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X RL+ RL x X

{y X : x y} x x, y R L

x y = L=1

(x y )2.

X x x x X > 0 y X x y < y x

k {1, . . . , L } ek RL k k x, y R L

x y

xi yi

i = 1 , . . . , L ,x > y xi > y i i = 1 , . . . , L .

k x X > 0 : x + ek x

x, y X x y x = y x y x, y X x y x y x > y x y

R 2+ k k = 1 k = 2

X = R L+ x y x = y x y

X = R L

R2+

(x1, x2) (y1, y2) (x1 x2)2 + x1 (y1 y2)2 + y1

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0

1

2

3

0 1 2 3

x 1

x 2

k

R L R L+

y X {x X : x y} y {x X : x y}

y

X

y X {x X : x y} {x X : x y}

y X {x X : x y} {x X : x y} {(x, y) X X : x y}

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(xn )n N (yn )n N X xn x yn y xn yn n N x y x, y X x y U x x U x

x U y y x y x U x , y U y

(xn , yn ) X X xn yn

x, y X x y m X x m y

U x = {z X : z m} U y = {z X : m z} x U x y U y x U x , y U y x m m y

x y m X x m y

U x =

{z

X : z

y

} U y =

{z

X : x

z

} x U x

y U y x U x , y U y x y

x x x x y x x y y x x y y x y

{(x, y) X X : x y} S = {(x, y) X X : x y}.

(x, y) S U x x U y y x y x U x , y U y

(x, y ) S : (x, y) U x U y S. (x, y) S

S = (x,y ) S U x U y . S

y X {x X : x y} {x X : x y}

S y X L (y) = {x X : x y} {x X : x y}

x L (y) (x, y ) S S X U x x U y y U x U y S

x U x (x , y ) U x U y x y

x L (y) : x U x L (y).

x L (y)

L (y) = x L ( y ) U x .

L (y)

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(xn )n N {x X : x y} x x (xn , y)n N

(x , y) x y

x

y

X X = R L+ x y

x y = L=1

(x y )2. Y X y Y Y y Y x X y Y

> 0 x X x y < x Y x X

X X

X = R 2+ X = R 2

Y 1 = R 2+ , Y 2 = {y R 2+ : y1 < 1}, Y 3 = {y R 2+ : y1 + 2 y2 < 4}

X = R L+ R L Y X

Y = X O O RL

Y 1, Y

2, Y

3

X = R2+

Y 1 = X R 2, Y 2 = X {y R 2 : y1 < 1}, Y 3 = X {y R 2 : y1 + 2 y2 < 4},

R 2, {y R 2 : y1 < 1}, {y R 2 : y1 + 2 y2 < 4} R 2

k x, y X > 0 x y x + ek y + ek

k

x, y X > 0 x y x y

X X

X

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(x1, x2) (y1, y2) 3x1 + exp x2 3y1 + exp y2,

x y x y x

y x y

f : RL+ R k R x RL+ > 0f (x ) = k f (x) x y f (x) f (y)

x y f (x) f (y) f (x ) = k f (x) k f (y) = f (y ) x y. f (x1, x2) = min {x1, x2} f (x1, x2) = x1x32

R2+

X = R L+ x y x > y x y x y

x1 , x2 0 min{x1 , x2} max{x1 , x2} min{x1 , x2} R2+

x, y R 2+ x y x y min{x1 , x2} > min{y1 , y2} x y min{x1 , x2} = min {y1 , y2}

max{x1 , x2}min{x1 , x2} max{y1 , y2}min{y1 , y2} x y x > y x y x y

y

X

{x

X : x y

}

X x, y X x y [0, 1] x + (1 )y y x y

y x

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x, y X x = y x y (0, 1) x + (1 )y y

x, y

X

12 x + 12 y

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u : X R x, y X :x y u(x) u(y).

u

X u : X R

u

x, y X x y, u(x) > u (y), x y, u(x) = u(y).

x, y X x y x y y x u(x) u(y) u(y) u(x)

u(x) > u (y) x y u(x) = u(y) x, y X

x y

u(x)

u(y).

x y x y x y u(x) > u (y) u(x) = u(y) u(x) u(y) u(x) u(y) x y y x

x y y x u(y) > u (x)

R

x, y R : x y x y + 1 u : R

R u(x) = x x

R

u

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X n : X N x X u(x) =

z X :x z

2 n (z) .

(2 n )n N n N 2 n = 1 u u

x, y X x y u(x) = u(y) x y u(x) u(y) 2 n (x) > 0

X z I

R b(z)

X g(z)

X

z I g(z) b(z).

z < z z z

z, z I : z < z b(z ) g(z). z < z

u(b(z)) < u (g(z)) < u (b(z )) < u (g(z )) .

z I

[u(b(z)) , u(g(z))]

z, z I

z = z

[u(b(z)) , u(g(z))] [u(b(z )) , u(g(z ))] [u(b(z)) , u(g(z))]

[u(b(z)) , u(g(z))] r (z) Q z

z = z r (z) = r (z ) r : I Q I Q

X = R 2

(x1, x2) (y1, y2) x1 > y1 (x1 = y1 x2 y2) .

z R b(z) = ( z, 0) g(z) = ( z, 1) g(z) b(z) z, z R , z < z g(z) = ( z, 1) (z , 0) = b(z )

z R

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x R x x < z x [z, )

x x < z [z, )

b(z) =

{{x

} : x < z

} {[z,

)

} R g(z) = {{x} : x z} {(z, )}

x x = z x z

P Q P Q g(z) b(z) g(z) b(z)

z < z b(z ) g(z) b(z ) g(z)

t

[0,

)

x x : [0, ) {0, 1} x(t) y(t) t

x y z [0, ) b(z) z

b(z)( t) = 1 t < z,

0

g(z) z

g(z)( t) = 1 t z,

0

g(z) b(z) z < z b(z ) g(z)

X (X, )

X

C X x, y X x y, c1, c2 C x c1 c2 y.

x, y X x y x c1 c2 y

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X X

u U = {u(x) : x X } u U (u1 , u2) U U u1 < u 2

(u1 , u2) U (u1 , u2)

U =

u

(u1 , u2) x(u1 , u2) u1 y(u1 , u2) u2 J = {x(u1 , u2), y(u1 , u2)} (u1 , u2) J

r 1 , r 2 Q r1 < r 2 (r 1 , r 2) U = x(r 1 , r 2) X (r 1 , r 2) U R x(r 1 , r 2) (r 1 , r 2) R C = J R

C X

X C n : C N u u(x) = c C :c x 2

n (c)

u

X C = X X

R L+

X

X

X y X {x X : x y} {x X : x y}

C X X C C c, c C

c = c c c c c C C C = {c1, c2, . . .} Q = (0 , 1) Q

(0, 1) Q = {q 1, q 2, . . .} f : C Qf (c1) := q 1 n N, n 2 f {c1, . . . , cn 1}

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{c1, . . . , cn} f (cn ) Q q Q q

k {1, . . . , n 1} : q > f (ck ) cn ck . a, b C a b C a b

(a, b) = {c C : a c b}, cm cm

(a, b) f f (cm ) (f (a), f (b)) Q

X x X u(x) = sup {f (c) : c C, c x} x 1

u x, y X x y u(x) = u(y) x y a, b C x a b y u(x) f (a) > f (b) u(y).

R

(, r ) (r, )

r u 1((, r ))

u 1(( r, )) r Q u 1((, r )) r inf f (C ) X r > sup f (C ) r = sup f (C )

r / f (C ) {x X : x f 1(r )} r f (C ) r / f (C ) inf f (C ) < r < sup f (C ) r

f (C ) f (C ) (f 1, f 2) f (C ) f (C ) f 1 < f 2 (f 1, f 2) f (C ) =

inf f (C ) < r < sup f (C ) a, b C f (a) < r < f (b) m N f (a), r, f (b) Q {q 1, . . . , q m } r

p, p f (C ) p < r < p n N {q 1, . . . , q n}

p1 = max f (C ) {q 1, . . . , q n} (, r ), ( p1, r ) {q 1, . . . , q n} = , p2 = min f (C ) {q 1, . . . , q n} (r, ), (r, p 2) {q 1, . . . , q n} = .

r ( p1, p2) r ( p1, p2) f (C )

p1, p2 f (C ) b1, b2 C p1 = f (b1), p2 = f (b2) ( p1, p2)f (C ) = p C f ( p1) < f ( p) < f ( p2) (b1, b2) C b1

b2 b f (b ) ( p1, p2) r r / f (C )

r (f 1, f 2) f (C ) u 1((

, r )) =

{x

X : x

f 1(f 2)} R L+

C Q (0 , 1)

R

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e = (1 , . . . , 1) R L+ x X x 0 x x e

x X max{x1, . . . , x L} e x 0e

{x R L+ : x1 = = xL}, x x 0 x x e

x u(x) = x u

x, y X x y x e ye u(x) = x y = u(y) u

u 1((, )) (, )

u 1((, )) = {x X : x e} {x X : x e}

x x e 0 x xe u(x ) = x = u(x)

