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20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33 221 33 222 34 223 3 35 1

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3 1 11 1 111 R x 1, x 2, x 3, ( x n (n = 1, 2, 3, 11 ( R ( x n x R ϵ N N n x n x < ϵ ( x n x ϵ > 0, N N, n > N : x n x < ϵ ( x n x n xn x lim x n = x x n x (n n x ( x n 11 x n = 1/n ( x n 0 ϵ N 1/ϵ N N n > N 0 < x n = 1 n < 1 N ϵ n > N : x n 0 < ϵ ( x n 0 12 ( x n 1 x n = n n 1 n R ( x n 0 11 11 ϵ > 0, N N, n > N : x n x < ϵ ϵ > 0, N N, n > N : x n x ϵ (11 ϵ N N n x n x ϵ ϵ = 1/2 n x n 0 = 1 0 = 1 > ϵ (12

4 ϵ 1/2 N n (11 ( x n 0 ( x n 1 n 11 12 ( x n R 12 112 1 f : R R ( x n f(x1, f(x 2, f(x 3, 13 f : R R f(x = x 3 + 1 (13 ( x n 11 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 = ( 1n n 3 + 1 f(x n 1 (n 1 13 ( x n lim n x n = 0 0 (13 f 0 ( x n f(x n 1 (n 12 ( 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

5, (x n x ( (f(x (f(x n lim f(x n n lim n f(x n = f (lim n x n 13 f 0 12 ϵ > 0, δ > 0, y R : x y < δ f(x f(y < ϵ 13 12 14 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

6 12 121 L R L x 1, x 2, x 3, ( x n (n = 1, 2, 3, Euclidean Norm 13 ( 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 (14 L = 2, 3 14 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 (15 n ( x n x ( xn x x n = ( x 1 n, x 2 n, x 3 n,, x L n 14 l = 1, 2, 3,, L : x l n x l (n (16 14 14 (16 15 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 (17 (18 (19 (110 xn 0 (n 0 R 2 0

7 122 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 15 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 ϵ - δ 15 ϵ > 0, δ > 0, x R L : x u < δ f(x f(u < ϵ 1 1 1 13 (x 1, x 2,, x L F

8 131 L = 2 M = 1 F (x 1, x 2 = F (x, y 16 (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 15 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 17 ( 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

9 17 1 F 16 (K, L (K +dk, L+dL Q = F (K, L F (K, L K K + dk Q F K (K, LdK L L + dl Q F L (K, LdL Q dq 4 dq = F F (K, LdK + (K, LdL K L 16 1 F F 17 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, 0h + (0, 0h = 0 (111 x y F (111 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

10 132 17 1 F 18 ( F : R 2 R (x, y R 2 (u, v R 2 g : R R g(ϵ = F (x + ϵu, y + ϵv g ϵ = 0 g(ϵ g(0 F (x + ϵu, y + ϵv F (x, y lim = lim ϵ 0 ϵ ϵ 0 ϵ (112 F (x, y (u, v (112 F (x, y 2 (u, v F 18 (u, v R 2 (u, v = (1, 0 (u, v = (0, 1 x y 17 F 2 (0, 0 (ϵ, ϵ F (u, v = (1, 1 F (0 + ϵ, 0 + ϵ F (0, 0 ϵ = ϵ 0 ϵ (0, 0 (u, v = (1, 1 F 1 = 1 133 (110 (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

11 1 16 1 f x 2 c R f(x + h ( f(x + ch lim = 0 (113 h 0 h (113 c c = f (x f (113 1 (113 19 ( F : R 2 R (x, y R 2 (c, d R 2 F (x + h, y + k ( F (x, y + ch + dk lim = 0 (114 (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 (114 lim (h n, k n = (0, 0 n {(h n, k n } 15 (17 (18 (19 (110 (114 16 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

