} : g(x) 0, x X. { f(x) h(x) g : X R m

Transcripción

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5 { f(x) } : g(x) 0, x X X R n f, h : X R g : X R m f, h, g

6 f/h f, h, g

7 { f(x) } : g(x) 0, x X X R n f, h : X R g : X R m f, h, g

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11 f/h f h f/h f h

12 m A = {a 1, a 2,, a m } n B = {b 1, b 2,, b n } P = (p ij ) p ij = b i a j ) i = 1 n, j = 1 m. p ij 0 p ij = 1 i j P t j > 0 a j x = (x j ) x j 0 x j = 1 T (x) = x j p ij log(p ij / x k p ik ) i j k t j x j j C T (x) j {T (x) : x j 0, j x j = 1} G = (E, V ) E = {x 1, x 2,, x n } n V = {e 1, e 2,, e m } m x 1 x n x i x j c ij t ij x ij 1 (x i, x j ) n m c ij x ij i=1 j=1 n m t ij x ij i=1 j=1 x 1j = 1, j i x ij = j j x ji x in = 1 i

13 n m x ij i j b ij c ij m m n m (b ij c ij )x ij i=1 j=1 n m x ij i=1 j=1 x ij a i j i ( ) P r x ij r j 1 l j j i x ij 0 i, j b ij N(µ b,ij, σ 2 b,ij ) c ij N(µ c,ij, σ 2 c,ij) a i i r j l j V X = (x 1, x 2,, x k ) Y = (y 1, y 2,, y k ) µ X = E(X), µ Y = E(Y ) Z Z = (z 1, z 2,, z k ) X Y L(Z) = a Z a a µ X a µ Y { (a µ X a µ Y ) 2 a a V a : a R n }

14 a = λv 1 (µ X µ Y ), λ 0 λx 1 + (1 λ)x 2 A λ [0, 1] A R n x 1, x 2 A f : A R n R A f A x 1, x 2 A x λ := λx 1 + (1 λ)x 2 λ [0, 1] f(x λ ) λf(x 1 ) + (1 λ)f(x 2 ) f : A R n R A f A x 1, x 2 A, x 1 x 2 x λ := λx 1 + (1 λ)x 2 λ (0, 1) f(x λ ) < λf(x 1 ) + (1 λ)f(x 2 ) {f(x) : x X} x x X β ϵ (x) f(x) f(x) x X β ϵ (x) x x X f(x) f(x) x X f X

15 f X f X f X, x x X f : X R n R X f(λx 1 + (1 λ)x 2 ) {f(x 1 ), f(x 2 )} λ [0, 1], x 1, x 2 X f : X R n R X f(λx 1 + (1 λ)x 2 ) < {f(x 1 ), f(x 2 )} λ (0, 1), x 1, x 2 X f(x 1 ) f(x 2 ) f : X R n R X f X f(λx 1 + (1 λ)x 2 ) < {f(x 1 ), f(x 2 )} λ (0, 1), x 1, x 2 X f X f X f X f X Estrictamente convexa Convexa F uertemente cuasiconvexa Explcitamente cuasiconvexa Cuasiconvexa Estrictamente cuasiconvexa f : A R n R A f A x 1, x 2 A x λ := λx 1 + (1 λ)x 2 λ [0, 1] f(x λ ) λf(x 1 ) + (1 λ)f(x 2 ) f X f X

16 f : X R n R, X x X f(x) f(x) = f(x) + f(x)(x x) + α(x, x x) x x α(x, x x) x x 0 f : X R n R, X x X f(x) H(x) f(x) = f(x) + f(x)(x x) (x x) H(x)(x x) + α(x, x x) x x α(x, x x) x x 0 f : X R n R, X f X f X H(x) x X H(x) X f X f(x)(x x) 0 x X X f(x) = 0 f X x X f f X x X f f : X R n R, X f X X x, x X f(x)(x x) 0 f(x) f(x) f : X R n R, X f X X x x X f(x)(x x) 0 f(x) > f(x) f X f(x) = 0 x