X u : X

R

u

u

X = R L+ L N X

x (0, . . . , 0) x X

x, y X x y v 0 x (y1 + v, y2, . . . , yL )

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u(x) = x1 + v(x2, . . . , x L )

x X (0, x2, . . . , x L ) (0, . . . , 0).

v(x2, . . . , x L ) 0(0, x2, . . . , x L ) (v(x2, . . . , x L ), 0, . . . , 0). x1 0

(x1, x2, . . . , x L ) (x1 + v(x2, . . . , x L ), 0, . . . , 0) . u : X R u(x) = x1 + v(x2, . . . , x L )

x, y X : x y (x1 + v(x2, . . . , x L ), 0, . . . , 0) (y1 + v(y2, . . . , yL ), 0, . . . , 0)

x1 + v(x2, . . . , x L ) y1 + v(y2, . . . , yL ),

x, y X : x y, v 0 x y + ve1

A X =AR + (a, m ) X a A m R +

X (a, m ) (a , m ) X

(a, m ) (a , m ) m 0 (a, m ) (a , m ) a A m, m R + m > m (a, m ) (a, m )

(a, m ) (a , m ) X c 0 (a, m ) (a , m ) (a, m + c) (a , m + c)

(a, m ) X a

a, a A

m, m R

(a, m ) (a , m ) a, a A m, m ,w,w R+ (a, m ) (a , m ) (a, w) (a , w ) m m = w w

a A v : A R a Av(a) = m m, m, m (a , m ) (a, m ).

m, m v m, m

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u : X R u(a, m ) = v(a) + m

u X

x, y

X

(0, 1)

u(x + (1 )y) min{u(x), u(y)}. x, y X x = y (0, 1)

u(x + (1 )y) > min{u(x), u(y)}.

X = R L+ L N X u : X

R

u u

x, y X (0, 1) x y u(x) u(y) min{u(x), u(y)} = u(y) x + (1 )y y

u(x + (1 )y) u(y) = min {u(x), u(y)},

u X r R X u (r ) = {x X : u(x) r}

u : X R X

R

f : R R f (x + y) = f (x) + f (y) x, y R u R f (xu ) = xf (u) x x N

x Z x Q u = 1 c = f (1) f (x) = cx x f

Q f : R R

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f R {(x, y ) R 2 | y = f (x)} f f

n F : R n R F (x + y) =F (x) + F (y) x, y R n

f i : R R i = 1 , . . . , n x R n F (x) = f 1(x1) + + f n (xn )

X = R L L N X x ,y,z X x y x + z y + z

x X R x (1, . . . , 1)

1, . . . , L R

++

u : X R

u(x) = 1x1 + + L xL x X u(x) R x u(x)e

u : RL R

u x, y X x u(x)e x + y u(x)e + y y u(y)e u(x)e + y u(x)e + u(y)e =

(u(x) + u(y))e x + y (u(x) + u(y))e u(x + y) = u(x) + u(y) u : RL R

u i : R R i = 1 , . . . , L u(x) =

Li=1 u i (x i )

u i ui 1, . . . , L R u(x) =

Li=1 i x i 1, . . . , L

i

x, y X x y

X = RL R L+

u : R L+ R u(x) = xa 11 xa LL =L

i=1

xa ii (L N , a 1, . . . , a L > 0)

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x, y R L++

x y L

i=1

a i ln x i L

i=1

a i ln yi .

R L+

X = R L+ L N X

i {1, . . . , L } x, y X t > 0 x y (x1, . . . , x i 1, tx i , x i+1 , . . . , x L ) (y1, . . . , y i 1, ty i , yi+1 , . . . , yL )

x X R + x (1, . . . , 1)

R L++ RL+

R L++ f : RL R L++ x RL f (x) = (exp x1, . . . , exp xL ) f f 1 : RL++ RL f 1(y) = (ln y1, . . . , ln yL )

RL++ f RL

x, y R L : x f y f (x) f (y).

f x RL x f (1, . . . , 1) L

x,y, t R L++ : x y (t1x1, . . . , t L xL ) (t1y1, . . . , t L yL ). (ln x1, . . . , ln xL ) f (ln y1, . . . , ln yL )

(ln x1, . . . , ln xL ) + (ln t1, . . . , ln tL ) f (ln y1, . . . , ln yL ) + (ln t1, . . . , ln tL ). f f

f RL a1, . . . , a L > 0 f x Li=1 a i x i x, y R L++

x y (ln x1, . . . , ln xL ) f (ln y1, . . . , ln yL ) L

i=1

a i ln x i L

i=1

a i ln yi .

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u u(x) = Li=1 xa ii R

n++

R L+ u RL+ x (0, . . . , 0) x R L+

x x + (1 /n )e R L++ n N (0, . . . , 0) x + (1 /n )e n > 0 x + (1 /n )e n e x + (1 /n )e 0 = lim

n u(x + (1 /n )e) = lim

n u(n e) = limn

a 1 + + a Ln .

a1 + + aL > 0 limn n = 0 x e 0 x x + (1 /n )e n e

n N limn n = 0

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X x X L(x) = {y X | y x}

Y X

Y

y Y : y y y Y. y Y y Y y y

{L(y) : y Y } Y Y Y {L(y ) : y Y } Y Y

y

L(y )

Y

y

Y

X = R L+ X

w > 0 w p RL++ B ( p, w) p wB ( p, w) = {x R L+ | p x w}.

B ( p, w) = Li=1 {x R L | xi 0} {x R L | p x w}. 0 xi w/p i i

R n {x R n : a x c} {x R n : a x c} a R n

a = 0 c R

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RL

B ( p, w)

R2

(x1 , x2) (y1 , y2) x1 > y1 (x1 = y1 x2 y2) .

Y R 2

(X, B , C ) X

B B B X B

C B B C (B ) B B

(X, B , C ) X B B C (B )

B B : C (B ) = {x B | x y y B}.

(X, B , C )

A, B B , x, y A B : x C (A), y C (B ), x C (B ). A B x y

x C (A) x y x y C (B ) y x x y

x B

(X, B , C )

A, B B : A B C (B ) A = , C (A) = C (B ) A. B

A A B A B

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(X, B , C )

B (X,B

, C )

A, B a C (A) b C (B ) A a C (B ) A, b C (A) C (A) A B a, b A B,a C (A),b C (B ).

a C (B ), b C (A) x, y X {x, y} B B x y x C ({x, y})

x, y

X x

C (

{x, y

}) y

C (

{x, y

})

x y y x x ,y,z X x y y z x C ({x, y}) y C ({y, z}) x z x C ({x, z})

x = y y = z x = z x z x x x,y,z {x,y,z } B

x C ({x,y,z }) x C ({x, z}) x / C ({x,y,z }) C C ({x,y,z }){y, z} =

y z y C ({y, z}) = C ({x,y,z }) {y, z} C ({x,y,z }) {x, y} = x y x C ({x, y}) = C ({x,y,z }) {x, y} x / C ({x,y,z })

(X,

B , C ) B

B z C (B ) z y y B y B {y, z} B , {y, z} B z C (B ) {y, z} = z C ({y, z}) z y z B z y y B z C (B )

y C (B ) {y, z} B , {y, z} B y C (B ) {y, z} = z yz C ({y, z}) = C (B ) {y, z} z C (B )

f : X

R X

(X, B , C )

B

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X = {1, 2, . . . , n } n N, n 3 B X C (X, B , C )

v : X R x X v(x) R r R B B

C (B )

B

B

X B B C (B ) = {x B | y B : y x}

(X, B , C ) X = R2+ B B1 = B((2 , 1), 2) p = (2 , 1) w = 2 B2 = B((1 , 2), 2) B1 B2 C (B1)

C (B2)

(X, B , C ) (X, B , C )

(X, B , C )

x p u(x, p)

( p, w)

R 3++

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L N X = R L+

p R L++ i {1, . . . , L } pi > 0 w > 0

X u : X R

B ( p, w) = {x R L+ : p x w}

B ( p, w)

u

max u(x) x B ( p, w)

p R L++ w > 0 x( p, w)x( p, w) = {x B ( p, w) : x y y B ( p, w)}

= {x B ( p, w) : u(x) = maxy B ( p,w) u(y)}.

u v : RL+1++ R p R L++ w > 0

v( p, w) = u(x ), x x( p, w),

( p, w) x x( p, w)

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u C 1 X

max u(x) p x w,x1

0,

xL 0,

x( p, w) ( p, w) RL+1++

L 2 (a1, . . . , a L ) R L++ x R L+ i

x i /a i i

u(x) = min {x1/a 1, . . . , x L /a L}, x

( p, w) RL+1++ B ( p, w)x( p, w) = x

x1/a 1 = = xL /a L . min{x1/a 1, . . . , x L /a L} < max{x1/a 1, . . . , x L /a L}

u(x ) =min{x1/a 1, . . . , x L /a L} i x i /a i = max {x1/a 1, . . . , x L /a L}

px =w ( p, w)

x = a1w

Li=1 a i pi

, . . . , aL w

Li=1 a i pi

.