12 134 19 (c, d (x, y (gradient vector ( F F F (x, y = (x, y, (x, y x y max F (x + h, y + k F (x, y (h,k subject to (h, k ϵ (x, y (x + h, y + k F (x, y (x, y ϵ F F 0 ϵ (h ϵ, k ϵ = F (x + h ϵ, y + k ϵ F (x + h ϵ, y + k ϵ (115 lim h ϵ = 0, lim k ϵ = 0 ϵ 0 ϵ 0 (115 ϵ ϵ 0 F 1 lim ϵ 0 ϵ (h 1 ϵ, k ϵ = F (x, y (116 F (x, y (116 (x, y F F (x, y (116 F F (x, y F (x, y lim (ϵh,ϵk (0,0 F (x + ϵh, y + ϵk ( F (x, y + F x (ϵh, ϵk F (x + ϵh, y + ϵk F (x, y lim = ϵ 0 ϵ (h, k F x F (x, yϵh + y (x, yϵk F (x, yh + y (x, yk (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 (117 F (x + ϵh, y + ϵk F (x, y lim = F (x, y (118 ϵ 0 ϵ (118 (h, k (117 (h, k 1 (118 F

13 14 141 110 F : R L R U R L k F U k C k 11 F : R 2 R R 2 F (x, y 1 F (x, y 142 F : R 2 R (x, y x (x, y (x, y F (x, t x F x : R2 R (119 (x, y F 2 y 2 F 2 (x, y x 2 F (x, y y x F y : R2 R (120 (x, y (120 2 F (x, y x y 2 F 2 (x, y y 2 F/ y x 2 F/ x y x y y x 12 ( F : R 2 R (x, y x y F x : R2 R F y : R2 R

14, (a, b 12 h, k R {0} 2 F x y (a, b = 2 F (a, b y x = F (a + h, b + k F (a + h, b F (a, b + k + F (a, b ϕ (x = F (x, b + k F (x, b = ϕ (a + h ϕ (a F (a, b x ϕ (x = F F (x, b + k (x, b x x (a, a + h ϕ (x θ (0, 1 ϕ (a + h ϕ (a h = ϕ (a + θh h = F F (a + θh, b + k (a + θh, b x x h = k F/ x F F (a + θh, b + k = x x (a, b + F θh 2 x 2 (a, b + h 2 F (a, b + o (h y x F F (a + θh, b = x x (a, b + F θh 2 (a, b + o (h x2 o ( lim h 0 o (h /h = 0 h 2 = 2 F o (h (a, b + y x h lim h 0 h 2 = 2 F (a, b y x x y lim h 0 h 2 = 2 F (a, b x y 13 ( F : R 2 R (x, y x y 2 2 F x y : R2 R 2 F y x : R2 R

15 (x, y (a, b 2 F x y (a, b = 2 F (a, b y x 13 h = F F (a + θh, b + k (a + θh, b x x 2 F/ x y y 1 θ (0, 1 2 F/ x y (a, b hk = 2 F y x (a + θh, b + θ k lim (h,k (0,0 hk = 2 F (a, b y x x y lim (h,k (0,0 hk = 2 F (a, b x y 2 F x y (a, b = 2 F (a, b y x (a, b 2 (x, y (a, b 143 1 14 ( 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

16 15 ( 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 16 ( 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 2 144 17 ( F : R 2 R c R (a, b R 2 F (a, b = c F (a, b 0 y a 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 (a, b x 18 F (K, L c K L F (K, L = Q (121 (K, L Q (K, L (121 F (K, L = Q L = g(k (122 L g (122

17 g g ( F dg dk (K = K K, g(k ( K, g(k F L (K, L 17 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 (1 145 111 ( F : R L ++ R x R L ++, t > 0 : F (tx = t m F (x F m 5 19 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 1 110 19 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

18 18 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 18 ( 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 18 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 111 Q = F (K, L F 1 K F (K, L + L F (K, L = Q (123 K L (K, L (K, L (Q (123 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 (124 F m x R n ++ : x F (x = mf (x

19 1 p F K (K, L p F L (K, L (124 15 151 1 1 f : R R f(x x x + h f f(x + h 19 (( f : R R x f k + 1 θ [0, 1] f(x + h = f(x + f (xh + 1 2! f 2 + + 1 k! f (k (xh k + R k+1 (h; x (125 R k (h; x = 1 (k + 1! f (k+1 (x + θhh 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 (xh + f 2 + + f (k (xh k f(x + h x f h f(x + h 1 k = 1 f(x + h f(x + f (xh 2 k = 2 f(x + h f(x + f (xh + 1 2 f 2 k = 0 f (x + h = f (x + f (x + θh h f (x + h f (x h = f (x + θh