17 Estrictamente convexa Estrictamente pseudoconvexa P seudoconvexa Convexa F uertemente cuasiconvexa Explcitamente cuasiconvexa f/h f h f/h f h R e x2 > 0 e x2 R f(x) = x x > 0 = x x > 0 x f h (f(x 2 ) f(x 1 ))(h(x 1 ) h(x 2 )) 0 x 1, x 2 X f h x 1, x 2 X λ [0, 1] f h X : f(λx 1 + (1 λ)x 2 )h(λx 1 + (1 λ)x 2 ) λf(x 1 )h(x 1 ) + (1 λ)f(x 1 )h(x 1 ) f h f(λx 1 + (1 λ)x 2 )h(λx 1 + (1 λ)x 2 ) (λf(x 1 ) + (1 λ)f(x 2 ))(λh(x 1 ) + (1 λ)h(x 2 )) = λf(x 1 )h(x 1 ) + (1 λ)f(x 2 )h(x 2 ) + λ(1 λ)((f(x 2 ) f(x 1 ))(h(x 1 ) h(x 2 ))) λf(x 1 )h(x 1 ) + (1 λ)f(x 2 )h(x 2 )

18 f, h (f(x 2 ) f(x 1 ))(h(x 1 ) h(x 2 )) 0 x 1, x 2 X h 1/h 1 1/h : x 1, x 2 X, λ [0, 1] 1 h(λx 1 +(1 λ)x 2 ) h(x 1 ) h(x 2 ) h(λx 1 + (1 λ)x 2 ) λh(x 1 ) + (1 λ)h(x 2 ) 1 h(λx 1 +(1 λ)x 2 ) 1 λh(x 1 )+(1 λ)h(x 2 ) 1 1 λh(x 1 )+(1 λ)h(x 2 ) h(x 1 ) h(x 2 ) h 1 λ 1 + (1 λ) 1 λh(x 1 )+(1 λ)h(x 2 ) h(x 1 ) h(x 2 ) h(x 1 )h(x 2 ) (λh(x 1 ) + (1 λ)h(x 2 ))(λh(x 2 ) + (1 λ)h(x 1 )) 0 λ(1 λ)(h(x 1 ) h(x 2 )) 2 1 h(λx 1 +(1 λ)x 2 ) λ 1 h(x 1 ) + (1 λ) 1 h(x 2 ) X R n f, h : X R f X h X (f(x 2 ) f(x 1 ))(h(x 1 ) h(x 2 )) 0 x 1, x 2 X f/h X f 1 h f/h X X f, h : X R 0 X f(x) = f(x) f(x) 2

19 f(x) f(x) f(x) f(x) 2 (x x) = 0 x x x x f(x) f(x) x x 1 x x f(x) f(x) 2 (x x) = x x f(x) f(x) f(x)(x x) x x f(x) (x x) x x 2 x x + f(x) x x (1 ) f h f(x) f(x) f(x) f(x) 2 (x x) x x x x = 0 A R m R k x A x 1 = (x 1, x 2,, x m ) R m y 1 = (y 1, y 2,, y k ) R k x = (x 1, y 1 ) a A, b A, a b a i b i i {1, 2,, m, m + 1,, m + k} R m R k x 1 x 2 x 1 i x 2 i i {1, 2,, m} ϕ : R m R k R {x 1 x 2, y 2 y 1 } ϕ(x 1, y 1 ) ϕ(x 2, y 2 ) x 1 x 2 ϕ(x 1, y) ϕ(x 2, y) y R k x 2 x 1 ϕ(x 1, y) ϕ(x 2, y) y R k X R n f : X R m, g : X R k, ϕ : R m R k R θ : X R θ(x) = ϕ(f(x), g(x)) ϕ R m R k θ X X f g X ϕ R m R k θ X ϕ R m R k θ X f X g X ϕ R m R k

