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n. xi i- x = (x1, x2,..., xn).

M Rn x M = { = (x1, x2,..., xn) |xi 0, i = 1,..., n}.

M x y y,, y x x M.

y x y,, y x x M.

y, x y,, y x x M.

y y y x x x, M ;

x x x M y, y y x z x z x z, M x x x M y, y y x z x z, M x z y y y x x x, M x x, x x.

y, y y x z x z, M.

y y y) x x z z ( ) (y y y y y x z z x ( ) ( ), x z z x x z,.

y y x x, M y y y y, y x..

x x x x M Ix x x Ix = {y M |y }.

Px x M x Px = {y M |y }, Nx, x M y} x Nx = {y M |.

Px Nx M x M.

Px Nx = Ix.

y x, M y y;

x x ( xi yi i = 1,..., n) y y y x x x =.

x y, y x.

y x, M y [0; 1] +(1 - ) y y x x x x, +(1 - ) xi + (1 - )yi i- x M p = (p1, p2,..., pn), pi i- p, x n p x = pixi, i=1 x.

b. Bp,b p b x x Bp,b = { M |p b}.

y p p y x x b,, Bp,b, [0; 1]. b, p p y p( +(1 - y)) b, +(1 - ) y x +(1 - ) b. x x Bp,b, Bp,b.

x p0 = min pi. Bp,b, i i = 1, 2,..., n xi b/p0, Bp,b.

xk Bp,b p p x0 x0. xk.

p p xk b, k, x0 b, x0 Bp,b, Bp,b.

u : M, R y;

x x u( ) u(y) y;

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

M x u( ) x f(u) f(u( )) x 1. u( ) u x ( ) > 0 (i = 1, 2,..., n), (1) xi u u u x x x x grad u( ) = ( ), ( ),..., ( ) x1 x2 xn u x ( ) x u x ( ) i- xi 2. i- u x lim ( ) =. (2) xi0 xi 3. i- u x lim ( ) = 0. (3) xi xi x 4. u( ) x u( ) 2u 2u 2u x x x ( ) ( )... ( ) x2 x1x2 x1xn 1 2u 2u 2u x x x ( ) ( )... ( ) 2u x2x1 x2 x2xn x 2 U = ( ) = x2...

2u 2u 2u x x x ( ) ( )... ( ) xnx1 xnx2 xn 2u x ( ) < 0. (4) xi n i x u( ) = aix, ai > 0, 0 < i < 1; (5) i i=n n i x u( ) = x, i > 0, i < 1; (6) i i=1 i=n x u( ) = ai ln (1 + bixi), ai > 0, bi > 0; (7) i=-/ n x u( ) = ix-, i > 0, 0 < < 1, -1 < = 0; (8) i i=xi x u( ) = min, ki > 0. (9) i ki n n i x u( ) = x, i > 0, i < i i=1 i= 1 - 4.

1.

x u( ) u j n j i i x ( ) = jx -1 x = x, i = 1, 2,..., n.

j i xj xj i=1 i i =j i > 0, xi > 0 i = 1, 2,..., n, u 1 n 2 n n n i i i x ( ) = x, x,..., x > 0.

x x1 i=1 i x2 i=1 i xn i=1 i 2. j = 1, 2,..., n u j n j i i x lim ( ) = lim x = lim jx -1 x =, j i xj0 xj0 xjxj xj i=1 i i =j n i < 1, j - 1 < 0.

i=3. j = 1, 2,..., n u j i x lim ( ) = lim jx -1 x = 0, j i xj xj xj i =j n i < 1, j - 1 < 0.

i=4.

n 2u j(j - 1) i x ( ) = x < 0, i x2 xj j i= j - 1 < 0 i = 1, 2,..., n.

n 1(1 - 1) 12 n 1n n i i i x x... x i x2 x1x2 i=1 i x1xn i=1 i i= n 21 n 2(2 - 1) 2n n i i i x x... x i 2u x2x1 i=1 i x2 x2xn i=1 i x U = ( ) = i=x...