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( p, w) R L+1++ : x( p, w) = a1w

Li=1 a i pi

, . . . , aL w

Li=1 a i pi

,

( p, w) R L+1++ : x( p, w) = a1w

Li=1 a i pi

, . . . , aL w

Li=1 a i pi

.

( p, w) R L+1++ : v( p, w) = u a1w

Li=1 a i pi

, . . . , aL w

Li=1 a i pi

= w

Li=1 a i pi

.

X = R2+ p = (8 , 4) w = 40 X

w = (1 , 1)

X = R L+ L N X

x( p, w) ( p, w) R L+1++

( pn , wn , xn )n N RL+1++ X ( p, w, x ) R L+1++ X xn x( pn , wn ) n N x x( p, w)

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( p, w) R L+1++ , > 0 : x(p, w ) = x( p, w) u x( p, w)

( p, w) R L+1++ u x( p, w)

( p, w) RL+1++

p x = w ( p, w) R L+1++ x x( p, w)

( pn , wn , xn )n N RL+1++ X ( p, w, x) RL+1++ X xn x( pn , wn ) n N x x( p, w)

x X pn xn wn p x w x B ( p, w) x / x( p, w) y B( p, w) y x

U x x U y y y x (x , y ) U x

U y y U y p y < w y B ( p, w) p y w

y = y y y

( pn , wn ) ( p, w) pn y wn n y B( pn , wn )U y xn x xn U x n n xn U x y B ( pn , wn ) U y

y xn xn pn wn B (p,w ) = {x R L+ : (p) x w} = {x R L+ : p x w} = B ( p, w)

x( p, w) = x( p, w) = x x( p, w) x( p, w) = B( p, w) {x X : x x }

x, y

x( p, w) x

= y 1

2x + 1

2y

B ( p, w) B( p, w) x y B( p, w)

x x( p, w) p x w x B( p, w) p x < w > 0 {y X : x y < }

y y x x

X = R L+ L N u : X R

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( p, w) R L+1++ , x( p, w) =

( p, w) R L+1++ > 0 v(p, w ) = v( p, w) i v i

v v

v r R : {( p, w) R L+1++ : v( p, w) r} u v

( p, w) R L+1++ i {1, . . . , L } p p i B( p , w)

B( p, w)

v( p , w) = maxy B ( p ,w)

u(y) maxy B ( p,w) u(y) = v( p, w),

p R L++ 0 < w < w v( p, w) < v ( p, w ) x x( p, w) x B ( p, w) p x w < w p x < w > 0

{y X : x y < } B( p, w ) y y x

v( p, w) = u(x) < u (y) maxz B ( p,w ) u(z) = v( p, w ).

r R {( p, w) RL+1++ : v( p, w) r} = ( p, w), ( p , w ) [0, 1] ( p , w ) = ( p, w) + (1 )( p , w ) v( p , w ) r u(x) r x B ( p , w )

x B( p , w ) x R L+ ( p x) + (1 )( p x) w + (1 )w p x w p x w p x w x B( p, w) u(x) v( p, w) r

i

v

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u : R L+ R p R L++ u R

u

min p xx R L+ ,u(x) u. p R L++

u h( p, u)

h( p, u) = {x R L+ : u(x) u p x p y y R L+ u(y) u}.

e( p, u)

e( p, u) = minx R L+ ,u (x) u

p x = p x x h( p, u).

U = {u(x) :x R L+ } u


h( p, u) ( p, u)

R L++

U

( p, u) R L++ U, > 0 : h(p,u ) = h( p, u) h( p, u)

( p, u) R L++ U

h( p, u) ( p, u) R L++ U u(x) = u ( p, u) RL++ U u u(0, . . . , 0) x h( p, u)

p , p RL++

u U

x h( p , u)

x h( p , u) ( p p ) (x x ) 0

u < u (0 , . . . , 0) p x 0 x RL+ h ( p, u ) = {(0 , . . . , 0)} u

R L+

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( p, u) RL+ U u(y) = u y X {x X : u(x) u} = {x X : x y}

{x R L+ : u(x) u}{x R L+ : p x p y}

x px x (p ) x x p x

( p, u) RL++ U h( p, u) = h( p, u) = y h( p, u)

h( p, u) = {x R L+ : u(x) u} {x R L+ : p x p y} u u(0, . . . , 0) h( p, u) = {(0, . . . , 0)} u > u (0, . . . , 0) h( p, u) x, x

(x + x )/ 2 (x + x )/ 2 x (0, . . . , 0) (x + x )/ 2 = (0 , . . . , 0)

x ( p, u) x

( p, u) R L++ U u u(0, . . . , 0) x h( p, u) u = u(0, . . . , 0) h( p, u) = {(0, . . . , 0)} u(x) = u(0, . . . , 0) = u

u > u (0, . . . , 0) u(x) > u x = (0 , . . . , 0) x u(y) > u y x

lim 1 u(x ) = u(x) > u u(x ) > u (0, 1) p (x ) =( p x) 0 : e(p,u ) = e ( p, u)

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u p RL++ u , u U u(0, . . . , 0) u 0

e( p, u) = 0 u = 0 ,

p(u + 1)

u > 0. p > 0 e( p, ) u = 0

e : RL++ U R

( p0, u0) R L++ U y R L+ u(y) > u 0 ( p0, u0) p x x {z RL+ : u(z) u, p z p y}

U = R+ x ( p, u) RL++ U

u(x ) = u ( p, u) x = ( a1u , . . . , a L u)

h( p, u) = ( a1u , . . . , a L u) e( p, u) = p (a1u , . . . , a L u) = u Li=1 a i pi h( p, u) e( p, u)

u : RL+ R p

R L++ u > u (0, . . . , 0)

= 1 , . . . , L p

= 1 , . . . , L : h ( p, u) = e( p, u)

p.

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h() p RL++ u U x = h( p, u) p u p R L++ ,

e( p , u) = minx R L+ ,u (x ) u

p

x

p

x,

p = p f : RL++ R f ( p ) = e( p , u) p x p = p p

= 1 , . . . , L : f ( p)

p=

e( p, u)p x =

e( p, u)p h ( p, u) = 0 ,

u : RL+ R v()

( p, w) p RL++ w > 0 = 1, . . . , L = 1 , . . . , L : x ( p, w) =

v( p, w)/pv( p, w)/w

.

f : R L++ R f ( p ) = v( p , p x) x = x( p, w) p = p

u : RL+ R p R L++

x w > 0 x u = u(x ) p x = w :

x x( p, w) x h( p, u(x )) e( p, u(x )) = w. x u U u > u (0, . . . , 0) x

w = p x u :x h( p, u) x x( p, p x ) v( p, p x ) = u

x x( p, w) p x = w x p u(x ) x h( p, u(x ))

e( p, u(x )) = p x p x = w u(x) u(x ).

u e(, u ) {x X : u (x ) u }

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x B ( p, w) x x( p, w) u(x) = u(x )

e( p, u(x )) = p x = p x = w. x h( p, u(x )) e( p, u(x )) = w

x h( p, u)

u(x ) = u

x

p p x x x( p, p x ) v( p, p x ) = u(x) u(x ) = u p x p x .

x ( p, u) p x = p x

v( p, p x ) = u(x) = u(x ) = u. x x( p, p x ) v( p, p x ) = u

e( p, v( p, w)) = w

v( p, e( p, u)) = u x( p, w) = h( p, v( p, w)) h( p, u) = x( p, e( p, u))

x x( p, w) v( p, w) = u(x ) e( p, v( p, w)) = e( p, u(x )) = w

x( p, w) h( p, v( p, w)) x x( p, w) u(x) = v( p, w) x h( p, u(x)) = h( p, v( p, w)) h( p, v( p, w)) x( p, w)

x h( p, v( p, w)) x x( p, px) x h( p, v( p, w)) p

x = e( p, v( p, w)) = w x

x( p, p

x) = x( p, w)

v( p, w) = w

Li=1 a i pi

x( p, w) = a1w

Li=1 a i pi

, . . . , aL w

Li=1 a i pi

.

u = v( p, e( p, u)) = e( p,u )Li =1 a i pi

e( p, u) = u Li=1 a i pi

= 1 , . . . , L : h ( p, u) = e( p, u)

p= a u,

h( p, u)

h( p, u) = x( p, e( p, u)) = a1e( p, u)

Li=1 a i pi

, . . . , aL e( p, u)

Li=1 a i pi

= ( a1u , . . . , a L u).

