 2 \ {0}, λ R (2.4) λ x (2.4) λ x (A λi) x = 0 (2.5) x x = 0 (A λ) (A λ) 1 (2.5) (A λi) 1 (A λi) x = 0 x = 0 (2.5) (2.5) (A λ) (A λ) det (A λi) = 0 A λ 1 det (A λi) = 1 λ = λ2 + 1 = 0 (2.6) (2.6) λ A 2.4. 2 2 ( ) 1 1 A = 0 1 Ax = λx, x R 2 \ {0}, λ R (2.7)

41 x 2.3 λ det (A λi) = 0 1 λ 1 0 1 λ = (λ 1)2 = 0 λ = 1 (2.7) ( ) ( ) 0 1 x 1 Ax = x (A I) x = = 0 0 0 x 2 ( ) x = t 1 0, t R \ {0} (2.8) A 1 2.3.2 n A Ax = λx (A λi) det (A λi) = 0 (2.9) λ (2.9) (2.9) λ det (A λi) = 0 λ det (A λi) = 0 λ n A n n n. λ Ax = λx Ax = λx (A λi) x = 0 x λ 2.4 2.4.1

42 2.22 ( ). n A n P Λ A = P ΛP 1 A, P, P 1 AP., AP = P Λ. λ 1 0 P = (v 1 v n ) Λ =... 0 λ n Av i = λ i v i i = 1,... n. 2.24 ( ). n A R n,, A. 2.24. i = 1, 2,..., n : Av i = λ i v i (2.10) v 1, v 2,..., v n R n c 1 c 2 x R n, c =. Rn : x = c 1 v 1 + c 2 v 2 + + c n v n (2.11) c n P P = (v 1 v 2... v n ) P n (2.11) x = P c c = P 1 x (2.12) A (2.11) (2.10) Ax = c 1 Av 1 + c 2 Av 2 + + c n Av n = c 1 λ 1 v 1 + c 2 λ 2 v 2 + + c n λ n v n λ 1 0... 0. 0 λ 2 Λ =.... 0 0... 0 λ n

43 c 1 λ 1 λ 1 0... 0 c 2 λ 2. Ax = P. = P 0 λ 2 c Ax = P Λc (2.13).... 0 c n λ n 0... 0 λ n (2.12) (2.13) c Ax = P ΛP 1 x A = P ΛP 1 2.24 2.6. 2.24 2.25. P 1 AP,, P R n. 2.25. : A n v 1,..., v n. P P = (v 1,..., v n ), P R n e 1,..., e n v 1,..., v n., P R n. : P i A λ i v i. (i = 1,..., n). P 1 AP i b i,., P 1 AP. b i = P 1 Av i = P 1 λ i v i = λ i e i 2.5. ( ) 5 3 D = P = 3 5 ( 1 1 1 1 ) 2.24 A R n A 2.26. n A l (l n) l v 1, v 2,..., v l 1, n. 2.26. λ 1,..., λ l A, v 1,..., v l. v 1,..., v l 1, v 1,..., v k 1 1, v 1,..., v k 1, v k 1 k., v k = α 1 v 1 + + α k 1 v k 1 ( )

44. A ( ),, ( ) λ k, Av k = α 1 Av 1 + + α k 1 Av k 1 λ k v k = α 1 λ 1 v 1 + + α k 1 λ k 1 v k 1. λ k v k = α 1 λ k v 1 + + α k 1 λ k v k 1. ( ) ( ) ( ), α 1 (λ 1 λ k )v 1 + + α k 1 (λ k 1 λ k )v k 1 = 0. v 1,..., v k 1 1, α 1 (λ 1 λ k ) = = α k 1 (λ k 1 λ k ) = 0. λ 1 λ k = = λ k 1 λ k 0, v k = 0., v k A., A l 1. 1 n R n 2.26 n A n R n A n 1 n n 1, I. 2.4.2 n = 2 2.6 ( 1 ). 2 A λ R A λ A λi = 0 ( ) x = x 1 x 2 : Ax = λx (2.14) R 2 λ R 2 2.2 A A λi 0 dim {Ker (A λi)} = 1 1 R 2 2.4.

45 v ( v / Ker (A λi)) v 1 = (A λi)v (2.15) v 2 = v (2.16) v 1 v 2 2.7. (1) (A λi) 2. (2) v 1, v 2 R 2. 2.7 (A λi)v 1 = (A λi) 2 v = 0 v 1 λ Av 1 = λv 1 (2.17) (2.15) (2.16) (2.17) Av 1 = λv 1 Av 2 = v 1 + λv 2 ( ) λ 1 A(v 1 v 2 ) = (v 1 v 2 ) 0 λ P = (v 1 v 2 ) R 2 2 (2.18) P P (2.18) ( ) P 1 λ 1 AP = 0 λ 2.7 ( 0 ). 0 det (A λi) = 0 λ 1 = µ + iν (2.19) λ 2 = µ iν i (i 2 = 1), µ ν ν 0 Ax = λ 1 x x C 2 x = u + iw (2.20)

46 Ax = λ 2 x x C 2 x = u iw u w R 2 w 0 (2.19) (2.20) Ax = λ 1 x A(u + iw) = (µ + iν)(u + iw) Au + iaw = µu νw + i(µw + νu) Au = µu νw (2.21) Aw = µw + νu (2.22). λ 2 = µ iν, x = u iw Ax = λ 2 x,. P = (u w) R 2 2 P (2.21) (2.22) ( ) P 1 µ ν AP = ν µ (2.23) (2.23) ν < 0 λ 1 λ 2 ρ = µ 2 + ν 2 ( ) 2 ( ) 2 µ ν + = 1 ρ ρ θ (0, π) (2.24) (2.25) (2.23) ( ) ( ) µ ν ρ cos θ ρ sin θ = ν µ ρ sin θ ρ cos θ ( ) cos θ sin θ = ρ sin θ cos θ µ = ρ cos θ (2.24) ν = ρ sin θ (2.25) θ ρ 2.5 2 2.5.1 2 m n A F (x) = Ax F (y, x) = y Ax

47 2.27. m n A R m n F : R m R n R F (y, x) = y Ax, y R m, x R n F ( 1 ) F : R m R n R A R m n F (y, x) = y Ax, y R m, x R n m = n y = x 2 2.23 (2 ). n x R n n n Q(x) = ij x i x j (2.26) j=1 i=1 Q : R n R 2 2.23 Q : R n R 2.27 n 11 12... 1n.... 21 A =..... R n n n1........ nn (2.26) n n ij x i x j = x Ax j=1 i=1 2.24 ( ). A, A = A, A. Q A., ( ) 1 0 0 1 Q(x) = x Ax = x 2 1 + x 2 2, ( ) 1 1., Q 1 1 Q(x) = x Ax = x 2 1 + x 1 x 2 x 1 x 2 + x 2 2 = x 2 1 + x 2 2 x Ax = x ( 1 2 A + 1 2 A ) x Q(x) = x Ax A. A.