20 f X g X ϕ ϕ R m R k θ X x 1, x 2 X, λ (0, 1), x 1 x 2, (f(x 1 ), g(x 1 )) (f(x 2 ), g(x 2 )) x λ = λx 1 + (1 λ)x 2 θ(x λ ) = ϕ(f(x λ ), g(x λ )) ϕ(λf(x 1 )+(1 λ)f(x 2 ), λg(x 1 )+(1 λ)g(x 2 )) = ϕ(λ(f(x 1 ), g(x 1 ))+ (1 λ)(f(x 2 ), g(x 2 ))) < ϕ λϕ(f(x 1), g(x 1 )) + (1 λ)ϕ(f(x 2 ), g(x 2 )) = λθ(x 1 ) + (1 λ)θ(x 2 ) ϕ θ x 1, x 2 X θ(x 2 ) (x 1 x 2 ) 0 θ(x 2 ) (x 1 x 2 ) = ϕ(f(x 2 ), g(x 2 )) (x 1 x 2 ) = 1 ϕ(f(x 2 ), g(x 2 )) f(x 2 )(x 1 x 2 ) + 2 ϕ(f(x 2 ), g(x 2 )) g(x 2 )(x 1 x 2 ) 0 1 ϕ(f(x 2 ), g(x 2 )) f(x 2 )(x 1 x 2 ) + 2 ϕ(f(x 2 ), g(x 2 )) g(x 2 )(x 1 x 2 ) f f(x 2 )(x 1 x 2 ) f(x 1 ) f(x 2 ) ϕ 1 ϕ(f(x 2 ), g(x 2 )) f(x 2 )(x 1 x 2 ) 1 ϕ(f(x 2 ), g(x 2 )) (f(x 1 ) f(x 2 )) g ϕ 1 ϕ(f(x 2 ), g(x 2 )) g(x 2 )(x 1 x 2 ) 1 ϕ(f(x 2 ), g(x 2 )) (g(x 1 ) g(x 2 )) 0 1 ϕ(f(x 2 ), g(x 2 )) f(x 2 )(x 1 x 2 ) + 2 ϕ(f(x 2 ), g(x 2 )) g(x 2 )(x 1 x 2 ) 1 ϕ(f(x 2 ), g(x 2 )) (f(x 1 ) f(x 2 )) + 1 ϕ(f(x 2 ), g(x 2 )) (g(x 1 ) g(x 2 )) = ϕ(f(x 2 ), g(x 2 )) ((f(x 1 ), g(x 1 )) (f(x 2 ), g(x 2 ))) ϕ ϕ(f(x 1 ), g(x 1 )) > ϕ(f(x 2 ), g(x 2 )) θ(x 1 ) > θ(x 2 )

21 ϕ R m R k θ X x 1, x 2 X, x 1 x 2 λ (0, 1) x λ = λx 1 + (1 λ)x 2 θ(x 1 ) < θ(x 2 ) θ(x λ ) < θ(x 2 ) ϕ(f(x 1 ), g(x 1 )) < ϕ(f(x 2 ), g(x 2 )) ϕ ϕ(λ(f(x 1 ), g(x 1 )) + (1 λ)(f(x 2 ), g(x 2 ))) < ϕ(f(x 2 ), g(x 2 )) ϕ(λf(x 1 ) + (1 λ)f(x 2 ), λg(x 1 ) + (1 λ)g(x 2 )) < ϕ(f(x 2 ), g(x 2 )) f ϕ g ϕ ϕ(f(x λ ), g(x λ )) ϕ(λf(x 1 ) + (1 λ)f(x 2 ), λg(x 1 ) + (1 λ)g(x 2 )) < ϕ(f(x 2 ), g(x 2 )) θ(x λ ) < θ(x 2 ) X R n f : X R m, g : X R k, ϕ : R m R k R θ : X R θ(x) = ϕ(f(x), g(x)) ϕ R m R k θ X X f g X ϕ R m R k θ X ϕ R m R k θ X f X g X ϕ R m R k

22 X R n f : X R, h : X R 0 x X f h f/h X > 0 < 0 > 0 < 0 > 0 < 0 f h f/h X > 0 < 0 > 0 < 0 > 0 < 0 f h f/h X > 0 < 0 > 0 0 < 0 > 0 < 0 f h f/h X > 0 < 0 > 0 0 < 0 > 0 < 0 ϕ : R R R ϕ(x, y) = x y