20 152 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 (xh n n 3 19 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

21 2 21 211 m n A R m n a 11 a 12 a 1n a 21 A = a m1 a mn a 1j a 2j A = (a 1 a 2 a n, a j = Rm, j = 1, 2, 3,, n a m1 A n x = x 1 x 2 Rn, x n a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1n x n a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2n x n Ax = = x 1a 1 + x 2 a 2 + + x n a n R m a m1 x 1 + a m2 x 2 + a m3 x 3 + + a mn x n R n x R m Ax x Ax 21 ( 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 21 A R m n F : R n R m F (x = Ax F

22 F : R n R m x R n, F (x = Ax A R m n 21 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 a 1j a 2j a m1 = x 1 F (e 1 + x 2 F (e 2 + + x n F (e n 0 = a j j = 1, 2, 3,, n, A = (a 1 a 2 a n 1 F (x = x 1 a 1 + x 2 a 2 + + x n a n = (a 1 a 2 a n x 1 x 2 = Ax x n F A R m n F (x = Ax 21 A F F (x = Bx B R m n A A = B 21 21

23 212 1 1 a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + a m3 x 3 + + a mn x n = b m a 11 a 12 a 1n a 21 A = a m1 a mn Ax = b x = x 1 x 2 b 1 b 2 Rn, b = Rm x n b m, x 1 a 1 + x 2 a 2 + + x n a n = b b A 1 1 1 1 Ax = b x 2 b A 1 213 1 22 (1 n m a 1, a 2,, a n x = x 1 x 2 x n Rn : x 1 a 1 + x 2 a 2 + + x n a n = 0 x = 0 a 1, a 2,, a n 1, 1,, x = x 1 x 2 x n Rn \ {0} : x 1 a 1 + x 2 a 2 + + x n a n = 0 a 1, a 2,, a n 1

24 a 1, a 2,, a n 1 1 22 x R n, k, x k 0 : x 1 a 1 + x 2 a 2 + + x k a k + x n a n = 0 a k = x 1 x k a 1 x 2 x k a 2 x k 1 x k a k 1 x k+1 x k a k+1 x n x k a n 21 1 a 1, a 2,, a n 1 j : a j 0 j, k : j k a j a k 1 j : a j = 0 j, k, j k : a j = a k a 1, a 2,, a n 1 22 1 1 a 1, a 2,, a n 1 a i1, a i2,, a ik (k n 1 1 1 a 1, a 2,, a n 1 k a 1, a 2,, a n, a n+1,, a n+k 1 22 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 23 n m n > m n x 1,, x n m 1 a 11 x 1 + + a 1n x n = 0, a m1 x 1 + + a mn x n = 0 x j 0 j (x 1,, x n m

25 23 a 1, a 2,, a n R m 1 n m 23 n > m a 1,, a n 1,, α 1 = = α n = 0, α 1 a 1 + + α n a n = 0 a 11 α 1 + + a 1n α n = 0, a m1 α 1 + + a mn α n = 0, a 1,, a n n, a 1,, a n 1, n m 214 23 ( R m V v V, w V : v V, t R : v + w V tv V V R m : R 2 R 3 24 n m a 1, a 2,, a n 1 a1, a 2,, a n R m a 1, a 2,, a n R m 24 ( R m V, V n a 1, a 2,, a n 1 a 1, a 2,, a n V a 1, a 2,, a n = V 2 a 1, a 2,, a n 1 a 1, a 2,, a n V 1 21 V = R 2 V ( ( 1 0 a 1 =, a 2 = 0 1