48 2.5.2 2 2.25 (n ). n A x R n \ {0} : x Ax > 0, A postive definite) x R n : x Ax 0, A (positive semidefinite) x R n \ {0} : x Ax < 0 A (negtive definite) x R n : x Ax 0 A (negtive semidefinite) 2.8. I 2.25 A = I x = x 1 x 2. x n Rn \ {0} : x Ax = x Ix = x 2 1 + x 2 2 + + x 2 n > 0 A = I R n n x = x 1 x 2. x n Rn \ {0} : x Ax = x Ix = x 2 1 x 2 2 x 2 n < 0 2.28 ( ). λ 1 0... 0 0 λ. 2 A = R. n n... 0 0... 0 λ n 1. A 2. i : λ i > 0 i : λ i 0 2., A i λ i < 0 (λ i 0).

49 2.28 2.26 ( ). R n v 1, v 2,..., v n 0 (i j ) v i v j = 1 (i = j ) (2.27) v 1, v 2,..., v n R n 2.27 ( ). v 1, v 2,..., v n. P, P = (v 1 v 2... v n ) R n n P P = P P = I P = P 1, P., P, P = P 1. 2.29 ( ). n A 1. A 2. A P 2 :,., A. 2.29. 1 2: A.,., n A P., α 1 0 P 1 AP =... 0 α n., α 1,..., α n A. n = 1,, n. α 1 A, v 1. v 1 W 1 W 1, W 1 A-. e 1, e 2,..., e n e 1 = v 1, e 2,..., e n = W1 A A ( ) A α 1 0 = 0 A 1,. {e i } {e i } P 1, P 1 A., A 1,, A 1., n 1 P P 1 A 1P.

50,, ( ) 1 0 P = P 1 0 P ( ) 1 ( ) ( ) P 1 1 0 α 1 0 1 0 AP = 0 P 0 A 1 0 P ( ) α 1 0 = 0 P 1 A 1P. 2 1: 2.8 2.8. n A, P 1 AP n P A 2.9. A n v A x R n v x = 0 v (Ax) = 0 2.30. A P Λ, Λ = P 1 AP 2.30. P. x R n : (P 1 x) Λ(P 1 x) = x Ax (P 1 x) Λ(P 1 x) = x (P 1 ) Λ(P 1 x) = x P ΛP 1 x = x Ax P (P 1 ) = P, 2.30., P = (v 1,..., v n ), z R n,,. λ 1 0 Λ =..., 0 λ n x = z 1 v 1 + + z n v n x Ax = λ 1 z 2 + + λ n z 2 2.30, ( ) ( ) 2.28

51 2.31 ( ). n A ( ) A ( )., n A ( ) A ( ). 2.28 ( ). A n i 1, i 2,..., i m 1 i 1 < i 2 < < i m n m B b 11 b 12... b 1m i1i 1 i1i 2... i1i m. b... 21.... i2i B M = = 1.......... b m1........ b mm im i 1......... im i m B M detb M A m principl minor i 1 = 1, i 2 = 2,..., i m = m A m leding principl minor m,., A n 2 n 1 n 2.9. 3 3 1 2 3 A = 2 4 5 3 5 6 A 1 2 3 ( ) ( ) ( ) 1 2 1 3 4 5 ( ) ( ) ( ) det 2 4 5, det, det, det, det 1, det 4, det 6 2 4 3 6 5 6 3 5 6 1 2 3 det 2 4 5, 3 5 6 ( ) 1 2 ( ) det, det 1 2 4 2.32. n A k detb k A k detbk 1. A 2. A m ( ) m ( )

52 3. A 4. A m ( ), m ( ) :. : 1, 2 A. : 3, 4 : A = ( 0 0 0 1, detb 1 = 0, detb 2 = 0 A. (, ( ) 1 x = 1, x Ax = 1 < 0.) : :, ) ( ) 1 1 A = 5 1 Q(x) = x Ax = x 2 1 4x 1 x 2 + x 2 2, ( ) 1 2 B = 2 1 Q(x) = x Ax = x 2 1 4x 1 x 2 + x 2 2., A det(1) > 0, deta > 0, B det(1) > 0, detb = 1 4 = 3 < 0.

53 3 3.1 3.1.1 3.1 ( ). R n C x C, ɛ > 0, y R n : x y < ɛ y C C 3.2 ( ). R n C C C c = { x R n x / C } C 3.1 ( ). C x C x (x n ) N n > N : x n C. 3.2 ( ). C x R n x (x n ) n : x n C x C. 3.1. 3.1 3.1 3.2 3.2,. 3.3 ( ). C x, y C, t [0, 1] : tx + (1 t)y C C., x, y C, [x, y] C. 3.1.2 3.4 ( ). C C F : C R x, y C, t [0, 1] : F (tx + (1 t)y) tf (x) + (1 t)f (y) F