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u : RL+

R

( p, w) RL +1++ u = v( p, w) > u (0, . . . , 0) k, {1, . . . , L } h ( p, u)

pk=

x ( p, w)pk

+ x ( p, w)

w xk ( p, w).

h ( p, u) = x ( p, e( p, u)) pk

B0 B1

B1 B0

B0 B1 B0 B1

( p0, w0) RL+1++ ( p1, w1) R L+1++

p0 = p1, w0 = w1 p0 = p1, w0 = w1 p0 = p1, w0 = w1

RL+ ( p0 , w0) ( p1 , w1) x0 x( p0 , w0) p1 x0 < w 1 ( p1 , w1)

( p0 , w0)

v v( p1, w1) > v ( p0, w0)

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p R L++ e( p, )

e( p, v( p1, w1)) > e ( p, v( p0, w0)) v( p1, w1) > v ( p0, w0).

e( p, v( p1, w1)) e( p, v( p0, w0))

( p0, w0) ( p1, w1)

( p, u) R L++ U y R L+ u(y) = u e( p, u) = min p x

x

R L

+u(x) u= min p x

x

R L

+x y

p p0 p1

u0 = v( p0, w0) u1 = v( p1, w1) e( p0, u0) = w0 e( p1, u1) = w1

EV (( p0, w0), ( p1, w1)) = e( p0, u1) e( p0, u0) = e( p0, u1) w0,CV (( p0, w0), ( p1, w1)) = e( p1, u1) e( p1, u0) = w1 e( p1, u0).

( p0, w0) ( p1, w1) ( p2, w2) EV (( p0, w0), ( p1, w1)) EV (( p0, w0), ( p2, w2))

p0 CV (( p0, w0), ( p1, w1)) p1 CV (( p0, w0), ( p2, w2))

p2

u0 = v( p0, w0) = w0

Li=1 a i p0i

u1 = v( p1, w1) = w1

Li=1 a i p1i

,

EV (( p0, w0), ( p1, w1)) = e( p0, u1) e( p0, u0) = w1Li =1 a i p0iLi =1 a i p

1i w0,

CV (( p0, w0), ( p1, w1)) = e( p1, u1) e( p1, u0) = w1 w0Li =1 a i p

1i

Li =1 a i p0i

.

( p0, w0) T (0, w0) ( p1, w1) = ( p0, w0T ) e( p0, u0) = e( p1, u0) = w0 e( p1, u1) = e( p0, u1) = w1 = w0T

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EV (( p0, w0), ( p1, w1)) = CV (( p0, w0), ( p1, w1)) = T T

T

R L+ p0 RL++ w > 0 t > 0

p1 = p0 + te e = (0 , . . . , 0, 1, 0, . . . , 0) RL 1 0 T = tx ( p1, w)

EV (( p0, w), ( p1, w)) = e( p0, u1) w 0, u1 = v( p1, w)

T T

x x B ( p0 + te , w) p0 x + tx w p0 x wtx =w

T x B( p0, w

T ) x

e( p0, u1) w T w T e( p0, u1) 0

u : R L+ R x R L+ u(x) = xa 11 xa 22 xa LL a1 , . . . , a L > 0

p p > 0 p p ( p , p )

( p0, w) {1, . . . , L } p1 = p0 ( p1, w) = (( p1, p0 ), w)

e( p, u)p

= h ( p, u) e( p1, u1) = w.

EV (( p0, w), ( p1, w)) = e( p0, u1) w= e( p0, u1) e( p1, u1)= p

0

p1

e( p , p0 , u1)

pdp

= p0

p1h ( p , p0 , u

1)dp .

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CV (( p0, w), ( p1, w)) = p0

p1h ( p , p0 , u

0)dp .

p0 > p 1 EV (( p0, w), ( p1, w)) CV (( p0, w), ( p1, w)) u0 = v( p0, w) u1 = v( p1, w) v p u0 u1 e u e( p , p0 , u0) e( p , p0 , u1)

p > 0 x ( p, e( p, u)) = h ( p, u)

h ( p , p0 , u0) = x ( p , p0 , e( p , p

0 , u

0)) x ( p , p0 , e( p , p0 , u1)) = h ( p , p0 , u1) p > 0

EV (( p0, w), ( p1, w)) CV (( p0, w), ( p1, w)) = p0

p1h ( p , p0 , u1)dp

p0

p1h ( p , p0 , u0)dp

= p0

p1h ( p , p0 , u

1) h ( p , p0 , u0) dp 0.

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y Y y y y Y y y y

y Y y = 0 y / Y

y Y [0, 1] y Y

y Y 1 y Y

y Y 0 y Y

y, y Y y + y Y y y y y

y + y

Y

y, y Y

[0, 1]

y + (1 )y Y Y y, y Y , 0 y + y Y

Y R L Y 0 Y Y

Y Y

Y Y

Y x, y Y (0, 1) z Y z x + (1 )y, z = x + (1 )y Y y Y [0, 1] y + (1 )0 = y Y

Y Y y Y 0 Y y = 12 y + 12 y Y Y

Y y, y Y , 0 2y Y 2y Y 12 (2y ) +

12 (2y ) = y + y Y

Y = = 1 Y

y, y Y , 0 ky Y ky Y k N k N /k 1 /k 1 Y (/k )y Y (/k )y Y y = k(/k )y Y y Y

y + y Y y Y, y = 0 y Y z Y z 12 y + 12 (y) = 0 , z = 0

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Y

Y

RL p

RL++

max p yy Y.

p R L++ ( p) = max { p y : y Y }.

y() p RL++

y( p) = {y Y : p y = ( p)}.

( p) = + Y R L

p R L++ p y 0 y Y ( p) = +

p R L++ p y > 0 y Y Y y Y 1 p(y ) = ( py)

Y R L

r R y r y Y {1, . . . , L } p R L++

p RL++ y Y P = Y

{y

R L : p

y

p

y

}P Y P P r

y P {1, . . . , L } p y p y

p y p y k=

pkyk p y k=

pkr,

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y p y k= pk r /p P Y

Y R L

RL+ y Y y

p

R L++

Y = Y {y R L : y } Y = Y {y R L : y } (yn )n N

Y yn zn = yn / yn n N zn Y zn + / yn 0

(zn )n N z = 0 Y

Y

y( p) = ( p) < Y R L

Y y( p) p R L++ p RL++ y( p) y

p ( p)/p = y = 1 , . . . , L

p, p R L++ y y( p) y y( p ) ( p p ) (y y ) 0.

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epi() = {( p, v) R L++ R : v ( p)} ( p1, v1), ( p2, v2) epi() [0, 1] v1 + (1 )v2 (p 1 + (1 ) p2)

y y(p 1 + (1 ) p2) pi y ( pi ) vi i = 1 , 2 (p 1 + (1 ) p2) = p1 y + (1 ) p2 y v 1 + (1 )v2.

p1, p2 RL++ [0, 1] : (p 1 + (1 ) p2) ( p1) + (1 )( p2) p1, p2 RL++ [0, 1] y y(p 1 + (1 ) p2) pi y ( pi ) i = 1 , 2

(p 1 + (1 ) p2) = p 1 y + (1 ) p2 y ( p1) + (1 )( p2). p RL++ y( p) = Y {y RL : p y = ( p)} Y

y( p) p

p RL++ : p y ( p ) p = p

h : R L++ R h( p ) = ( p ) p y p p

= 1 , . . . , L : h( p)/p = ( p)/p y = 0 ,

( p p ) (y y ) = ( p y p y ) + ( p y p y) 0, p

y = ( p)

p

y

p y = ( p ) p y

Y = {y R L : F (y) 0}, F p RL++

y Y

max p yF (y) 0 0 = 1 , . . . , L :

p = F (y )

y.

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c(w, q )

c(w, q ) = minz R L 1+ ,f (z) q

w z = w z z z(w, q ).

min p xx R L+u(x) u

f z 0

q f (z) 0 0

z

0 = 1 , . . . , L

1 :

w = f (z )

z+ z = 0 .

z > 0 = 0

w = f (z )

z

wwk

= f (z )/zf (z )/z k

,

f : RL 1+ R+ w R L 1++ p > 0

max pq w zq f (z),z R L 1+ .

y( p, w) ( p, w) (z, q ) p > 0 q = f (z)

pq w z < pf (z) w z.

maxz R L 1+

pf (z) w z.

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f : R L 1+ R + q 0 {z R L 1+ : f (z) q } w R L 1++ p > 0

max z R L 1+ pf (z) w z max q 0 pq c(w, q )

z R L 1+ q z 0 pf (z) w z pq z c(w, q z ) q 0 zq R L 1+ pf (zq) w zq pq c(w, q )

maxz R L 1+

pf (z) w z = maxq 0 pq c(w, q ).

q 0 q 0

p c(w, q )

q = q = 0 .

q > 0 = 0 q

f : R + R f (z) = z z 0

Y = {(z, q ) R 2 : q f (z), z 0} = {y R 2 : y1 0, y2 y1}. w > 0 p > 0

maxz 0

p z wz. z = 0 z > 0

p2 z w = 0 ,

z = p2w2

z = p2w p22w p

2

4w = p24w > 0

y( p, w) = p2w

2, p2w

Y

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( p, w) = p2

4w q

min wzz 0, z q.

z z = q z = z(w, q ) = q 2 c(w, q ) = wq 2

maxq 0

pq c(q, w) = maxq 0 pq wq 2. q = p2w

y Y y Y y y y = y

Y R L y Y p R L++ y y( p) y Y y Y p R L+

y p

y y Y y y, y = y p y > p y y y

Z = {y RL

: y > y }

y Z Y =

p RL , p = 0 p y p y y Z y Y p RL+ p < 0

p y 0 y p y Y p y p y

n N yn = ( y1 + 1 /n , . . . , y L + 1/n ) Z p yn p y yn y p y p y

Y R2 y Y p R2+ p = (0 , 0) y p y Y R2 y Y

p R 2++ Y R2 y Y

p R 2+

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E = ( h , h )h H H

h H h R L

+ L

N

h R L+ L = h H h x = ( xh )h H h H xh R L+ x

h H xh

h H xh =

p h H p h h x R L+ p x p h h Bh ( p, p h )

xh ()