n n1 n n2 n n(n - 1) i i i x x... x i xnx1 i=1 i xnx2 i=1 i xn i= x, x u( ) = const, n u du = dxi = 0. (10) xi i=x u( ).

u u u u x x x x ( ) = ( ), ( ),..., ( ).

x x1 x2 xn u(x1, x2) = x1x2. MN.

x0 = (x0, x0). x1 x x1 = (x1, x1) = (x0 + x1, x0 + x2) 1 2 1 x0.

|x2/x1| dx2 u/x- =.

dx1 u/xdx M1 = - dx Mik i- k- dxk u/xi Mik = - =. (11) dxi u/xk k Ei xi xk xk/xk k Ei = lim -, xixi/xi Mik k Ei =. (12) xk/xi xi k Ei xk.

Mik, i, k = 1, 2;

k Ei, i, k = 1, 2.

Mik i k u/xi i n k n i xk j j Mik = = x : x = .

u/xk xi j=1 j xk j=1 j k xi k Ei i k xk i xk xi i k Ei = Mik : = : =.

xi k xi xk k xi xk, Mik Ix y y x Ix. Ix = Iy.

p x x u( ) = = p1x1 + p2x2 + + pnxn.

x u( ) = min{2x1, x2}.

b, x Bp,b, x x x, Bp,b, x, Bp,b, x max u( ). (1) x x Bp,b x = (x, x,..., x), 1 2 n x u( ) x Ip,b, x x Ip,b = { M |p = b}. (2) Ip,b Bp,b x u( ) x u( ) Bp,b x x Ip,b. x Ip,b, Bp,b, / x p x < b, p y, b - x y > 0.

y p y y z x = +. b, Bp,b, x u(y) > u( ), x.

x u( ) y) x x u( +(1 - ) > u( ) + (1 - )u(y), 0 < < 1.

x u( ) y x, x Bp,b, u( ) = u(y), p p y x = = b.

p p(1 +1 y) = b, z x z x z = +1 y, = 2 2 2 Ip,b.

1 z x x u( ) > u( ) + u(y) = u( ) = u(y), 2 y x,.

x u( ) max. (3) p x =b x u( ).

L(x1, x2,..., xn, ) = u(x1, x2,..., xn) - (p1x1 + p2x2 + + pnxn - b).

x (, ) L x (, ) L L u = - pi = 0, i = 1, 2,..., n, xi xi (4) n L = pixi - b = 0.

i= (n + 1) (n + 1) n u u x x ( ) : ( ) = pi : pj, i, j = 1, 2,..., n. (5) xi xj (u/xi) : (u/xj) i- x j-, u u x x Mij = ( ) : ( ) = pi : pj, i, j = 1, 2,..., n.

xi xj, u x = ( ) :, xi pi i, i, u x = ( ), (6) b x = (x, x) 1 u(x1, x2) p1x1 + p2x2 = b,, u u ptg = - (x, x) : (x, x) = -.

x1 1 2 x2 1 2 p u - pi = 0, i = 1, 2,..., n, (7) xi p1x1 + p2x2 + + pnxn = b x, x,..., x, p1, p2,..., 1 2 n pn, b, 2u 2u 2u x x x ( ) ( )... ( ) -p x2 x1x2 x1xn 2u 2u 2u x x x ( ) ( )... ( ) -p p x2x1 x2 x2xn U J = =...

p 2u 2u 2u x x x ( ) ( )... ( ) -pn xnx1 xnx2 xn p1 p2... pn p x U u( ), p J U J J- q V J-1 =.

q - q V n n, = (q1, q2,..., qn) q, V, JJ-1 = In+1, In+ (n+1)(n+1). In nn, n = (0, 0,..., 0) n p q U - V In n JJ-1 = =.

p q 0 - n p q UV + = In, q p U - = n, p V = n, p q = 1, p q), V = U-1(In q p = U-1, (8) = 1/(p U-1 p ), U-1 U.

x x = (p, b), (9) = (p, b), x (p, B) x = x(p1, p2,..., pn, b), i = 1, 2,..., n.

i i p1 = 1/p x = x(1, p2/p1,..., pn/p1, b/p1), i = i i 1, 2,..., n p2/p1,..., pn/p1 b/p1.

n 1/ pi.

i= x1, x2, u(x1, x2) = x1x2, (p = p1, p2), b x2 : x1 = p1 : p2, x2p2 = x1p1, p1x1 + p2x = b, p1x1 = p2x = b/2.

b b x1(p1, b) =, x1(p2, b) =.