23 θ(x) = ϕ(f(x), ) = f(x) h > 0 ϕ D = {(x, y) R R : y > 0} ϕ ϕ f/h { } f(x) : g(x) 0, x X > 0 x X R n z 0 = 1/, z = xz 0 x X W = {(z 0, z) R R n : z 0 > 0, z/z 0 X} {z 0 f(z/z 0 ) : g(z/z 0 ) 0, z 0 h(z/z 0 ) = 1, (z 0, z) W } X f, g X h X W (z 1 0, z 1 ), (z 2 0, z 2 ) W, λ [0, 1] (z λ 0, z λ ) = λ(z 1 0, z 1 ) + (1 λ)(z 2 0, z 2 ) z λ 0 = λz (1 λ)z 2 0 > 0 z λ z λ 0 = λ = λz1 +(1 λ)z 2 λz 1 0 +(1 λ)z2 0 z0 1 z 1 λz0 1+(1 λ)z2 0 z0 1 z = λ 1 z + (1 λ) 2 λz0 1+(1 λ)z2 0 λz0 1+(1 λ)z2 0 + (1 λ) z0 2 z 2 λz0 1+(1 λ)z2 0 z0 2 z 0 f(z/z 0 ) X X z = λ 1 z λz z + (1 λ) 2 z +(1 λ)z2 0 z0 1 λz (1 λ)z2 0 z0 2

24 (z0, 1 z 1 ), (z0, 2 z 2 ) W, λ [0, 1] ( ) ( (λz0+(1 λ)z 1 0)f 2 λz 1 +(1 λ)z 2 = (λz λz 0+(1 λ)z 1 0)f 2 λ 0 1+(1 λ)z2 0 ( (λz0 1 + (1 λ)z 2 λz0 0) 1 λz 0 1+(1 λ)z2 0 (1 λ)z 2 0f(z 2 /z 2 0) z0 1 λz0 1+(1 λ)z2 0 z 1 z 1 0 f(z 1 /z 1 0) + (1 λ)z2 0 λz 1 0 +(1 λ)z2 0 f(z 2 /z 2 0) z0 2 λz0 1+(1 λ)z2 0 + (1 λ) ) = λz0f(z 1 1 /z0) 1 + h g ) z 2 z0 2 X f, h, g X{f(x) : x X} < 0 {f(x) : x X} < 0 {z 0 f(z/z 0 ) : g(z/z 0 ) 0, z 0 h(z/z 0 ) 1, (z 0, z) W } z 0 h(z/z 0 ) 1 f(x) 0 { f(x) f(x) f(x) h(z/z 0 ) x X f(x) 0 h > 0 X } : g(x) 0, x X < 0 f(x) < 0 z 0 h(z/z 0 ) 1

25 X R n f, h : X R g : X R m > 0 x X { f(x) } : g(x) 0, x X S = {x X : g(x) 0} f, h, S x S h = 1 f, h g f, h, g f, h, g f, h g i

26 x X, u R m, u 0 K(x, u) = f(x) + ug(x) x X, r 0 R, r R m f(x), (r 0, r) 0, (r 0, r) 0 F(x, r 0, r) = r 0 + rg(x) (x, u) x X u 0 K(x, u) K(x, u) K(x, u) x X, u 0 (x, r 0, r) x X, (r 0, r) 0, (r 0, r) 0 F(x, r 0, r) F(x, r 0, r) F(x, r 0, r) x X, r 0 (x, u) (x, r 0, r) r 0 > 0 ] (x, u) (x, 1, u) ] (x, r 0, r) r 0 > 0 (x, r r 0 ) (x, u) x ug(x) = 0 (x, u)

27 f(x) + ug(x) (1) f(x) + ug(x) (2) f(x) + ug(x) x X, u 0 u = 0 ug(x) 0 u g(x) 0 x ug(x) 0 ug(x) 0 ug(x) = 0 f(x) f(x) + ug(x) x S ug(x) 0 f(x) f(x) f(x) + ug(x) x X x f(x) = x 1+x 2 x 1 X = {(x +1 1, x 2 ) R 2 : x 1 0, x 2 0} g(x) = x x 2 S = {(0, 0)} g(x) = 0 (x, u) x x = (0, 0) f(x) + ug(x) (1) f(x) + ug(x) (2) f(x) + ug(x) x X, u 0 0 x 1+x 2 x u(x x 2 ) (x 1, x 2 ) 0, u 0 x 2 = 0 { } 0 x 1 x 1 +1 u(x2 1) x 1 0, u 0 { 1 u x x 1 (x 1 +1) 1 0 } S X g(x) 0 r 0 > 0