E = ( h , h )h H ( p, x) p R L+ , p = 0 , x = ( xh )h H

h H xh p xh xh ( p, p h )

z p

z( p) =h H

xh ( p, p h ) {h} =h H

xh ( p, p h ) {}.

p z z( p)

z z( p) R L

h H, p R L+ , > 0 : Bh ( p, p h ) = B h (p, (p ) h ). p p > 0

= { p RL+ :

L=1 p = 1}

L

z z : R L

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S xh xh h S E

x xh h h H

( p, x)

E x

S H x (xh )h S xh h xh h S xh p xh B h ( p, p h ) p xh > p h h S

p h S xh > p h S h (xh )h S h S xh = h S h ( p, x)

E x

x

x xh h xh

h H

xk k xk k H p xh p xh = p h h H p xk > p xk = p k p h H xh > p h H h xh H x

h h H h

E

x h = xh h H x

( p, x) h H xh xh B h ( p, p xh ) xh xh

x xh h xh h H xh = xh h H h H xh = xh h

(xh + xh )/ 2 xh xh

E = ( h , h )h H , (Y f )f F , (hf )h H,f F ,

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H F f F Y f R L L N

h H h R L+

h R L+ L hf [0, 1] f F h H hf = 1 f F

(x, y) = (( xh )h H , (yf )f F ) h H xh R L+ f F yf Y f (x, y)

h H

xh h H

h +f F

yf .

p (yf )f F h H

x

R L+ : p

x

p

h +

f F

hf yf ,

p h h hf p yf f F

xh () h H yf () f F f ()

E ( p, x, y) p R L+ , p = 0 (x, y) = (( xh )h H , (yf )f F )

h

H xh p

xh xh p, p h +f F

hf yf ,

f F yf p yf yf ( p) f ( p) = p yf

z p

z( p) =h H

xh p, p h +f F

hf f ( p) f F

yf ( p) {},

p z( p) R L =

z Z R L z( p) Z p

z( p) = p

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z( p) p {( p, z) Z : z z( p)}

p z 0 p z z( p) p z( p) R L =

z p z p z

Z

(z) = { p : p z = max p p z}. p pz

z Z p0 (z) (z) = { p R L : p z = p0 z}

Z ( p, z) = (z) z( p)

z ( p, z) Z ( p, z) ( p, z) = (z)z( p)

z z( p) pz 0 p (z) pz p z p {1, . . . , L } p = e z = p z p z 0 z 0

E

Z Z z( p)

Z f F y

f + 0

x S = H x

E ( p, x) x

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E p z = 0 p z z( p)

p R L+ , z z( p) R L , {1, . . . , L} : p > 0 z = 0 . L 1

p R L++ , z z( p), {1, . . . , L } : zk = 0 k = z = 0.

h H

h X = R L+

h h

x, y R L+ : x R L++ , y / R L++ x h y,

h X = R L++

E (x, y ) (x, y) xh h xh h H

xh h xh h H

( p, x, y) E (x, y)

E

x [0, 1] uT : [0, 1] R uT (x) = x x {0, 1} uT (x) = 1

uL : [0, 1] R uL (x) = x (T , L ) {z R 2+ : z1 + z2 = 1}

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A = {a1, . . . , a n} pi

a i A g = ( p1 a1, , pn an ).

G1 = ( p1 a1, , pn an ) : p1, . . . , pn 0,n

i=1

pi = 1 .

H T A = {H, T } ( 12 H, 12 T )

( 12 a1, 12 an ) 12 a1, 0 a2, , 0 an 1,

12 an .

ai (1a i) ai

G0 = A m N Gm

G0, . . . , G m 1

Gm = ( p1 g1, , pk gk ) : k N , p1, . . . , pk 0,k

i=1 pi = 1 , g1, . . . , gk m 1=0 G .

G = m =0 Gm .

A A = {a1, a 2} g a1 1

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a1 a2 1 a1 + (1 ) a2

(1 )(1 ) g (( + (1 ) ) a1, (1 )(1 ) a2).

g G pi ai A g g ( p1 a1, , pn an ) G1

g

G

A =

{a1, . . . , a n

} g

G1

( p1, . . . , p n ) R n G1 n = { p Rn+ : i pi = 1} Rn

G1 G1

A

g G

( p1 a1, , pn an )

g g ( p1 a1, , pn an )

g G2562 g g

g

( g, (1 ) g )

( g , (1 ) g ), g 1

g g

g, g , g G (0, 1)g g ( g, (1 ) g ) ( g , (1 ) g ).

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G

g g G1 g G1 : g g g

g G

g [0, 1]

g ( g g, (1 g) g). k N p1, . . . , p k > 0 g1, . . . , gk , h1, . . . , h k G

gi hi i = 1 , . . . , k ( p1 g1, , pk gk) ( p1 h1, , pk hk).

g g ,

[0, 1] >

( g, (1 ) g) ( g, (1 ) g).

n g G gs G1 g gs g gs g g g g

p, p n g g

g p + (1 g) p g

g ( g g, (1 g) g) k N k = 1 k N , k 2

k k

( p1 g1, , pk gk ) ( p1 g1, (1 p1) ( p21 p1 g2, , pk1 p1 gk)) ( p

1 h

1, (1

p

1)

( p2

1 p1 h

2,

, pk

1 p1 h

k))

( p1 h1, , pk hk )

g g , [0, 1] > = 1 = 0 1 > > > 0

( g, (1 ) g) ( g, (1 ) g) g

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( g, (1 ) g) g. g

( g, (1 ) g) = g ( g, (1 ) g)

( g, (1 ) g) ( g, (1 ) g)

( g, (1 ) g) ( g, (1 ) g),

u : G R u : G R

G

g, h G : g h u(g) u(h),

g Gu(g) =

n

i=1

pi u(a i ),

( p1 a1, , pn an ) g

G

g g G1 g g

g g g G g [0, 1]

g ( g g, (1 g) g) u(g) = g.

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g, h G g h (g g, (1 g) g) ( h g, (1 h ) g)

u(g) = g h = u(h),

u g G gs = ( p1 a1, , pn an )

g g gs u(g) = u(gs ) ai A u(a i )

a i (u(a i ) g, (1 u(a i )) g). i = 1 , . . . , n hi = ( u(a i) g, (1 u(a i )) g)

gs = ( p1 a1, , pn an ) ( p1 h1, , pn hn ). h

1, . . . , h

n

g ( p1 h1, , pn hn )

n

i=1

piu(a i) g, 1 n

i=1

pi u(a i ) g .

g gs ( p1 h1, , pn hn ) n

i=1 piu(a i ) g, 1

n

i=1 pi u(a i ) g .

u(g) [0, 1]

g (u(g) g, (1 u(g)) g). u(g) = ni=1 pi u(a i )

G

u : G R a, b R a > 0 au + b v : G R

G a, b R a > 0 v = au + b

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g g a > 0 b

v(g) = au(g) + b,v(g) = au(g) + b.

g G g (u(g) g, (1 u(g)) g) u(g) = u(g)u(g) + (1 u(g))u(g),

v(g) = u(g)v(g) + (1 u(g))v(g)= u(g)[au (g) + b] + (1 u(g))[au (g) + b]= a[u(g)u(g) + (1 u(g))u(g)] + b= au(g) + b,

A

G = n =0 Gn {a1 , . . . , a k} R k 2 Gn n

G G

g G ( p1 a1 , , pk ak )

L(g) = {a i : pi pj j = 1 , . . . , k } |L(g)|

G g, h Gg h

1

|L(g)|a i L (g )

a i

1

|L(h)|a i L (h )

a i .

G g G n g Gn ( p1 a1 , , pk ak ) u(g) =

km =1 pm am n

G g G ( p1 a1 , , pk ak ) u(g) = i :a i > 5 pi

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A R

u

g = ( p1 w1, , pn wn ) u(g) = ni=1 pi u(wi )

A E (g) = ni=1 pi wi u(E (g)) = u( ni=1 pi wi )

g u(g) u (E (g)) . G

g A

A R u

u A

u A

u A.

( p1 w1, , pn wn ) u( p1 w1, , pn wn ) =

n

i=1

pi u(wi) 0

ni=1 pi = 1 :

ni=1 piu(wi) < u (

ni=1 pi wi )

g

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g CE (g) g CE (g)

u(g) = u(CE (g)) .

A R A g w A

g w g w A g w CE (g) A g CE (g)

CE (g)

g P (g) u(g) = u(E (g)P (g))

P (g) = E (g)

CE (g).