54 3.5 ( ). C C F : C R, F, x, y C, t [0, 1] : F (tx + (1 t)y) tf (x) + (1 t)f (y) F 3.4 3.5 3.3. C C F : C R F x C, h R n : F (x + h) F (x) + F (x) h (3.1) 3.2. 3.3. ( :.) 3.3. F : R L R, F x R L y R L F (y) F (x) + F (x)(y x) 3.3 (3.1) F (x + h) { F (x) + F (x) h } 0 F (x + h) = F (x) + F (x) h + h 2 F (x)h + R 2 (h; x) lim h 0 R 2 (h; x) h 2 = 0 (3.1) h 2 F (x)h + R 2 (h; x) 0 (3.2) 2 ( ) ( ) 1 1 h h 2 F (x) h h + R 2(h; x) h 2 0., h = tv, v = 1, t 0, v 2 F (x)v 0. h 0 h 2 F (x)h R 2 (h; x) (3.2) 2 v 2 F (x)v 0. 3.4. C C 2 F : C R 1. F 2. x C F 2 F (x)

55 3.4. F : R L R, F 2 x R L 2 F (x) 3.3 3.4 3.5. C C F : C R F x C, h R n : F (x + h) F (x) + F (x) h 3.6. C C 2 F : C R 1. F 2. x C F 2 F (x) 3.1.3 3.6 ( ). C C F : C R x, y C, t [0, 1] : F ( tx + (1 t)y ) mx {F (x), F (y)} F (qusi-convex function) 3.5. F,. 3.7 ( ). C C F : C R, F, x, y C, t [0, 1] : F ( tx + (1 t)y ) min {F (x), F (y)} F (qusi-concve function) 3.7. C C F : C R 1. F 2. z R : { x C F (x) z } 3.8. C C F : C R 1. F 2. z R : { x C F (x) z }

56 3.9. C C F : C R R G : R R G F : C R 3.6. F (x 1, x 2 ) = x α 1 1 xα 2 2, α 1, α 2 > 0 α 1 + α 2 1 C. 3.10. C C F : C R F y R n : F (x) y = 0 y 2 F (x)y 0 (3.3) F (x) F (x) 3.10 (3.3) 2 F (x) F (x). 9 3.11. C C F : C R 1. F 2. y R n : F (x) y = 0 y 2 F (x)y 0, C. C 1 F : C R F x y : F (y) F (x) F (x) (y x).,. x y: (i) F (x) (y x) > 0 F (y) > F (x). (I.e., F (y) F (x) F (x) (y x) 0.) (ii) F (x) (y x) 0 F (y) F (x). (I.e., F (y) < F (x) F (x) (y x) < 0.) (i)., (ii). 3.8 ( ). C C F : C R x, y C : F (x) (y x) 0 F (y) F (x) F (pseudo-convex function) 3.7. (i). 9 F 2 F (x) R n.

57 3.1. 1. (i) (ii)., F F. 2. x F (x) 0, (i) (ii)., (i) (ii), F (x) 0. C = R, F (x) = x 3 3.1. (i) : x C, y C, F (x) (y x) > 0, ε > 0 z = (1 ε)x + εy = x + ε(y x) F (z) > F (x) (3.4) F (z) (y z) > 0. (3.5), (3.5) (ii), F (y) F (z)., (3.4) F (y) > F (x). (i). (ii) : x C, y C, F (x) (y x) 0, ε > 0 z = y + ε F (x), C z C, F (x) 0 F (x) (z x) > 0., (i) F (z) > F (x). ε 0, z y F (y) F (x)., (ii). F : C R (i) F F F (ii) x C : F (x) 0, F F. 3.2 3.2.1 3.12 ( ). J N A R J N n b R N 1. z R+ J : b = z A 2. x R N : Ax R+ J b x < 0 2

58 1 A z = b z z z 1 z 2 z =. RJ + A = 2. RJ N, 1 j = ( j1 j2... jn ), j = 1, 2,..., J z J J z A b = z A b = z A = z 1 1 + z 2 2 + + z J J (z j 0, j = 1, 2,..., J) 1 b A (cone) b 2 Ax R J + 2 Ax R J + j x 0, j = 1, 2,..., J 1 x 0, 2 x 0,..., J x 0 b x < 0 x R N A 90 b 90 x b 3.9 ( ). R N b R N b 0 x R N = { x R N x = 0 } x = v + λb v λ R v x b 3.8. x b v v = x x b b 3.12. z R J + x RN 1 z b = z A x b x = z Ax (3.6)

59 z 2 z Ax = z 1 1 x + z 2 2 x + + z J J x 0 b x < 0 (3.6) 1 2 J J = 1 J 2 J 1 1 2 A = 1 2. J 1 J R J N, A = 1 2. J 1 R(J 1) N A A A 1 A 1 z R J 1 + b = z A (3.7) A x R J 1 + b x < 0 x R N J x 0 x Ax R J + b x < 0 2 J x < 0 1, 2,..., J 1 b x J â 1, â 2,..., â J 1 b   = â 1 â 2. â J 1 R(J 1) N b  b = w  (3.8) w R+ J 1 w RJ 1 + w = w 1 w 2. w J 1 RJ 1 + b = w  = J 1 w j â j j=1

60 â j j x J 3.8 â j = j x j x J J, j = 1, 2,..., J 1 b = = = J 1 w j â j j=1 J 1 ( ) w j j x j x J J j=1 J 1 w j j j=1 ( J 1 j=1 w j x j x J ) J (3.9) b b x J (3.9) b = b x b x J J ( ) J 1 b x J 1 b x j = w j j + x w j j=1 J x J j=1 J ( ) = w A x J 1 b x j + x w j J x J (3.10) J J x < 0 w j 0, j = 1, 2,... J 1 j=1 A x R J 1 + b x < 0 (3.10) x J 1 b x j x w j J x 0 J j=1 w J = x J 1 b x j x w j J x J j=1 z = w 1 w 2. w J 1 w J z z b = w A + w J J = z A

61 A 1 b  b = w  w R+ J 1  1  2  x R+ J 1 (3.11) x R N x J x x b x < 0 (3.12) Ax R J + b x < 0 A 2 â j, j = 1, 2,... J 1 b λ R : â j = j + λ J (3.13) â j x = 0 (3.14) λ R : b = b + λ J (3.15) b x = 0 (3.16) x λ R : x = x + λx (3.17) (3.13) (3.18) J x = 0 (3.18) j x = â j x (3.19) (3.14) (3.17) (3.11) â j x = â j x 0 (3.20) (3.19) (3.20) (3.18) (3.21) j x 0 j = 1, 2,... J 1 (3.21) Ax R+ J (3.15) (3.18) b x = b x (3.22) (3.16) (3.17) (3.12) b x = b x < 0 (3.23) (3.22) (3.23) b x < 0