28 f f h f h h f h X f, g, h X {f(x) : x X} < 0 x (x, r 0, r) rg(x) = 0 x (z 0, z) z 0 = 1 z = z 0 x (x, r 0, r, s) ( ) ( ) ( ( )) z z z r 0 z 0 f z 0 + rg z 0 + s 1 z 0 h z 0 ( ) ( ) ( ( )) z z z r 0 z 0 f z 0 + rg z 0 + s 1 z 0 h z 0 ( ) ( ) ( ( )) z z z r 0 z 0 f z 0 + rg z 0 + s 1 z 0 h z 0 x X, (r 0, r) 0, (r 0, r, s) 0 r 0 f(x) + rg(x) r 0 f(x) + rg(x) r 0 f(x) + rg(x), x X, (r 0, r) 0, (r 0, r) 0 (x, r 0, r) rg(x) = 0 r = 0 0 rg(x) x r rg(x) 0 rg(x) = 0 r 0 > 0

29 r 0 > 0 (x, r 0, r) x X : g( x) < 0 p 0, p 0 : pg(x) 0 x X x 1, x 2 λ (0, 1) : g(λx 1 + (1 λ)x 2 ) < λg(x 1 ) + (1 λ)g(x 2 ) g(x) < 0 X g X p 0, p 0 : pg(x) 0 x X p 0, p 0 : pg(x) 0 x X x 1, x 2 λ (0, 1) : g(λx 1 + (1 λ)x 2 ) < λg(x 1 ) + (1 λ)g(x 2 ) 0 x = λx 1 + (1 λ)x 2 X (x, r 0, r) r 0 = 0 rg(x) rg(x) rg(x) x X, r 0, r 0 rg(x) 0 rg(x) x X, r 0, r 0 X f, g, h X {f(x) : x X} < 0 x (x, u) ug(x) = 0

30 f, h, g X (x, u) x K(x, u) = 0, x X, g(x) 0, u R m, u 0, ug(x) = 0 f(x) + m i=1 u i g i (x) = 0 x X, g(x) 0, u 0 m u i g i (x) = 0 i=1 (x, r 0, r) x F(x, r 0, r) = 0, x X, g(x) 0, (x, r 0, r) 0, (x, r 0, r) 0, rg(x) = 0 r 0 f(x) + m i=1 r i g i (x) = 0 x X, g(x) 0, (r 0, r) 0, (r 0, r) 0 m r i g i (x) = 0 i=1 x d f(x) d < 0 f(x)d < f(x) d x d x + λd S 0 < λ < λ 0 x d = k λ k (x k x) {x k } X, {x k } x

31 F 0 x D x T x I = {i : g i (x) = 0, x X} G 0 = {d : g i (x)d < 0 i I} G = {d : g i (x)d 0 i I} x F 0 D = d F 0 D f(x) d < 0 0 < λ < λ 0 x + λd X f(x+λd) = f(x) h(x+λd) f(x+λd) h(x+λd) + λ f(x) d }{{} < 0 < f(x) +α(x, λd) λd α(x, λd) 0 λ 0 0 < λ x F 0 T = d T, d = λ k (x k x), λ k > 0, {x k } x f/g X f(x k) = f(x) + f(x) (x h(x k ) k x) + α(x, x k x) x k x ϵ > 0 k x x B ϵ (x) f(x) f(x k) h(x k ) f(x) (x k x) + α(x, x k x) x k x 0 λ k k f(x) λ k(x k x) 0 d / F 0 x (X) F 0 G 0 = x g i x i I x d G 0 d / F 0 x (X) x + λd X 0 < λ < λ 0 d G 0 g i (x)d < 0 i I g i (x + λd) < g i (x) = 0 0 < λ < λ 1 g i x i / I g i (x) < 0 g i (x + λd) 0 < λ < λ 2 0 < λ {λ, λ 1, λ 2 } x + λd S d D F 0 G 0 = f/h X

32 X f, h X{f(x) : x X} < 0 g i X (x, u) x W g i (z 0, z, u, v) (z 0, z) x x f, h, g i x i I x i / I x (X) r 0 0, r i 0 i I r 0 f(x) + r i g i (x) = 0 i I F 0 G 0 = { f(x) d < 0 g i (x)d < 0 i I r 0 0, r i 0 i I r 0 f(x) + i I r i g i (x) = 0 x f, h, g i x, i I g i x x (X) (r 0, r) 0, (r 0, r) 0 (x, r 0, r) r 0 0, r i 0, i I r 0 f(x) + i I r i g i (x) = 0