26 a 1, a 2 1 ( x = x 1 x 2 V : x = x 1 a 1 + x 2 a 2 a 1, a 2 = V a1, a 2 V ( ( 1 1 a 1 =, a 2 = 0 1 a 1, a 2 V a 1, a 2 1 ( x = x 1 x 2 a 1, a 2 = V V : x = (x 1 x 2 a 1 + x 2 a 2 25 a 1,, a k b 1,, b l V k = l 25 k > l a 1,, a k b 1,, b l V, a 1,, a k b 1,, b l, a 1 = β1b 1 1 + + βl 1b l, a k = β1 k b 1 + + βl kb l a 1,, a k 1, ( α 1 a 1 + + α k a 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, (, a 1,, a k 1, k l, k l, k = l 25 ( V V dimv 26 F : R n R m A = (a 1 a 2 a n R m n F (x = Ax 1 F a 1, a 2,, a n = R n 2 F a 1, a 2,, a n 1 3 F a 1, a 2,, a n R n

27 26 ( m n A = (a 1 a 2 a n, a j = a 1j a 2j Rm, j = 1, 2, 3,, n A R m A ColA 27 ( m n a 1 a 2 A = a m a m1 ColA = a 1, a 2,, a n, ai = (a i1 a i2 a in R n, i = 1, 2, 3,, m A A RowA RowA = a 1, a 2,, a m 28 ( m n 1 Ax = 0 x R n A KerA KerA R n KerA = { x R n Ax = 0 } 27 m n A R m n dim (ColA + dim (KerA = n 27 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

28 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 a 1, a 2,, a 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

29 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 29 ( R n V V n V = { x R n x z = 0, z V } V R n 28 R n V dimv + dimv = n

30 28, 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 29 m n A R m n dim (ColA = dim (RowA dim (ColA = dim (ColA, dim (RowA = dim (RowA 29 x Ax = 0 x KerA a 1 x = a 2 x = = a m x = 0 z RowA : z x = 0 x (RowA KerA = (RowA dim (KerA = dim ( (RowA 27 28 dim (KerA = n dim (ColA dim ( (RowA = n dim (RowA dim (ColA = dim (RowA 27, 28 29 28 29 m n A R m n dim (KerA = n dim (RowA (21 1 Ax = 0 dim (KerA n dim (RowA 8 (21 ( = ( ( 210 (rank A A dim (ColA = dim (RowA = ranka 7,,,,, p121 8, a m = c 1 a 1 + + c m 1 a m 1 a 1 x = = a m 1 x = 0, a m x = 0

31 215, 211 ( A A 212 ( A R n n B R n n AB = I BA = I B A 210 A R n n 1 A, A 1 211 A B n AB = I BA = I, B A, AB = I BA = I 24 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 25 a 2 3 A [ 1 2 3 4 8 a ] 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

32 212 A R n n F : R n R n 3 1 F 2 F 3 F 212 1 2: F, a 1,, a n = R n, dimr n = dim(cola = n, dim(kera = 0, 0 = α 1 a 1 + + α n a n, α 1 = = α n = 0, a 1,, a n 1, F 2 1: dimr n = n, F, a 1,, a n n 1, R n x a 1,, a n, a 1,, a n = R n, F, a 1,, a n R n, 3 213 F : R n R n F 1 : R n R n F F 1 213 ( A R n n A, F : R n R n F (x = Ax, F F 1 214 A R n n F : R n R n A 214 A R n n F : R n R n F (x = Ax F, F 1 : R n R n, 213, 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 (22 F F 1 F F 1 : R n R n (F F 1 (x = x (23 (22 (23 AB I A B n 211 BA I B A B = A 1

33 22 m n m n A R n n det : R n n R deta deta A 221 2 214 (2 2 ( a 11 a 12 A = R 2 2 a 21 a 22 det : R 2 2 R deta = a 11 a 22 a 12 a 21 A 214 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 = a 11 a 21, Ae 2 = A a 1 a 2 a 1 a 2 e 1 e 2 ( a 21 a 22

34 e 1 e 2 e 1 e 2 180 a 1 a 2 a 1 a 2 e 1 e 2 a 1 a 2 1 222 215 ( 1 ( ( ( a 11 a 12 + a 12 a 11 a 12 a 11 a 12 det = det + det a 21 a 22 + a 22 a 21 a 22 a 21 a 22 2 t R ( ( a 11 ta 12 a 11 a 12 det = tdet a 21 ta 22 a 21 a 22 ( ( ta 11 a 12 a 11 a 12 det = tdet ta 21 a 22 a 21 22 216 ( ( ( a 11 a 12 a 12 a 11 det = det a 21 a 22 a 22 a 21 215 det : R 2 2 R 1 2 3 1 214 216 1 2