E (g) g

g

E (g)

u

CE (g) < E (g) g S P (g) > 0 g S

P (g) = E (g) CE (g)

g S u(g) < u (E (g)) CE (g) u(CE (g)) < u (E (g)) CE (g) < E (g) u

A = R ++ u(w) = ln( w) w A u w0 g

h (0, w0) :g = ((1 / 2) (w0 h) , (1/ 2) (w0 + h)) .

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E (g) = 12 (w0 h) + 12 (w0 + h) = w0 CE (g)

u(CE (g)) = u(g) = 12

ln(w0 h) + 12

ln(w0 + h) = ln w20 h2, CE (g) =

w20 h2 < w 0 = E (g)

P (g) = w0 w20 h2 > 0

u

w : u (w) > 0 u (w) < 0.

w

Ra (w) = u (w)u (w) .

u v

R1a (w) = u (w)u (w) > v (w)v (w) = R2a (w) w, P 1(g) u

P 2(g) v g S. Ra (w)

Ra () w

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w0 h :

g = ((1 / 2)

(w0

h) , (1/ 2)

(w0 + h)) .

E (g) = w0 P = P (g) > 0 g :

u(g) = 12

u(w0 h) + 12

u(w0 + h) = u(E (g) P ) = u(w0 P ). u(w0 P ) w0 :

u(w0 P ) u(w0) u (w0)P. u(w0 h) u(w0 + h) w0 :

u(w0 h) u(w0) u (w0)h + 12u (w0)h

2,

u(w0 + h) u(w0) + u (w0)h + 12

u (w0)h2.

12

u(w0 h) + 12

u(w0 + h) u(w0) + 12

u (w0)h2.

u(w0) + 12

u (w0)h2 u(w0) u (w0)P.

P 12

h2 u (w0)u (w0)

.

P h

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w0 > 0 u

w1 u(w0) < 12 (u(0)+ u(w1)) ( 12 0, 12 w1)

w1 ( 12 0, 12 w2)

w2 u(w1) < 12 (u(0) + u(w2))

g1 = (1 $1, 000, 000),g2 = ((0 .10) ($5, 000, 000), (0.89) ($1, 000, 000), (0.01) ($0)) ,g3 = ((0 .11) ($1, 000, 000), (0.89) ($0)) ,g4 = ((0 .10) ($5, 000, 000), (0.90) ($0)) .

g1 g2 g4 g3 u

g1 g2 u($1, 000, 000) > 0.10u($5, 000, 000) + 0 .89u($1, 000, 000) + 0 .01u($0).

g1 g2 0.11u($1, 000, 000) > 0.10u($5, 000, 000) + 0 .01u($0) 0.11u($1, 000, 000) + 0 .89u($0) > 0.10u($5, 000, 000) + 0 .90u($0) g3 g4,

g3 g4

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$1

3/ 4 (3/ 4)2 + (1 / 4)2 = 10 / 16

((10/ 16) $1, (6/ 16) $0),

((3/ 4) $1, (1/ 4) $0). 3/ 4 > 10/ 16

11 10

100 150

100 1, 500

100 1, 000, 000

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r > 0 1+ r

c = ( c0, c1, . . .) = ( ct )t=0 ct t

U

U (c) = (0)u(c0) + (1)u(c1) + (2)u(c2) + =

t=0 (t)u(ct ),

(0) = 1

ct t N u(ct ) (t) (0, 1) (0) = 1

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t =0 (t) 0 (1 + r ) 1

= (1 + r ) 1

(ct )t=0 (dt )t=0 c0 = d0

(c0, c1, c2, . . .) (d0, d1, d2, . . .) (c1, c2, . . . ) (d1, d2, . . . ).

(, , 0, 0, . . .)

u( ) = u( ) (t) = u( ) u( )u ( ) u( )

t

t t = 0 t N ( ) = u( ) u( )u ( ) u( )

< t

( 0, . . . , 0

t 1

, , , 0, 0, . . .) ( 0, . . . , 0

t 1

, , , 0, 0, . . .),

(t 1)(u() u( )) + (t)(u( ) u( )) = 0 ,

(t) = (t 1) u() u( )u( ) u( )

= u() u( )u( ) u( )

t

,

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t = 0 t = 1 t {0, 1}

t = 1 1/ 2 1/ 2

t S = {h,d,D } h d D A = {

k,

} k

u : S A R

u(s, a ) =

1 (s, a ) = ( h, ),

1 (s, a ) = ( h, k ), (s, a ) = ( d, ),0

> 0 D

t = 0 0 t = 1 0 < < 1

t = 1 t = 1

t = 1

t = 0

t = 1 t = 0 t = 1

(, ) , (0, 1) (0) = 1 , (1) = , (2) = 2, (3) = 3, . . .

U (c) = u(c0) +

t=1 t u(ct ).

u u(0) = 0

u(1) > u (2).

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k 3 22k 2 2

2k 1

2/ 3

3 22k 1

1/ 2 1/ 3

1/ 3 2/ 3

(x t )t=0 t st = inf {xs : s t} t

(s t )t=0 t (s t )t=0

(x t )t=0

lim inf t

x t = limt (inf {xs : s t}) . lim inf t x t = (x t )t=0

lim inf t x t = + (x t )t=0

c R liminf t x t = c > 0 T N c < x t t T > 0 T N t

T x t < c +

c c + > 0

x = ( x t )t=0 y = ( yt )t=0 R x y x L y

lim inf T 1T

T 1

t=0(x t yt ) > 0.

> 0 x y

1T

T 1

t=0(x t yt ) > T .

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U

U (1, x t ) U (0, x t ) t, x t ,U (1, x t ) > U (0, x t ) t, x t .

(0, 0, . . .) (1, 1, . . .) U

T N

x

U (x) =3 xt = 1 t 0 xt = 0 t

t x t 2 t 0 1 t

2 t > 0

U (0, 0, . . .) = 3 = maxx

U (x) > minx

U (x) = 0 = U (1, 1, . . .),

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S a A S \ {a}

a T b T P {a,b}(a) = 0 a T

T A a T b T P {a,b}(a) = 0 P T (S ) = P T \{ a}(S \{a})

S T S = T \ {a} P T (T \ {a}) = P T \{ a}(T \ {a}) = 1 P T (a) = 0

a a S T

a T S S a

S T A a S P {a,b}(a) / {0, 1} b T P T (a) = P T (S )P S (a).

T A P {a,b}(a) / {0, 1} a, b T , a = b

P {a,b}(a) / {0, 1} a, b A u : A R ++

P S (a) = u(a)b S u(b)

S A u

P A(a) > 0 a A P A(a) = 0 a A

b A \ {a} : 0 = P A(a) = P A({a, b})P {a,b}(a). P {a,b}(a)

= 0 P A(

{a, b

}) = P A(a) + P A(b) = 0 b

A

\ {a

}

b A : P A(b) = 0 , b A P A(b) = 1 P A(a) > 0 a A u(a) = P A(a)

S A :P S (a) =

P A(a)P A(S )

= P A(a)

b S P A(b) =

u(a)

b S u(b).

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P A( ) = P A( ) P A( ) = P A( )

P A( ) = P A( = P A( ) = 1/ 3.

1/ 2 1/ 3

A = {1, . . . , n } i A (i) > 0 i A

P A(i) = e (i)/

j A e ( j )/ =

exp((i)/ )

j A exp(( j )/ ).

i A u(i) = exp( (i)/ ) > 0

> 0

n = p R n+ :n

i=1

pi = 1 .

A

i j (i) ( j ) arg max i A (i)

p n c( p) R .

p n

ni=1 pi(i) > 0 c( p)

p

max p n

n

i=1 pi(i) c( p).

c( p) =n

i=1

pi ln ( pi ) ,

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0ln0 = 0 (1/n , . . . , 1/n )

max p n

n

i=1

pi(i) c( p),

i A : pi = exp((i)/ )

j A exp(( j )/ ),

c ni=1 pi (i) c( p)

p n pi > 0 i R ni=1 pi = 1

i = 1 , . . . , n : (i) (ln pi + 1) + = 0 , p ni=1 pi (i) c( p) i

(i) c( p)p i

= (i) (ln pi + 1) .

i = 1 , . . . , n pi = c exp((i)/ ), c = exp(( )/ )

n j =1 p j = 1

i = 1 , . . . , n : pi = exp((i)/ )

j A exp(( j )/ ),

0

i, j A, i = j

P A(i)P A( j )

= exp((i)/ )exp( ( j )/ )

= exp(i) ( j )

,

1/n

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A = {1, 2}, (1) = 4 , (2) = 0 > 0

0

A = {1, 2, 3}, (1) = 0 , (2) = 2 , (3) = 8 . > 0

> 0

R n c : Rn R+ r : {1, . . . , n } {1, . . . , n } x R n

c(x1 , . . . , x n ) = c(x r (1) , . . . , x r (n ) ) A = {1, . . . , n } n 2

: A R 0

P ( ) : max p n

n

i =1

pi (i) c( p(1/n , . . . , 1/n )) .

p P ( )

(i) > ( j )

pi pj

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X |X | A B A B

A B A = B A BN =

{1, 2, 3, . . .