2p1 2p x1x2 max.

x1+2x2= x1x2 max.

p1x1+p2x2=b u(x1, x2) = u1(x1) + u2(x2).

ai i- p = (p1, p2,..., pn) b n b0 = piai, i=a = (a1, a2,..., an).

i > 0 i- n i u(x) = (xi - ai), ai, i > 0, i = 1,..., n. (1) i=n i u(x) = (xi - ai) max, (2) p x =b i= u(x) u j n j i = (xi - ai) = u(x).

xj xj - aj i=1 xj - aj j u(x) - pj = 0, j = 1, 2,..., n, xj - aj (3) p1x1 + p2x2 + + pnxn = b.

xj pi b.

n xj, j u(x) xj = aj + , j = 1, 2,..., n. (4) pj pj j = 1, 2,..., n, n n n pjxj - pjaj - u(x) j = 0, j=1 j=1 j=n pjxj = b, j=n b - pjaj u(x) j==. (5) n j j=u(x) n j b - piai i=xj = aj +, j = 1, 2,..., n. (6) n pj i i=n aj, b0 = b - piai, i=n j- j/ i.

i= j- j- i = j, i, j = 1, 2,..., n, ai = 0, b xi(p, b) =. (7) npi i- j- xi xi 0, i = j, j Ei = : = -1, i = j, pj pj i- i- xi xi 1 b j Ei = : = : = 1, b b npi b npi i- u(x1, x2) = xx- (x1 + - )- max, (10) 1 p1x1+p2x2=b, 1 u(x1, x2) 1 u(x1, x2) = , (11) p1 x1 p2 x p1x1 + p2x2 = b.

u(x1, x2) u(x1, x2) - = - u(x); = u(x) x1 x1 x1 + - x2 x x1 x2, 1 1 - - = , p1 x1 x1 + - p2 x(I) x2 = (b - p1x1), pb x1(p1, b) =. (12) b + p, p b 0 x1 = -, b + px1 =.

x2 x1, 1 1 - - = , p1 x1 x1 + - p2 x(II) x1 = (b - p2x2), pb (b + p1( - )) x2(p1, p2, b) = (13) p1.

p2 (b + ) 0, 0 b p1( - ), x2(p1, p2, b) = (14) b (b + p1( - )) p1, p1( - ) < b.

p2 (b + ) b b + y(b) = b , 2(b + 3) 1 = p1( - ), 2 = p2, 3 = p1, + b +, y = kb + m, y(b) b + 1 k = lim = lim = > 5;

b+ b+ b 2(b + 3) b + 1 m = lim [y(b) - kb] = lim b - b = b+ b+ 2(b + 3) b b + 1 1 - 3 b 1 - = lim - 1 = lim = < 0, b+ 2 2(b + 2) b+ 2 b + 3 p1 - 3 p1 - p1 - p = = - < 0.

2 p2 p x2(b) 1 p x(b) = b -.

p2 p u(x) u(tx) = tu(x) t > 0. (8) t t = 1, n u(x) xi = u(x). (9) xi i= x = (x1, x2,..., xn).

x (p, b) (p, b) u x ( (p, b)) - pi(p, b) = 0, i = 1, 2,..., n, xi (1) p1x(p, B) + p2x(p, b) + + pnx(p, b) = b.