35 3 1 3 F : R 2 2 R 215 216 1 223 3 3 216 n 217 1 2 3 1 3 F : R n n R, 217 n 2 1 224,, 217 ( n 1, 2,, n M = { 1, 2, 3,, n } π : M M π M 218 ( 2 k = 1, 2, 3,, n i k = j σ(k = j k = i k k i, j σ : M M 218 218

36 219 π π = τ 1 τ 2 τ r π = π 1 π 2 π s 2 r s r s 219 219 ( sgn(π 1 π sgn(π = 1 π 220 M = { 1, 2, 3,, n } Π n a 11 a 12 a 1n a 21 A = R n n a n1 a nn (sgnπa π(11 a π(22 a π(nn = deta π Π (,, deti=1, 221 n A A deta = deta 221 deta = π Π(sgnπa π(11 a π(22 a π(nn π Π, π π 1 Π, deta = π Π(sgnπ 1 a π 1 (11a π 1 (22 a π 1 (nn π 1 (1, π 1 (2,, π 1 (n, π 1 (i = k i = π(k, deta = π Π(sgnπa 1π(1 a 2π(2 a nπ(n, deta

37 220 ( n a 11 a 12 a 1n a 21 A = R n n a n1 a nn i j n 1 a 11 a 1 j 1 a 1 j+1 a 1n A ij = a i 1 1 a i 1 j 1 a i 1 j+1 a i 1 n a i+1 1 a i+1 j 1 a i+1 j+1 a R (n 1 (n 1 i+1 n a n1 a n j 1 a n j+1 a nn A ij det (A ij A n 1 222 n A n a ij ( 1 i+j deta ij = deta j=1 n a ij ( 1 i+j deta ij = deta i=1 i j 223 ( A n, b R n, Ax = b x x j = det(a 1,, a j 1, b, a j+1, a n deta 23 231 22 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

38 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

39 221 ( n A Ax = λx, x R n \ {0}, λ R λ A x λ 22 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 23 2 ( 0 1 A = 1 0 Ax = λx, x R 2 \ {0}, λ R (24 λ x (24 λ x (A λi x = 0 (25 x x = 0 (A λ (A λ 1 (25 (A λi 1 (A λi x = 0 x = 0 (25 (25 (A λ (A λ det (A λi = 0 A λ 1 det (A λi = 1 λ = λ2 + 1 = 0 (26 (26 λ A 24 2 2 ( 1 1 A = 0 1 Ax = λx, x R 2 \ {0}, λ R (27

40 x 23 λ det (A λi = 0 1 λ 1 0 1 λ = (λ 12 = 0 λ = 1 (27 ( ( 0 1 x 1 Ax = x (A I x = = 0 0 0 x 2 ( x = t 1 0, t R \ {0} (28 A 1 232 n A Ax = λx (A λi det (A λi = 0 (29 λ (29 (29 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 24 241

41 222 ( 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 224 ( n A R n,, A 224 i = 1, 2,, n : Av i = λ i v i (210 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 (211 c n P P = (v 1 v 2 v n P n (211 x = P c c = P 1 x (212 A (211 (210 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

42 c 1 λ 1 λ 1 0 0 c 2 λ 2 Ax = P = P 0 λ 2 c Ax = P Λc (213 0 c n λ n 0 0 λ n (212 (213 c Ax = P ΛP 1 x A = P ΛP 1 224 26 224 225 P 1 AP,, P R n 225 : 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 25 ( 5 3 D = P = 3 5 ( 1 1 1 1 224 A R n A 226 n A l (l n l v 1, v 2,, v l 1, n 226 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 (

43 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 226 n A n R n A n 1 n n 1, I 242 n = 2 26 ( 1 2 A λ R A λ A λi = 0 ( x = x 1 x 2 : Ax = λx (214 R 2 λ R 2 22 A A λi 0 dim {Ker (A λi} = 1 1 R 2 24