}Z = {. . . , 2, 1, 0, 1, 2, . . .}Q = { p/q : p, q Z , q = 0}

R L N :

R L R L+ = {x R L : x1, . . . , x L 0} R L R L++ = {x R L : x1, . . . , x L > 0}

Q L++ x, y R L x y = x1y1 + + xL yL

x

y xi

yi i = 1 , . . . , L ,

x > y xi > y i i = 1 , . . . , L .

< k {1, . . . , L } ek RL k k

ek = (0 , . . . , 0, 1

k , 0, . . . , 0).

e = (1 , . . . , 1) R L

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t

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k x y z(k 1) z(k) k x = z(0) z(L) = y

R2+

x, y R 2+ : x y xk > yk k {1, 2} k k = 1 k = 2

(2, 2) (1, 1) (0, . . . , 0) R L+ (0, . . . , 0) x

x R L+ x = (0 , . . . , 0)

x R L > 0 y = x 2 e1 RL x y = 2 < y x

y x

R 2+

x y (x1 + 1)( x2 + 1) (y1 + 1)( y2 + 1) x = (1 , 2)

y = (2 , 1) (x1 + 1)( x2 + 1) = ( y1 + 1)( y2 + 1) = 6 , x y.

x y > 0

(x1 + + 1)( x2 + 1) = 3(2 + ) > 2(3 + ) = ( y1 + + 1)( y2 + 1) , x + e1 y + e1.

R2+ x y x, y

R 2+

x y 4x1 + 3 x22 4y1 + 3 y22 (1, 0) (0, 1)

4 1 + 3 02 > 4 0 + 3 12 2(1, 0) 2(0, 1) 4 2 + 3 02 < 4 0 + 3 22

x, y R L+ x y n N xn = x + (1 /n , . . . , 1/n ) R L+ xn > y xn y xn y n limn xn = x y

x, y R L+ x > y min{x1, x2} > min{y1, y2} x y y x x y x = (2 , 1) y = (1 , 1) x x y x y

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n N f (nu ) = nf (u) n f (0) = f (0 + 0) = f (0) + f (0) f (0) = 0 f (0u) = 0 f (u)

n N f (nu ) = nf (u) 0 = f (0) = f (nu + ( nu )) = f (nu ) + f (nu ) f (nu ) = f (nu ) = nf (u)

f (xu ) = xf (u) x

Z

x Q x = p/q p, q Z , q = 0 xu = ( p/q )u q (xu ) = pu f (q (xu )) = f ( pu) qf (xu ) = pf (u) f (xu ) = ( p/q )f (u) = xf (u)

f x, y R \{0} f (x)/x = f (y)/y a = ( x, f (x)) b = ( y, f (y)) a + b , R R2

a + b , Q R2 f , Q

(x + y,f (x + y )) = ( x + y,f (x ) + f (y )) = ( x + y,f (x) + f (y)) = a + b.

i

{1, . . . , n

} f i : R

R

x i R : f i (x i ) = F (x i ei ). F (n 1) x R n

F (x) = F n

i=1

x iei =n

i=1

F (x i ei ) =n

i=1

f i (x i ).

f i xi , yi R F f i (x i + yi ) = F (x i ei + yi ei ) = F (x i ei ) + F (yiei) = f i (x i ) + f i (yi).

f X

x, y X : x y f (x) f (y) x X

L(x) = {y X : y x} = {y X : f (y) < f (x)} = f 1((, f (x))) X

f

x, y X : x y f (x) f (y).

(X, B , C ) X A, B B x, y A B x C (A) y C (B ) x C (B ) = {z B : z z z B}

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y A x C (A) = {z A : z z z A} x y z B y C (B ) y z x y x z x z z B x C (B )

X =

{a,b,c,d

},

B =

{{a,b,c

},

{b,c,d

}}, C (

{a,b,c

}) =

{b

}, C (

{b,c,d

}) =

{c

}.

A, B B A B b c c

b b C ({b,c,d})

C ({a,b,c}) = {b} b c b a C ({b,c,d}) = {c} c b c d b c c b a

c a c y y {a,b,c} c C ({a,b,c}) X = {a,b,c}, B = {{a, b}, {b, c}, {a, c}}, C ({a, b}) =

{a}, C ({b, c}) = {b}, C ({a, c}) = {c} a

b, b

c, c a

A, B B x, y A B x C (A) y C (B )x C (B )

x = y C

B B : B C (B )

v(x) < r v(y) < r x C (A) x A y

A y

x y

C (B ) x

y

x = y C (B ) v(x) r v(y) r x C (A) x

A y A x y y C (B ) y x x = y C (B )

B B x (B ) B C C (B ) = B \ {x (B )} B B

B = X, A = C (B ) B A B C (B ) A = C (B ) = C (A) = C (B ) A = C (B ) C (A) = A \ {x (A)} A = C (B )

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B1 p1 = 2 w = 2 C (B1) = {(1, 0)} C (B2) = {(0, 1)}

x = (1 , 0) y = (0 , 1) B1 B2 x C (B1) y C (B2) x C (B2)

C (B1) = {x}

x y

C (B2) = {y}

y x u(x, p) =

x1 p1 > p 2,x2 p2 > p 1,x1x2 p1 = p2.

u y X

{x X : x y} = {x X : u(x) u(y)} = u 1

([u(y), )) u

{x X : x y} x, y R L+ x y i {1, . . . , L }

u(x) = min {x1/a 1, . . . , x L /a L} = xi /a i x y u(x) = xi /a i yi /a i min{y1/a 1, . . . , yL /a L} = u(y), x y.

x > y x y u(0, . . . , 0) = u(1, 0, . . . , 0) = 0 ,

y R L+ u(y) =

{x R L+ : x y} = {x R L+ : min{x1/a 1, . . . , x L /a L} }= L=1 {x R L+ : x /a }

x = ( a1 +1 , a 2, . . . , a L ), y = ( a1, . . . , a L ) (0, 1)

x y x + (1 )y,

u

B ( p, w) Z 2+

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B ( p, w) {x R 2+ : x1 3} {x R2+ : p1x1 + 4 min {x2, 5}+ 2 max {x2 5, 0} w}

p2 = 4 2 {x R 2+ : x2 5, p x w} {x R 2+ : x2 > 5, 8x1 + 2 x2 40}

{x R 2+ : p x p }

B ( p, w) {x R2+ : x1 = x2} B ( p, w) {x R2+ : x1 1/p 1, p1(x1 1/p 1) + p2x2 w}

1/p 1

(30/ 8, 5) u(x) = min {4x1, 3x2}

( p, w)

R L+1

++ > 0

x x( p, w) x x( p, w) z B( p, w) z x y := (1 / )z B( p, w) x x( p, w) x y x y = z z x

u(x) = x1 + x2 p1 > p 2 v( p, w) = w/p 2

p1 > p2 p1

( pn , wn )n N RL+1++ ( p, w)

RL+1++ v( pn , wn )

v( p, w) n N xn x( pn , wn ) xn B ( pn , wn ) n ( pn , wn ) ( p, w) (xn )n N

B ( p, w + 1) (xn )n N x X

( pn , wn , xn )n N x x( p, w) limn v( pn , wn ) = lim n u(xn ) = u(x) u

U : R +

R U (x) =

min{x, 1} u : R + R

u(x) = x x 1,

2 x > 1.

x( p, w) = {w/p } w p,[1,w/p ] w > p.

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h ( p, u) = x ( p, e( p, u)) pk

h ( p, u)pk

= x ( p, e( p, u))

pk+

x ( p, e( p, u))w

e( p, u)pk

.

e ( p,u )p k = hk( p, u)

h ( p, u)pk

= x ( p, e( p, u))

pk+

x ( p, e( p, u))w

hk( p, u).

u = v( p, w) e( p, u) = e( p, v( p, w)) = w u = v( p, w) h( p, u) = h( p, v( p, w)) = x( p, w)

h ( p, u)pk

= x ( p, w)

pk+

x ( p, w)w

xk ( p, w).

= (1 , 1)

p1 x0 < w 1 x0 B ( p1, w1) x0 p1 y w1 y x y y

x0

A = Li=1 a i

x( p, w) =a1A

w p1

,..., aLA

w pL

,

v( p, w) =wA

A L

i=1

a i pi

a i,

h( p, u) = u1/AL

i=1

pia i

a i /A a1 p1

, . . . , aL pL

,

e( p, u) = Au1/AL

i=1

pia i

ai/A

,

EV (( p0, w0), ( p1, w1)) = A(u1)1/AL

i=1

p0ia i

a i /A

w0,

CV (( p0, w0), ( p1, w1)) = w1 A(u0)1/AL

i=1

p1ia i

a i /A

.