62 3.2.2 =, b j, j = 1, 2,..., J b 3.13 ( ). R N C R N C b R N \ C c C : x c d > x b x R N d R 3.9. 3.13 (d = 0.) 3.14. A R J N 1. z R+ J \ {0} : z A = 0 2. x R N : Ax R J ++ 2 3.10. 3.14 1 2 3.14. 3.10 1 2 R J e à 1 1 1 e =. RJ 1 à = (A, e) R J (N+1) z à = (0, 0,..., 0, 1) }{{} N z R J + 1 x R N+1 à x R J + (3.24) (0, 0,..., 0, 1) x < 0 (3.25) x = ( x x N+1 ) R n

63 x R N Ã x = (A, e) ( ) x x N+1 = Ax + x N+1 e (3.26) (3.25) x N+1 = (0, 0,..., 0, 1) x < 0 (3.26) Ã x < Ax (3.24) Ax R++ J 2 3.14 1 A z = 0 z z, A, 1 2 1 x > 0, 2 x > 0,..., J x > 0 x R N A 90 x R N,, A,,. 3.3 L M N R L X N f n : X R (n = 1, 2,..., N) M g m : X R (m = 1, 2,..., M) (X, f 1, f 2,..., f N, g 1, g 2,..., g M ), mx (f 1(x), f 2 (x),..., f N (x)) x X subject to g 1 (x) 0, g 2 (x) 0,. g M (x) 0. (3.27) x X m : g m (x) 0, n : f n (x) f n (x ), n : f n (x) > f n (x )

64 x X x X f n g m 3.3.1 3.15 ( ). x X (3.27) N + M (µ 1, µ 2,..., µ N, λ 1, λ 2,..., λ M ) R N+M + 1. µ 1,..., µ N, λ 1,..., λ M 1 ; 2. m : g m (x ) > 0 λ m = 0; 3. N M µ n f n (x ) + λ m g m (x ) = 0. n=1 m=1 3 3.15 ( ). M (g m (x ) = 0) (g m (x ) > 0) K ( M) n : f n (x ) v > 0 m K : g m (x ) v > 0 v R L f 1 (x ). f N (x ) g 1 (x ) v R++ N+K. g K (x v R L 3.14 f 1 (x ). (µ 1,..., µ N, λ 1,..., λ K ) f N (x ) g 1 (x ) = 0. N µ n f n (x ) + n=1 g K (x K λ m g m (x ) = 0 m=1 (µ 1, µ 2,..., µ N, λ 1, λ 2,..., λ K ) R N+K + \ {0}

65 M K λ m = 0, m > K N + M (µ 1,..., µ N, λ 1,... λ K, λ K+1,..., λ M ) R N+K + N + M 3.15 3.1. µ λ 0 (µ 1,..., µ N, λ 1,..., λ M ) 3.15 t > 0 (tµ 1,..., tµ N, tλ 1,..., tλ M ) (µ 1,..., µ N, λ 1,..., λ M ) 0 1 µ 1 = 1 µ 1 0 µ 1 = 1 10 3.15 x L µ N λ K 0 L + N + K 1 L N µ n f n (x ) + n=1 K λ m g m (x ) = 0 m=1 g 1 (x ) = 0, g 2 (x ) = 0,. g K (x ) = 0. K L + K N 1 3.16 ( ). (3.27) X f n g m x X (µ 1,..., µ N, λ 1,..., λ M ) R N+M + 1. m : g m (x ) 0; 2. (µ 1,..., µ N, λ 1,..., λ M ) R N ++; 3. m : g m (x ) > 0 λ m = 0; 4. N M µ n f n (x ) + λ m g m (x ) = 0. n=1 m=1 x (3.27) 3.16 ( ). M (g m (x ) = 0) (g m (x ) > 0) 10 3.3.3

66 K ( M) x n : f n (x ) v 0 n : f n (x ) v > 0 m K : g m (x ) v 0 v R L 1 2 3 N µ n f n (x ) v + n=1 ( N µ n f n (x ) + n=1 M λ m g m (x ) v > 0 m=1 ) M λ m g m (x ) v m=1 4 x 3.11. x 3.16 2 3 3.16 3.1. 2 u : R 2 + R u(x 1, x 2 ) = (x 1 + 1) (x 2 + 1) p 1, p 2 w mx x R 2 + (x 1 + 1) (x 2 + 1) subject to w p 1 x 1 p 2 x 2 0. (3.28) R 2 + X = { x R 2 x 1 > 1, x 2 > 1 } x 1 0 x 2 0 mx x X (x 1 + 1) (x 2 + 1) subject to w p 1 x 1 p 2 x 2 0, x 1 0, x 2 0. (3.29) (3.29) (3.28) 3.1

67 3.2. 3.1 2 u : R 2 + R u(x 1, x 2 ) = x 1 + x 2 p 1, p 2 w mx x R 2 + u(x 1, x 2 ) = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0. (3.30) R 2 + u (x 1, x 2 ) x 1 (x 1 0) u (x 1, x 2 ) x 2 (x 2 0) x = (x 1, x 1) x 1 > 0 x 1 > 0 x R 2 ++ R 2 ++ mx x R 2 ++ u(x 1, x 2 ) = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0. (3.31) (3.30) (3.31) 3.1 3.2 3.3. 2 1 w(> 0) u : R+ 2 R u(x 1, x 2 ) = x 1 2 1 e x 2 mx x R 2 + u(x) subject to w x 1 x 2 0. (3.32) 3.12. x R+ 2 (3.32) x 1 3.12 { x R 2 x 1 > 0, x 2 0 } u X = { x R 2 x 1 > 0 }