33 r i = 0 i / I x f, h X f, h, g i x i I g i x x (X) { g i (x) : i I} u i 0, i I f(x) + i I u i g i (x) = 0 (r 0, r) 0, (r 0, r) 0 r 0 f(x) + i I r i g i (x) = 0 u i = r i r 0 r 0 = 0 i I r i g i (x) = 0 r i (x, u) u i = 0 i / I f h } x 1 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 { (x1 3) 2 +(x 2 2) 2 X = R 2 f(x 1, x 2 ) = (x 1 3) 2 + (x 2 2) 2 h(x 1, x 2 ) = x g 1 (x 1, x 2 ) = x x g 2 (x 1, x 2 ) = x 1 + 2x 2 4 (x, u) x x 1 x x u x x x 1 + 2u 2 x 2 = 0 2x u x u 2 = 0 x 1 0, x 2 0 x x x 1 + 2x u 1 0, u 2 0 u 1 (x x 2 2 5) + u 2 (x 1 + 2x 2 4) = 0

34 x 1 3 x 2 g 1 y g 2 u x 1 0, x 2 0 x x = 0 x 1 + 2x 2 4 = 0 x = (2, 1) u 1, u 2 u 1 0, u 2 0 4u 1 + 2u = 0 u 1 + 2u = 0 u = ( 2, ) ((2, 1), (2/27, 8/27)) x 2/3 2 1 S S R n f, h : S R > 0 S

35 { } f(x) : x S (P P ) q {f(x) q : x S} F (q) := {f(x) q : x S} F (q) F (q) R F (q) R F (q) R F (q) = 0 q, p R, λ [0, 1] F (λq + (1 λ)p) = {f(x) (λq + (1 λ)p) : x S} = = {λf(x) + (1 λ)f(x) λq (1 λ)p : x S} = = {λ(f(x) q) + (1 λ)(f(x) p) : x S} {λ(f(x) q)} + {(1 λ)(f(x) p) : x S} = = λ {f(x) q} + (1 λ) {f(x) p : x S} = λf (q) + (1 λ)f (p) G(q, x) : R S R G(q, x) := f(x) q F (q) = x {G(q, x) : x S} G(q, x) f, h a, b R (, a), (b, ) R F ((, a)) 1 F ((b, )) 1 R F ((, a)) 1 π 1 : R S R π 1 (q, x) = q G((, a)) 1 = {(q, x) : G(q, x) < a} = {(q, x) : F (q) < a} π 1 G((, a)) 1 = {q : F (q) < a} = F ((, a)) 1 F ((, a)) 1 F ((b, )) 1 F ((b, )) 1 (F ((b, )) 1 ) C = {q : F (q) b} = {q : x f(x) q b} = π 1 G((, b]) 1 G S π : R S R (F ((b, )) 1 ) C q 1 > q 2 F (q 2 ) = {f(x) q 2 : x S} = f(x 2 ) q 2 h(x 2 ) > f(x 2 ) q 1 h(x 2 ) {f(x) q 1 : x S} = F (q 1 )

36 F F (q) = {f(x) q : x S} = f(x q) qh(x q ) q q q f(x f) qh(x h ) = q f(x f ) = {f(x) : x S}, h(x h ) = { : x S} F (q) = {f(x) q : x S} f(x) q = x S q q q F (q) = 0 F q = f(x) = { f(x) : x S } F (q) = 0 ] x f(x) f(x) x S q f(x) x S f(x) q 0 x S q = f(x) f(x) q = 0 f(x) q f(x) q x S F (q) = {f(x) q : x S} = f(x) q = 0 ] F (q) = 0 {f(x) q : x S} = f(x) q = 0 f(x) q x S f(x) f(x) x S q f(x) f(x) q 0 x S F (q) f, h S q F (q) = {f(x) q} = 0 x x q F (q) { } (x1 3) 2 +(x 2 2) 2 x 1 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 x = (2, 1) F (q) = {f(x) q : x S} q n+1 = q n q 0 = 0 F (q 0 ) = F (0) = {f(x) : x S} = qn q n 1 F (q n) F (q n 1 ) F (q n)