44 v ( v / Ker (A λi v 1 = (A λiv (215 v 2 = v (216 v 1 v 2 27 (1 (A λi 2 (2 v 1, v 2 R 2 27 (A λiv 1 = (A λi 2 v = 0 v 1 λ Av 1 = λv 1 (217 (215 (216 (217 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 (218 P P (218 ( P 1 λ 1 AP = 0 λ 27 ( 0 0 det (A λi = 0 λ 1 = µ + iν (219 λ 2 = µ iν i (i 2 = 1, µ ν ν 0 Ax = λ 1 x x C 2 x = u + iw (220

45 Ax = λ 2 x x C 2 x = u iw u w R 2 w 0 (219 (220 Ax = λ 1 x A(u + iw = (µ + iν(u + iw Au + iaw = µu νw + i(µw + νu Au = µu νw (221 Aw = µw + νu (222 λ 2 = µ iν, x = u iw Ax = λ 2 x, P = (u w R 2 2 P (221 (222 ( P 1 µ ν AP = ν µ (223 (223 ν < 0 λ 1 λ 2 ρ = µ 2 + ν 2 ( 2 ( 2 µ ν + = 1 ρ ρ θ (0, π (224 (225 (223 ( ( 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 (224 ν = ρ sin θ (225 θ ρ 25 2 251 2 m n A F (x = Ax F (y, x = y Ax

46 227 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 2 223 (2 n x R n n n Q(x = a ij x i x j (226 j=1 i=1 Q : R n R 2 223 Q : R n R 227 n a 11 a 12 a 1n a 21 A = R n n a n1 a nn (226 n n a ij x i x j = x Ax j=1 i=1 224 ( 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

47 252 2 225 (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 (negative definite x R n : x Ax 0 A (negative semidefinite 28 I 225 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 < 0 228 ( 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

48 228 226 ( R n v 1, v 2,, v n 0 (i j v i v j = 1 (i = j (227 v 1, v 2,, v n R n 227 ( 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 1 229 ( n A 1 A 2 A P 2 :,, A 229 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

49,, ( 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: 28 28 n A, P 1 AP n P A 29 A n v A x R n v x = 0 v (Ax = 0 230 A P Λ, Λ = P 1 AP 230 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, 230, 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 2 230, ( ( 228

50 231 ( n A ( A (, n A ( A ( 228 ( A n i 1, i 2,, i m 1 i 1 < i 2 < < i m n m B b 11 b 12 b 1m a i1i 1 a i1i 2 a i1i m b 21 a i2i B M = = 1 b m1 b mm a im i 1 a im i m B M detb M A m principal minor i 1 = 1, i 2 = 2,, i m = m A m leading principal minor m,, A n 2 n 1 n 29 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 4 232 n A k detb k A k detbk 1 A 2 A m ( m (

51 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

52 3 31 311 31 ( R n C x C, ϵ > 0, y R n : x y < ϵ y C C 32 ( R n C C C c = { x R n x / C } C 31 ( C x C x (x n N n > N : x n C 32 ( C x R n x (x n n : x n C x C 31 31 31 32 32, 33 ( C x, y C, t [0, 1] : tx + (1 ty C C, x, y C, [x, y] C 312 34 ( C C F : C R x, y C, t [0, 1] : F (tx + (1 ty tf (x + (1 tf (y F

53 35 ( C C F : C R, F, x, y C, t [0, 1] : F (tx + (1 ty tf (x + (1 tf (y F 34 35 33 C C F : C R F x C, h R n : F (x + h F (x + F (x h (31 32 33 ( : 33 F : R L R, F x R L y R L F (y F (x + F (x(y x 33 (31 F (x + h { F (x + F (x h } 0 F (x + h = F (x + F (x h + h 2 F (xh + R 2 (h; x lim h 0 R 2 (h; x h 2 = 0 (31 h 2 F (xh + R 2 (h; x 0 (32 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 (xv 0 h 0 h 2 F (xh R 2 (h; x (32 2 v 2 F (xv 0 34 C C 2 F : C R 1 F 2 x C F 2 F (x