68 e x 2 x 2 R X g 1 : X R g 1 (x) = w x 1 x 2 mx x X u(x) subject to g 1 (x) 0. (3.33) X (3.33) (3.33) 3.13. mx x X u(x) subject to g 1 (x) 0, g 2 (x) 0. (3.34) (3.32) g 2 : X R 3.14. 3.13 (3.34) (3.32) 3.3.2 K L M X R L P R K f : X P R g m : X P R (m = 1, 2,..., M) X P 2 p P mx x X subject to g 1 (x, p) 0, f(x, p) (3.35) g 2 (x, p) 0,. g M (x, p) 0. (3.35) p P p P (3.35) 3.10 (Policy Function). p P (3.35) P : P X (p) Policy Function 3.11 (Vlue Function). Policy Function b : P R b(p) = f((p), p) b(p) Vlue Function

69 3.17. p P (1, λ 1,..., λ K ) R 1+K ++ x X (3.35) = (L+M) (L+M) M 2 xf(x, p ) + λ m 2 xg m (x, p ) x g 1 (x, p )... x g M (x, p ) m=1 x g 1 (x, p ) 0... 0...... x g M (x, p ) 0... 0 p Q P Policy Function (p) Vlue Function b(p) Q 3.17. 3.18 ( ). p P (1, λ 1,..., λ K ) R 1+K ++ x X (3.35) Policy Function (p) Vlue Function b(p) M b(p ) = p f(x, p ) + λ m p g m (x, p ) m=1 3.18. p P : g m ((p), p) = 0, m = 1, 2,..., M p p = p x g(x, p ) (p ) + p g m (x, p ) = 0, m = 1, 2,..., M (3.36) (3.36) λ m m M λ m x g(x, p ) (p ) + m=1 M λ m p g m (x, p ) = 0 (3.37) (1, λ 1,..., λ K ) R 1+K ++ x X (3.38) (3.37) x f(x, p ) + m=1 M λ m x g m (x, p ) = 0 (3.38) m=1 x f(x, p ) (p ) + Vlue Function M λ m p g m (x, p ) = 0 (3.39) m=1 p P : b(p) = f ((p), p) p p = p b(p ) = x f(x, p ) (x ) + p f(x, p ) (3.40) (3.40) (3.39)

70 3.3.3 3.3.1 mx x V f(x) (3.41) subject to g j (x) 0, j = 1, 2,..., m V R n f : V R g j : V R (j = 1, 2,... m) 3.19 ( ). x (3.41) m + 1 λ 0, λ 1,..., λ m m λ 0 f(x ) + λ j g j (x ) = 0 (3.42) j=1 λ j g j (x ) = 0, j = 1, 2,..., m (3.43) λ 0, λ 1,..., λ m 3.19 (3.42) 1 First Order Condition: FOC (3.43) Complementry Slckness λ 0 > 0 (3.42) (3.43) λ 0 λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 (3.44) j=1 g j (x ) = 0, j = 1, 2,..., m (3.45) (3.44) (3.45) 3.19 λ 0 = 0 Constrint Qulifiction 3.20 ( ). x (3.41) x m λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 j=1 g j (x ) = 0, j = 1, 2,..., m λ 1, λ 2,..., λ m 3.20 11 1. g j 2. x 0 R n j = 1, 2,..., m g j (x 0 ) > 0 11 3.19 λ 0 0 λ j > 0 g j (x ) 1.

71 3.20 2 Slter Condition 3.20 3.21 ( ). x (3.41) 1 g j 2 m λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 j=1 g j (x ) = 0, j = 1, 2,..., m λ 1, λ 2,..., λ m 3.22 ( ). x x m λ 1, λ 2,..., λ m (3.44) (3.45) g j f 12 x (3.41) 13 3.20 3.21 3.23 ( ). R n f (3.41) 3.23 3.20 3.21 (3.44) (3.45) 14 (3.41) m L(x, λ) = f(x) + λ j g j (x) L : V R m + R x V j=1 λ R m + L(x, λ ) L(x, λ ) L(x, λ) (x, λ ) V R m + L(x, λ) sddle point 3.24 ( ). (3.41) (x, λ ) V R+ m L(x, λ) 1. x i L(x, λ ) 0, i = 1, 2,..., n 12 h φ > 0 φ : R R f(x) φ`h(x) 13 4 14 6 214-223

72 2. x i 3. x i L(x, λ ) = 0, λ j L(x, λ ) 0, 4. λ j λ j L(x, λ ) = 0, 3.24 x i x i L(x, λ ) = x i L(x, λ ) = x i i = 1, 2,..., n j = 1, 2,..., m j = 1, 2,..., m m f(x ) + x i j=1 ( f(x ) + x i g j (x ) 0, i = 1, 2,..., n (3.46) x i ) m λ j g j (x ) = 0, i = 1, 2,..., n (3.47) x i λ j j=1 L(x, λ ) = g j (x ) 0, j = 1, 2,..., m (3.48) λ j λ j L(x, λ ) = λ j g j (x ) = 0, j = 1, 2,..., m (3.49) λ j (3.48) (3.49) i x i > 0 (3.46) (3.47) x i f(x ) + m j=1 λ j f(x ) + g j (x ) = 0, x i m λ j g j (x ) = 0 j=1 i = 1, 2,..., n 1 (3.41) x f(x ) + m λ j g j (x ) = 0 (3.50) j=1 g j (x ) = 0, j = 1, 2,..., m (3.51) g j (x ) 0, j = 1, 2,..., m (3.52) 15 3.25 ( ). f g j (j = 1, 2,..., m) x (3.41) λ R m + (x, λ ) V R+ m L(x, λ) 3.25 f g j (j = 1, 2,..., m) (3.41) 3.24 (3.41) 15 (3.50) (3.51) (3.52) (3.50) (3.51)

73 3.26 ( ). f g j (j = 1, 2,..., m) (x, λ ) V R+ m 3.24 (x, λ ) L(x, λ) (3.41) (3.41) 3.27 ( ). (x, λ ) V R+ m L(x, λ) x (3.41) L(x, λ) x 3.15. f : R R 2 f > 0 f > 0 f 0 min x R g(x) := 1 x f(x) x > 0 g g (x) = 0 x g (x) = 0 x