1 2 n b, n x 2u j x ( (p, b)) (p, b) - pi (p, b) = 0, i = 1, 2,..., n, xixj b b j=(2) n x pj j = 1.

b j=x j, b b x p U - = 0, b b (3) x p = 1.

b J, J- x (p, b) = U-1 p, b (4) (p, b) = .

b x b.

du/db u 2u = = = (x(p, b)).

b b b b k pk, n x 2u j x ( (p, b)) (p, b) - pi (p, b) = (p, b)ik, i = 1, 2,..., n, xixj pk pk j=(5) n x pj j = -x(p, b), k pk j= ik x p U - = In, p p (6) x p = - x (p, b).

p J- x p p x = U-1 In - U-1 - U-1p, (7) p n x(p, b) i, pk i,k= xi(p, b) pk.

b(p), x du( (p)) = 0, n x u( (p)) x i = 0. (8) xi pk i= (x(p), (p)) x u( (p)) - pi(p) = 0, i = 1, 2,..., n. (9) xi x u( (p)) xi n x i pi(p) = 0. (10) pk i= (p) pk n x i pi = 0, (11) pk i= pk, (k = 1, 2,..., n), n x x j 2u( ) - = ik, i, k = 1, 2,..., n, (12) xixj pk pk i= ik- x p = In, U p p x(p) p = 0, p p U - In =. (13) p 0 J-1, x p p = U-1 In - U-1. (14) p comp x k p comp dpk db = xdpk.

k b(p), x du( (p)) = 0, n x u( (p)) x i = 0. (8) xi pk i= (x(p), (p)) x u( (p)) - pi(p) = 0, i = 1, 2,..., n. (9) xi x u( (p)) xi n x i pi(p) = 0. (10) pk i= (p) pk n x i pi = 0, (11) pk i= pk, (k = 1, 2,..., n), n x x j 2u( ) - = ik, i, k = 1, 2,..., n. (12) xixj pk pk i= x p = In, U p p x(p) p = 0, p p U - In =. (13) p 0 J-1, x p p = U-1 In - U-1. (14) p comp x k p comp dpk db = xdpk.

k n pixi = b b, i=n xi(p, b) pi = 1. (16) b i=xi(p, b) > 0, b xi(p, b) < 0, b u(x) xi(p, b) < 0, pi F (x1, x2) = 0; F (x1, x2,..., xn, y) = 0; F (x1, x2,..., xn; y1,... ym) = 0.

x1x2 max, 10x1+2xxx max.

1 p1x1+p2x2b x x u( ) =.

k p =.

k p p p H = U-1 In - U-1, > 0, (1) > x p H z Rn z z (2) H 0, z p z.

= R z p z = 0.

= p)H( p) = 2 (p U-1 p - p p p p ).

z z U-1 U-H = ( p p = 1/ U-p z z 2(p U-1 p - U-1 p ) =, H = n- p z.

= R p z +, v = v v U-1 p = 0, (3) p)U-1 p = U-1 p - p p = 0.

z ( - z U-p p = 1/, = U-1 p.

U-1 z z z v v v v - U-1 p p U-1.

H = H = U-1 v v v v U-1, U, U-1 < 0, z z H < 0.

x(p) i < 0. (4) pi z i- z = (0, 0,..., 1,..., 0), z z H = hii < 0, H xi(p, b) x(p) xi i = - xi. (5) pi pi b xi > 0, pj x i < 0.

pj x i > 0.

pj u(x) p > 0, x (p) p = U-1 p -U-1 p p p = 0, U-p p p = 1/ U- i = 1, 2,..., n n x(p) i pj = 0, (6) pj j=x(p) i j = i > 0, pj i j i j xi(p, b) xi j Ei = :, (7) pj pj xi(p, b) xi Ei = :. (8) b b n j Ei = 0. (9) j=n n 1 xi xi j Ei = pj + b. (10) xi j=0 pj b j=xi x xi i = - xj, j = 1, 2,..., n.

pj pj b pj, n pjxj = b, j=n n xi x xi n xi i pj = pj - xjpj = - b.

pj j=0 pj b b j=0 j= (x1, x2) (p1, p2), b x1 = (x1, x1) L1 I1.

1 p q I2 0x x, I2 L2.

I0 I = tg -q1/p2. I0 L x0.

I0, L1, x1 x x0 x x1 x x2 x x2 x x0 x 2 b x1(p1, b) = .

3 px.

p1 comp u(x1, x2) = x1x2 max 10x1+2x ars xr-xs




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