54 34 F : R L R, F 2 x R L 2 F (x 33 34 35 C C F : C R F x C, h R n : F (x + h F (x + F (x h 36 C C 2 F : C R 1 F 2 x C F 2 F (x 313 36 ( C C F : C R x, y C, t [0, 1] : F ( tx + (1 ty max {F (x, F (y} F (quasi-convex function 35 F, 37 ( C C F : C R, F, x, y C, t [0, 1] : F ( tx + (1 ty min {F (x, F (y} F (quasi-concave function 37 C C F : C R 1 F 2 z R : { x C F (x z } 38 C C F : C R 1 F 2 z R : { x C F (x z }

55 39 C C F : C R R G : R R G F : C R 36 F (x 1, x 2 = x α 1 1 xα 2 2, α 1, α 2 > 0 α 1 + α 2 1 C 310 C C F : C R F y R n : F (x y = 0 y 2 F (xy 0 (33 F (x F (x 310 (33 2 F (x F (x 9 311 C C F : C R 1 F 2 y R n : F (x y = 0 y 2 F (xy 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 (Ie, F (y F (x F (x (y x 0 (ii F (x (y x 0 F (y F (x (Ie, F (y < F (x F (x (y x < 0 (i, (ii 38 ( C C F : C R x, y C : F (x (y x 0 F (y F (x F (pseudo-convex function 37 (i 9 F 2 F (x R n

56 31 1 (i (ii, F F 2 x F (x 0, (i (ii, (i (ii, F (x 0 C = R, F (x = x 3 31 (i : x C, y C, F (x (y x > 0, ε > 0 z = (1 εx + εy = x + ε(y x F (z > F (x (34 F (z (y z > 0 (35, (35 (ii, F (y F (z, (34 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 32 321 312 ( 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

57 1 A z = b z z z 1 z 2 z = RJ + A = a 2 RJ N, a 1 a j = (a j1 a j2 a jn, j = 1, 2,, J z J a J z A b = z A b = z A = z 1 a 1 + z 2 a 2 + + z J a J (z j 0, j = 1, 2,, J 1 b A (cone b 2 Ax R J + 2 Ax R J + a j x 0, j = 1, 2,, J a 1 x 0, a 2 x 0,, a J x 0 b x < 0 x R N A 90 b 90 x b 39 ( a R N b R N a b 0 x R N a = { x R N a x = 0 } x = v + λb v a λ R v x a b 38 x a b v v = x a x a b b 312 z R J + x RN 1 z b = z A x b x = z Ax (36

58 z 2 z Ax = z 1 a 1 x + z 2 a 2 x + + z J a J x 0 b x < 0 (36 1 2 J J = 1 J 2 J 1 1 2 A = a 1 a 2 a J 1 a J R J N, A = a 1 a 2 a J 1 R(J 1 N A A A 1 A 1 z R J 1 + b = z A (37 A x R J 1 + b x < 0 x R N a J x 0 x Ax R J + b x < 0 2 a J x < 0 a 1, a 2,, a J 1 b x a J â 1, â 2,, â J 1 b   = â 1 â 2 â J 1 R(J 1 N b  b = w  (38 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

59 â j a j x a J 38 â j = a j x a j x a J a J, j = 1, 2,, J 1 b = = = J 1 w j â j j=1 J 1 ( w j a j x a j x a J a J j=1 J 1 w j a j j=1 ( J 1 j=1 w j x a j x a J a J (39 b b x a J (39 b = b x b x a J a J ( J 1 b x J 1 b x a j = w j a j + x w j a j=1 J x a J a j=1 J ( = w A x J 1 b x a j + x w j a J x a J (310 a J a J x < 0 w j 0, j = 1, 2, J 1 j=1 A x R J 1 + b x < 0 (310 x J 1 b x a j x w j a J x 0 a J j=1 w J = x J 1 b x a j x w j a J x a J j=1 z = w 1 w 2 w J 1 w J z z b = w A + w J a J = z A