74 4 4.1 16 2 x 2 mx (c 0,c 1 ) subject to u(c 0 ) + δu(c 1 ) (4.1) c 0 + c 1 x c 0 0 c 1 0 u c 0 c 1 0 1 δ (4.1) (c 0, c 1) u (c 0) = λ δu (c 1) = λ λ u (c 0) + δu (c 1) = 0 c 0 + c 1 = x x T (4.1) mx (c 0,...,c T 1 ) subject to u(c 0 ) + δu(c 1 ) + δ 2 u(c 2 ) + + δ T 1 u(c T 1 ) (4.2) c 0 + c 1 + + c T 1 x c 0 0 c 1 0. c T 1 0 2 (4.2) (c 0,..., c T 1 ) u (c 0) = λ δu (c 1) = λ. δ T 1 u (c T 1 ) = λ 16 Stokey-Lucs, chpeter4

75 δ t 1 u (c t 1) + δ t u (c t ) = 0, t = 1, 2,..., T 1 c 0 + c 1 + + c T 1 = x T T x mx {c t :t=0,1,... } subject to δ t u(c t ) (4.3) t=0 c t x t=0 c t 0, t = 0, 1,... Dynmic Progrmming: DP 4.2 X R n + x t X δ 0 < δ < 1 f : X X R 4.2.1 DP( x 0 ) mx {x t :t=1,2,... } δ t f (x t, x t+1 ) t=0 subject to Γ (x t, x t+1 ) 0, t = 0, 1,... x 0 = x 0 x X Γ (x, y) 0 y X 4.1 ( ). x z u (z) δ {c t } (4.3) t {x t } c t = x t x t+1 x = x 0

76 (4.3) mx {x t:t=0,1,... } subject to δ t u (x t x t+1 ) t=0 x 0 = x x t x t+1 0, t = 0, 1,... f (x t, x t+1 ) := u (x t x t+1 ) Γ (x t, x t+1 ) := x t x t+1 f (x t, x t+1 ) Γ(x t, x t+1 ) mx {x t:t=0,1,... } subject to δ t f (x t, x t+1 ) t=0 x 0 = x Γ (x t, x t+1 ) 0, t = 0, 1,... DP 4.1 ( ). β (β > 0) t t + 1 x t+1 = β(x t c t ) DP 4.2 ( ). t x t c t s t i t t + 1 c t + s t = x t x t = (1 + i)s t δ t u (c t ) t=0 DP 4.1 ( ). {x t : t = 0, 1,... } DP( x 0 ) {x t : t = 0, 1,... } x 0 Γ (x t, x t+1 ) > 0 t 4.2.2 X f : X X R

77 Euler eqution f (x t 1, x t ) + δ f (x t, x t+1 ) = 0, t = 1, 2,... (4.4) x t x t DP( x 0 ) 4.1 ( ). {x t : t = 0, 1, 2,... } DP( x 0 ) (4.4) 4.1. {x t } DP( x 0 ) δ t f (x t, x t+1 ) = f (x 0, x 1 ) + δf (x 1, x 2 ) + + δ t 1 f (x t 1, x t ) + δ t f (x t, x t+1 ) +... (4.5) t=0 t x t V (z) = f (x 0, x 1 ) + δf (x 1, x 2 ) + + δ t 1 f (x t 1, z) + δ t f (z, x t+1 ) +... (4.6) V (z) mx z V (z) (4.7) subject to Γ (x t 1, z) 0 Γ (z, x t+1 ) 0 z = x t x t V (x t) > V (x t ) f (x 0, x 1 ) + + δ t 1 f (x t 1, x t) + δ t f (x t, x t+1 ) + > δ t f (x t, x t+1 ) {x 0, x 1,... x t,... } {x 0, x 1,... x t,... } DP( x 0 ) (4.8) z = x t z = x t (4.8) 1 d dz V (z) = 0 z=xt δ t 1 f (x t 1, x t ) + δ t f (x t, x t+1 ) = 0 x t x t δ t 1 f (x t 1, x t ) + δ f (x t, x t+1 ) = 0 (4.8) x t x t t (4.4) t=0

78 4.2 ( ). mx {x t :t=0,1,... } δ t u (x t x t+1 ) t=0 subject to x t x t+1 0, t = 0, 1,... x 0 = x u (x t 1 x t ) + δu (x t x t+1 ) = 0 c t u (c t 1 ) + δu (c t ) = 0 δu (c t ) u (c t 1 ) = 1 (4.9) t 1 δ t 1 u (c t 1 ) t δ t u (c t ) (4.9) t 1 t t 1 t 17 (4.9) 1 4.2.3 DP( x 0 ) 1 lim t δt f (x t, x t+1 ) x t = 0 (4.10) x t Trnsverslity Condition: TC 4.2 ( ). f (x t, x t+1 ) X X R R x t f (x t, x t+1 ) 0 X X {x t : t = 0, 1,... } 1. (4.4) 2. (4.10) {x t : t = 0, 1,... } DP( x 0 ) 4.2. {x t : t = 0, 1,... } (4.4) (4.10) {x t : t = 0, 1,... } δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 t=0 t=0 17

79 δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 t=0 t=0 f (x t, x t+1 ) ( T lim T ( T T = lim δ t f (x t, x t+1 ) δ t f ( x t, x ) ) t+1 T t=0 t=0 ( T = lim δ t( f (x t, x t+1 ) f ( x t, x t+1) )) (4.11) T f ( ) x t, x t+1 f (xt, x t+1 ) Df (x t, x t+1 ) (x t x t, x ) t+1 x t+1 = f 1 (x t, x t+1 ) (x t x t ) + f 2 (x t, x t+1 ) ( x ) t+1 x t+1 f (x t, x t+1 ) f ( x t, x t+1) f1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) t+1 t=0 δ t( f (x t, x t+1 ) f ( x t, x ) )) t+1 lim T t=0 ( T δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x t+1) )) (4.12) t=0 x 0 = x 0 T t=0 δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) ) t+1 = f 1 (x 0, x 1 ) (x 0 x 0) + = f 2 (x 0, x 1 ) (x 1 x 1) + δf 1 (x 1, x 2 ) (x 1 x 1) + δf 2 (x 1, x 2 ) (x 2 x 2) + δ 2 f 1 (x 2, x 3 ) (x 2 x 2) + δ 2 f 2 (x 2, x 3 ) (x 3 x 3) + δ 3 f 1 (x 3, x 4 ) (x 3 x 3) +. δ T 1 f 2 (x T 1, x T ) (x T x T ) + δ T f 1 (x T, x T +1 ) (x T x T ) + δ T f 2 (x T, x T +1 ) ( x T +1 x ) T +1 T 1 δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x t+1) + δ T f 2 (x T, x T +1 ) ( x T +1 x ) T +1 t=0 (4.12) lim T ( T t=0 δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) )) t+1 = lim T ( T 1 t=0 (4.4) δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x ) ) t+1 + lim T δt f 2 (x T, x T +1 ) ( x T +1 x ) T +1 (4.13) f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 ), t = 0, 1, 2,...