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1 = 2 = (1 , 1) ( p, x) = ((1 , 1), (1, 1), (1, 1))

p RL+ , z z( p) R L p z = k: pk > 0 pkzk = 0 z = 0 p > 0 p, z, zk = 0 k = p z = p z = 0 p > 0 z = 0

{1, . . . , L } h h

h RL++

( p, x, y) E (x, y) (x, y)

h H :xh h xh p xh p xh = p h + f F hf yf ,xh h xh p xh > p xh = p h + f F hf yf .

h

H

h H (yf )f F p

p h H

xh > p h H

xh

=h H

p h +f F

hf yf

= p + p f F

yf

p + p f F

yf .

p h H xh > p + p f F yf (x, y)

E = ( 1, 2, 1, 2)

1

2 R 2+ 1 = (1 , 0) 2 = (0 , 1)

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p 1 ( p2/p 1, 0)

1 u1(x) = max

{min

{2x1, x2

}, min

{x1, 2x2

}} 2 u2(x) = x1 + x2 1 = 2 = (1 , 1)

p1 > p 2 2x1 = x2 ( p 1/ ( p1 + 2 p2), 2 p 1/ ( p1 + 2 p2))

(0, p 2/p 2)

p2 > p1

p = (1 / 2, 1/ 2) {(2/ 3, 4/ 3), (4/ 3, 2/ 3)} {x R2+ : x1 + x2 =2} ((2/ 3, 4/ 3), (4/ 3, 2/ 3))

((4/ 3, 2/ 3), (2/ 3, 4/ 3))

1, 2 1 = 2 = (1 , 1) ( p, x) p x = ( x1, x2) R2+ R2+ xh B h ( p, p h ) h = 1 , 2

{0}

R 2

{(xT , xL ) R 2+ : xT + xL 1}

xT / (0, 1) 0

(0, 1) (1, 0) (T , L )

x = ( xT , xL )

xL

< L

xL

L

T = 0 xT {0, 1} T (0, 1)

T = 1 xT = 1

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xT (0, 1) xT {0, 1}

{(1, 0)} (T , L ) = (1 , 0)

{(0, x) : L

x 1}

T

(0, 1)

{(0, 1)} (T , L ) = (0 , 1) T (0, 1)

(T , L )

p > 0 0 T [0, 1) 1 T = 1

L

{( p, xT , xL ) R 3 : p > 0, xT = 0 , xL = L} T [0, 1) {( p, xT , xL ) R 3 : p > 0, xT = 1 , xL = 0} T = 1 .

G max{a1, . . . , a k} G

min{a1, . . . , a k} u(g) = 1|L(g)| a i L(g) a i

a1 > a 2 a1 ( pa1, (1 p) a2) p > 1/ 2 a1 a1 ( 12 a1, 12 a2) p = 1 / 2

a 1 + a 22 < a 1

a1 u(g) = u(gs )

a1 > a 2 g = a1 g = a2

( g, (1 ) a1) ( g , (1 ) a1) (0, 1) a1

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G max{a1, . . . , a k} G g G ( 12 g, 12 g)

g

G1 u(g) = km =1 pm am

1

a1 G0 ( 12 a1, 12 a1) G1 (1 a1) G0 G1

g = a1 G0,g = ( 12 a1, 12 a1) G1,g = ( 12 g , 12 g ) G2. (0, 1)

(

g, (1

)

g ), (

g , (1

)

g )

G3.

u(g) = a1 0,u(g ) = a1 1,

u( g, (1 ) g ) = a1 3,u( g , (1 ) g ) = a1 3,

G

min{a1, . . . , a k} 5 < max{a1, . . . , a k} G am > 5

G am 5

ak ak

G

i = 1 , . . . , k u(a i ) = 0 ai 5 u(a i ) = 1 g G ( p1 a1, , pk ak ) u(g) = i:a i > 5 pi =k

i=1 piu(a i)

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u (u(ct ))t=0 u(ct ) > 1/ (t) t (t)u(ct ) > 1 t

t=0 t u(ct )

u B

R c = ( ct )t=0

t | (t)u(ct )| B (t) t=0 B (t) = B t=0 (t)

t=0 (t)u(ct )

k u(h, k ) = 1 u(h, ) = 1

k u(d, k) = 0 u(d, ) = k

k u(d, k) + 0 = 0 u(d, ) + 12 (u(h, ) + u(d, k)) = + 12 (1 + 0) = 12 k 1

2

< 0,

k 12 = 0 ,

12 > 0.

12 > 0

u(1) > (1 + ) / u(2).

(1 + 366 ) / u(2) > (1 + 365) / u(1) .

11 +

/ 0 limt inf {xs : s t} = c T N c < inf {xs : s t} < c + ,

t T t = T c < inf {xs : s T },

c < x t

t T T N t = max {T, T }inf {xs : s max{T, T }} < c + ,

t T x t < c + lim inf t x t = c > 0 T N

c / 2 < x t t T

c < c / 2 inf {xs : s T }. T t T

c < inf {xs : s t}. t T s t

xs < c + / 2 < c + ,

inf {xs : s t} c + / 2 < c + . > 0 T N

c < inf {xs : s t} < c + , lim inf t x t = c

(x t )t=0

lim inf t

x t > 0 > 0 : xt > t.

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liminf t x t > 0 inf {xs : s t} T N inf {xs : s T } 1 xt 1 t T = c/ 2 T N xt > c = c/ 2

t T > 0 xt > t T N

x t >

t T

inf {xs : s t}

t T

x = ( x t )t=0 x0 = 1 t N xt =(t + 1) 2 t 1k=0 xk 1T T 1t=0 xt = T x = ( x t )t=0 y = ( yt )t=0

liminf T

1T

T 1

t=0(x t yt ) > 0 liminf T

1T

T 1

t=0x t > liminf

T

1T

T 1

t=0yt .

x = (0 , 0, . . .) z = ( zt )t=0 liminf t zt = limsup t zt

limsup t zt = lim t (sup{zs : s t})

lim supT

1T

T 1

t=0yt < 0 liminf T

1T

T 1

t=0yt < 0.

1/ 2 1/ 3 1/ 2 = 1/ 6 < 0 2/ 3 1/ 2 = 1/ 6 > 0

c ni=1 pi (i) 12 c( p) i

(i) 12

c( p)p i

= (i) 12

2 pi 1n

= (i) 1

pi 1n

.

p i

0

pi 0 R ni=1 pi = 1 i = 1 , . . . , n :

(i) 1

pi 1n

+ i + = 0 i pi = 0 .

i = 1 , . . . , n : pi = (i) + ( i + ) + 1n

.

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p pi > 0 i = 0

j A pi p j = (i) + +

1n ( j ) + ( j + ) +

1n

= ((i) ( j )) j ((i) ( j )) ,

> 0 j 0 p

pi > 0 p j > 0

pi p j = ((i) ( j )) ,

1

pi 1n (i) = 1 p j 1n ( j ). i {1, . . . , n } pi > 0

= 1

pi 1n (i) R ,

= 1

p j 1n ( j )

j p j > 0 k :

k = 0 pk > 0,

1 pk 1n (k) pk = 0 .

k 0 pk = 0 j p j > 0

p j pk (( j ) (k)) ,

(( j ) (k)) 1

p j pk 0.

k = 1

pk

1n (k)

= 1

pk

1n (k)

1

p j 1n

+ ( j )

= ( ( j ) (k)) 1

p j pk 0,

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1 2

P A(1) P A(2) = ((1) (2)) = 4 . P A(1) + P A(2) = 1

P A(1) = 4 + 1

2 , P A(2) =

14 2

.

1/ 4 (0, 1/ 4] > 1/ 4

P A(1) = 1 , P A(2) = 0 .

> 0 i A

P A(i) = exp((i)/ )

j A exp(( j )/ ).

P A(1) = exp(0/ )

exp(0/ ) + exp(2 / ) + exp(8 / )

= 1

1 + exp(2 / ) + exp(8 / ),

P A(2) = exp(2/ )

1 + exp(2 / ) + exp(8 / ),

P A(3) = exp(8/ )

1 + exp(2 / ) + exp(8 / ).

(0, 1)

1/ 3

P A(i) i A > 0

P A(i) > 0 P A(i) P A( j ) ((i) ( j )) j A

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(3) > (2) > (1)

P A(i) > 0 i A P A(3), P A(2) > 0, P A(1) = 0 P A(3) > 0, P A(2) = P A(1) = 0

P A(3) = 1

P A(3) P A(2) = ((3) (2)) = 6 ,P A(3) P A(1) = ((3) (1)) = 8 ,

P A(1) + P A(2) + P A(3) = 1 .

P A(2) = P A(3) 6,P A(1) = P A(3)

8,

3P A(3) 14 = 1 .

P A(3) = 1+14 3 ,P A(2) = 1+14 3 6 = 1 43 ,P A(1) = 1+14 3 8 = 1 103 .

(0, 1/ 10) (0, 1/ 10)

P A(3)

P A(2) = ((3)

(2)) = 6 ,

P A(3) P A(1) = P A(3) ((3) (1)) = 8 ,P A(1) = 0 ,

P A(1) + P A(2) + P A(3) = 1 .

P A(2) = P A(3) 6,P A(1) = 0 ,P A(3) 8,

2P A(3) 6 = 1 .

P A(3) = 1+6 2 ,P A(2) = 1+6 2 6 = 1 63 ,P A(1) = 0 .

P A(2) P A(3)

P A(3) = 1 + 6

2 8,

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