60 A 1 b  b = w  w R+ J 1  1  2  x R+ J 1 (311 x R N x a J x x b x < 0 (312 Ax R J + b x < 0 A 2 â j, j = 1, 2, J 1 b λ R : â j = a j + λa J (313 â j x = 0 (314 λ R : b = b + λa J (315 b x = 0 (316 x λ R : x = x + λx (317 (313 (318 a J x = 0 (318 a j x = â j x (319 (314 (317 (311 â j x = â j x 0 (320 (319 (320 (318 (321 a j x 0 j = 1, 2, J 1 (321 Ax R+ J (315 (318 b x = b x (322 (316 (317 (312 b x = b x < 0 (323 (322 (323 b x < 0

61 322 =, b a j, j = 1, 2,, J b 313 ( R N C R N C b R N \ C c C : x c d > x b x R N d R 39 313 (d = 0 314 A R J N 1 z R+ J \ {0} : z A = 0 2 x R N : Ax R J ++ 2 310 314 1 2 314 310 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 + (324 (0, 0,, 0, 1 x < 0 (325 x = ( x x N+1 R n

62 x R N Ã x = (A, e ( x x N+1 = Ax + x N+1 e (326 (325 x N+1 = (0, 0,, 0, 1 x < 0 (326 Ã x < Ax (324 Ax R++ J 2 314 1 A z = 0 z z, A, 1 2 a 1 x > 0, a 2 x > 0,, a J x > 0 x R N A 90 x R N,, A,, 33 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, max (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 (327 x X m : g m (x 0, n : f n (x f n (x, n : f n (x > f n (x

63 x X x X f n g m 331 315 ( x X (327 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 3 315 ( 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 314 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}

64 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 315 31 µ λ 0 (µ 1,, µ N, λ 1,, λ M 315 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 10 315 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 1 316 ( (327 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 (327 316 ( M (g m (x = 0 (g m (x > 0 10??

65 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 311 x 316 2 3 316 31 2 u : R 2 + R u(x 1, x 2 = (x 1 + 1 (x 2 + 1 p 1, p 2 w max x R 2 + (x 1 + 1 (x 2 + 1 subject to w p 1 x 1 p 2 x 2 0 (328 R 2 + X = { x R 2 x 1 > 1, x 2 > 1 } x 1 0 x 2 0 max 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 (329 (329 (328 31

66 32 31 2 u : R 2 + R u(x 1, x 2 = x 1 + x 2 p 1, p 2 w max x R 2 + u(x 1, x 2 = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0 (330 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 ++ max x R 2 ++ u(x 1, x 2 = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0 (331 (330 (331 31 32 33 2 1 w(> 0 u : R+ 2 R u(x 1, x 2 = x 1 2 1 e x 2 max x R 2 + u(x subject to w x 1 x 2 0 (332 312 x R+ 2 (332 x 1 312 { x R 2 x 1 > 0, x 2 0 } u X = { x R 2 x 1 > 0 }

67 e x 2 x 2 R X g 1 : X R g 1 (x = w x 1 x 2 max x X u(x subject to g 1 (x 0 (333 X (333 (333 313 max x X u(x subject to g 1 (x 0, g 2 (x 0 (334 (332 g 2 : X R 314 313 (334 (332 332 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 max x X subject to g 1 (x, p 0, f(x, p (335 g 2 (x, p 0, g M (x, p 0 (335 p P p P (335 310 (Policy Function p P (335 P a : P X a(p Policy Function 311 (Value Function Policy Function b : P R b(p = f(a(p, p b(p Value Function

68 317 p P (1, λ 1,, λ K R 1+K ++ x X (335 = (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 a(p Value Function b(p Q 317 318 ( p P (1, λ 1,, λ K R 1+K ++ x X (335 Policy Function a(p Value Function b(p M b(p = p f(x, p + λ m p g m (x, p m=1 318 p P : g m (a(p, p = 0, m = 1, 2,, M p p = p x g(x, p a(p + p g m (x, p = 0, m = 1, 2,, M (336 (336 λ m m M λ m x g(x, p a(p + m=1 M λ m p g m (x, p = 0 (337 (1, λ 1,, λ K R 1+K ++ x X (338 (337 x f(x, p + m=1 M λ m x g m (x, p = 0 (338 m=1 x f(x, p a(p + Value Function M λ m p g m (x, p = 0 (339 m=1 p P : b(p = f (a(p, p p p = p b(p = x f(x, p a(x + p f(x, p (340 (340 (339

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