80 (4.13) 1 lim T ( T 1 δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x t+1) ) = 0 (4.14) t=0 2 lim T δt f 2 (x T, x T +1 ) ( x T +1 x ) T +1 = lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 (4.11) (4.12) (4.13) (4.14) (4.15) δ t f (x t, x t+1 ) δ t f ( x t, x t+1) lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 t=0 t=0 (4.16) lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 TC (4.15) (4.16) x t f (x t, x t+1 ) 0 x T +1 = lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 lim T δt f 1 (x T, x T +1 ) x T = 0 + lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 (4.17) (4.17) 0 (4.16) (4.17) δ t f (x t, x t+1 ) δ t f ( x t, x t+1) 0 (4.18) t=0 f (x t, x t+1 ) (4.18) t=0 t=0 δ t f ( x t, x t+1) < t=0 δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 {x t : t = 0, 1,... } DP( x 0 ) 4.1. 4.2 f {x t : t = 0, 1,... } δ t f ( x t, x t+1) t=0 f f t=0 f(x t, x t+1 ) = u (g(x t ) x t 1 ) u = ln(z) f R X x X R X [0, M]

81 4.3 ( ). u (z) = ln (z) mx {x t:t=0,1,... } δ t ln (x t x t+1 ) t=0 subject to x t x t+1 0, t = 0, 1,... x 0 = x c t = x t x t+1 1 1 + δ = 0, t = 0, 1,... x t x t+1 x t+1 x t+2 (4.19) t = 0, 1, 2,... 1 c t + δ 1 c t+1 = 0 c t+1 = δc t (4.19) c t = δ t c 0, t = 0, 1,... (4.20) (4.20) (4.20) c 0 c 0 x 0 (4.20) (4.20) ( c t = c 0 1 + δ + δ 2 +... ) t=0 = c 0 1 1 δ c 0 1 1 δ = x 0 x 0 c 0 = δx 0 c 0 = x 0 x 1 c 1 = x 1 x 2 x 1 = x 0 c 0 = δx 0 x 2 = x 1 c 1 = δ (x 0 c 0 ) = δ 2 x 0

82 x t = δ t x 0, t = 0, 1,... (4.21) x 0 (4.21) {x t : t = 0, 1,... } f(z) = ln(z) 4.2 f x t f (x t, x t+1 ) 0 f (x t, x t+1 ) TC TC x t f (x t, x t+1 ) = 1 x t x t+1 = 1 c t = 1 δ t c 0 lim t δt f (x t, x t+1 ) x t = lim δ t 1 x t t δ t δ t x 0 c 0 = lim δ t x 0 t c 0 = lim δ t 1 t 1 δ = 0 TC (4.21) 4.2.4 Bellmn eqution 18 4.2 ( ). v : X R { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) (4.22) v (4.22) v 4.3 (Policy Function). v(x) { } φ(x) f(x, y) + δv(y) mx y: Γ(x,y) 0 φ(x) Policy Function 18

83 4.3 ( ). DP( x 0 ) x 0 X {x t : t = 0, 1,... } v ( x 0 ) = t=0 δ t f ( x t, x ) t+1 v (4.22) (4.23) (4.23) v Vlue Function x 0 = x 0 4.3 {x t : t = 0, 1,... } DP( x 0 ) Vlue Function 4.3. x 0 X DP( x 0 ) { x 0, x 1, x 2, x 3,... } x 0 X Vlue Function v v ( x 0 ) = t=0 δ t f ( x t, x ) t+1 v ( x 0 ) = f ( x 0, x 1) + δf (x 1, x 2) + δ 2 f (x 2, x 3) +... (4.24) v ( x 0 ) < f ( x 0, ˆx 1 ) + δv (ˆx 1 ) (4.25) ˆx 1 {x X Γ( x 0, x) 0} ˆx 1 mx {x t :t=2,3,... } δ t f (x t+1, x t+2 ) t=0 subject to Γ (x t, x t+1 ) 0, t = 1, 2,... x 1 = ˆx 1 {ˆx 1, ˆx 2, ˆx 3,... } Vlue Function v (ˆx 1 ) = f (ˆx 1, ˆx 2 ) + δf (ˆx 2, ˆx 3 ) + δ 2 f (ˆx 3, ˆx 4 ) +... (4.26) (4.25) (4.24) (4.26) f ( x 0, x 1) + δf (x 1, x 2) +... < f ( x 0, ˆx 1 ) + δv (ˆx 1 ) = f ( x 0, ˆx 1 ) + δf (ˆx 1, ˆx 2 ) + δ 2 f (ˆx 2, ˆx 3 ) +... { x 0, ˆx 1, ˆx 2,... } { x 0, x 1, x 2,... } DP( x 0 ) (4.25) x 1 {Γ( x 0, x 1 ) 0}, v ( x 0 ) f ( x 0, x 1 ) + δv (x 1 ) (4.27) x 0 X { } x 0 X, v ( x 0 ) = mx x 1 : Γ( x 0,x 1 ) 0 f( x 0, x 1 ) + δv(x 1 ) v (4.22)

84 4.4 ( ). 4.3 u (z) = ln (z) Vlue Function x t = δ t x 0 c t = (1 δ)δ t x 0 v (x) = δ t ln ( (1 δ)δ t x ) t=0 { x X, v (x) = mx ln(x y) + δ y: Γ(x,y) 0 mx y ln(x y) + δ δ t ln ( (1 δ)δ t y )} (4.28) t=0 δ t ln ( (1 δ)δ t y ) t=0 subject to Γ(x, y) 0 (4.29) y 1 1 y ln ( (1 δ)δ t y ) = lny + ln(1 δ)δ t 1 x y + δ δ t 1 y = 0 t=0 y = δx (4.29) { mx ln(x y) + δ y: Γ(x,y) 0 δ t ln ( (1 δ)δ t y )} = ln(x δx) + δ δ t ln ( (1 δ)δ t δx ) t=0 = ln ( (1 δ)x ) + = t=0 δ t+1 ln ( (1 δ)δ t+1 x ) t=0 δ t ln ( (1 δ)δ t x ) t=0 = v (x) (4.28) 4.3. u (z) = ln (z) 4.1 4.2 Vlue Function

85 4.3 {x t : t = 0, 1,... } DP( x 0 ) Vlue Function v DP( x 0 ) Vlue Function 4.4 ( ). v 1. v (4.22) 2. {x t : t = 0, 1,... } lim t δ t v(x t ) = 0 19 φ { x 0, φ( x 0 ), φ 2 ( x 0 ),... } DP( x 0 ) 4.4. v { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) (4.30) mx y f(x, y) + δv(y) subject to Γ(x, y) 0 φ(x) (4.30) φ x X, v(x) = f ( x, φ(x) ) + δv ( φ(x) ) (4.31) φ(x) X (4.31) x = φ(x) x X, v ( φ(x) ) = f ( φ(x), φ φ(x) ) + δv ( φ φ(x) ) x X, v(x) = f ( x, φ(x) ) + δv ( φ(x) ) = f ( x, φ(x) ) + δf ( φ(x), φ 2 (x) ) + δ 2 v ( φ 2 (x) ). = δ t f ( φ t (x), φ t+1 (x) ) t=0 φ 0 (x) = x φ t (x) = φ φ φ(x), }{{} t 1 t x x = x 0 v( x 0 ) = δ t f ( φ t ( x 0 ), φ t+1 ( x 0 ) ) (4.32) t=0 (4.32) { x 0, φ( x 0 ), φ 2 ( x 0 ),... } 19

86 v Γ( x 0, x 1 ) 0 x 1 v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) (4.33) x 1 Γ(x 1, x 2 ) 0 x 2 (4.33) v(x 1 ) f(x 1, x 2 ) + δv(x 2 ) v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 v(x 2 ) {x t : t = 0, 1,... } v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 v(x 2 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 f(x 2, x 3 ) + δ 3 v(x 3 ).. T δ t f(x t, x t+1 ) + δ T v(x T ) t=0 δ t f(x t, x t+1 ) (4.34) t=0 ( T ) δ t f(x t, x t+1 ) + δ T v(x T ) lim T t=0 = lim = T T t=0 δ t f(x t, x t+1 ) t=0 δ t f(x t, x t+1 ) + lim T δt v(x T ) (4.32) (4.34) {x t : t = 0, 1,... } δ t f ( φ t ( x 0 ), φ t+1 ( x 0 ) ) δ t f(x t, x t+1 ) t=0 t=0 φ { x 0, φ( x 0 ), φ 2 ( x 0 ),... } DP( x 0 ) Policy Function Vlue Function v v

87 v : X R T v(x) T v(x) := mx { } f(x, y) + δv(y) y:γ(x,y) 0 mx y f(x, y) + δv(y) subject to Γ(x, y) 0 (4.35) x X, T v(x) = v(x) (4.36) x X, v(x) = mx { } f(x, y) + δv(y) y:γ(x,y) 0 v v (4.36) (4.35) T v v T 2 v(x) := mx { } f(x, y) + δt v(y) y:γ(x,y) 0 T 2 v(x) (4.37) x X, T 2 v(x) = T v(x) (4.38) x X, T v(x) = mx { } f(x, y) + δt v(y) y:γ(x,y) 0 T v (4.38) (4.37) T 2 v T n v v v v 4.5. DP( x 0 ) f X x X T n v(x) v (x) := lim n T n v(x) x X, v (x) = mx { f(x, y) + δv (y) } y:γ(x,y) 0 v v 4.5 v T n v v v v

88 4.3 v : X R v { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) v mx y f(x, y) + δv(y) subject to Γ(x, y) 0 Policy Function φ(x) 1 x X, f 2 (x, φ(x)) + δv (φ(x)) = 0 (4.39) x d dx v(x) = { } f(x, y) + δv(y) x y=φ(x) v (x) = f 1 (x, φ(x)) (4.39) x X, f 2 (x, φ(x)) + δf 1 (φ(x), φ 2 (x)) = 0 (4.40) {x 0, x 1, x 2,... } Policy Function φ(x t 1 ) = x t, φ 2 (x t 1 ) = x t+1, t = 1, 2,... (4.40) x = x t 1 X f 2 (x t 1, x t ) + δf 1 (x t, x t+1 ) = 0, t = 1, 2,... 4.4 x x 1 1 p > 0 c > 0 y + cy δ DP t x t X X X = {0, 1, 2,... } x t 1 t 1 t x t 1 t + 1 x t+1 t

89 x t+1 (x t 1) t + c(x t+1 x t + 1) f 1 π(x t ) ( + c(x t+1 x t + 1) ) x t+1 x t + 1 > 0, f(x t, x t+1 ) = π(x t ) x t+1 x t + 1 0. p x t > 0, π(x t ) = 0 x t 0. Γ(x t, x t+1 ) = x t+1 x t + 1 0 4.4. f, π, Γ DP 1 1 2 1 0 Vlue Function n n r n := p(δ + δ 2 + + δ n ) ( + cn) = p 1 δn 1 δ ( + cn) n ( r n 1 + δ n + ( δ n) ) 2 1 +... = r n 1 δ n = p 1 1 δ + cn 1 δ n (4.41) 1 pδ + c pδ > + c n = 1 (4.41) (4.41) n n 4.5. n n (4.41) pδ > + c (4.41) Π Π := p 1 1 δ + cn 1 δ n