37 = {(x 1 3) 2 + (x 2 2) 2 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 S} ((2, 1), (1/3, 2/3)) x = (2, 1) F (0) = 2 q 1 = 1 F (q 1 ) = F (1) = {f(x) : x S} = {(x 1 3) 2 + (x 2 2) 2 x 1 1 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 S} ((2, 1), (2/3, 1/3)) (2, 1) F (1) = 1 q 2 = q 1 q 1 q 0 F (q F (q 1 ) F (q 0 ) 1) = 1 1( 1) = F ( ) { 2 3 = f(x) 2 : x S} = 3 { (x 1 3) 2 (x 2 2) 2 2x 1+2 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 S } ((2, 1), (5/9, 4/9)) F ( 2 3) = 0 x = (2, 1) q = 2 = f(x) 3 F q Ax b { f(x) : Ax = b, x 0 }

38 S = {x : Ax = b, x 0} x R n Ax b A µ m n (R), b R m F R n x F F x 1, x 2 F, λ (0, 1) x = λx 1 + (1 λ)x 2 F = {x R n : Ax = b, x 0} F x F A = [B, N] B B 1 b 0 x = (x B, x N ) x B = B 1 b, x N = 0 x + λd F x F, λ 0 ( ) n m F d 0 F d F F d d 1, d 2 F d = αd 1 + βd 2 α, β > 0 F = {x R n : Ax = b, x 0} F d A = [B, N] B ( ) B 1 B 1 a j a j 0 a j N d = α, α > 0 e j F = {x R n : Ax = b, x 0} {x 1, x 2,, x r } F {d 1, d 2,, d s } F F

39 x = r λ i x i + s µ j d j λ i 0 i, i=1 j=1 r λ i = 1, µ j 0 j i=1 f(x) f(x i) h(x i ) f(x) f(x i) h(x i ) x x i = 1,, r x x x i S f/h { } S = x R n : x = r r λ i x i, λ i 0 i, λ i = 1 S i=1 f S f S x S, S = { i=1 f(x) {f(x i ), i = 1,, r} = f(x i ) x i x R n : x = r λ i x i + s µ j d j, λ i 0 i, i=1 j=1 } r λ i = 1, µ j 0 j f S { S = x R n : x = r λ i x i, λ i 0 i, i=1 r i=1 } λ i = 1 x f f(x) f(x) x S f S F S \ S, x r x = λ i x i + s µ j d j µ j i=1 j=1 }{{} x 0 S x = x s µ j d j S j=1 x = 1 2 x x f( x) < f(x) f( x) > {f(x 0 ), f( x)} f(x 0 ) f( x) f(x 0 ) f( x) f( x) f f(x 0 ) = f( x) f( x) {f(x 0 ), f( x)} = f(x 0 ) x 0 S i=1

40 f( x) < f(x) x f(x) f( x) x S { } f(x) : Ax = b, x 0 > 0 S = {x : Ax = b, x 0} f(x) S S f(x) S S S f(x) S f(x) S f/h S S ( ) A µ m n (R) S = {x : Ax = b, x 0} n m B 1 b 0 n m n m { } 3log(x1 )+8log(x 2 ) 2x 1 : x x 2 10, x 1 1, x 2 2 f(x) = 3log(x 1 ) + 8log(x 2 ) S

41 S = {x 1 + x 2 10, x 1 1, x 2 2} ( ) 3 = 3 2 ( ) 5 x 3, x 4, x 5 = 10 3 x 1 x 2 x 3 x 4 x 5 b x x x R 1 = {(3, 1, 2)} W 1 = {(4, 1, 2), (5, 1, 2)} x 4, x 1, x 2 x 1 x 2 x 3 x 4 x 5 b x x x R 2 = {(3, 1, 2), (4, 1, 2)} W 2 = {(5, 1, 2)} (5, 1, 2) x 1 x 2 x 3 x 4 x 5 b x x x R 3 = {(3, 1, 2), (4, 1, 2), (5, 1, 2)} W 3 = R 3 v 1 = (1, 2), v 2 = (8, 2) v 3 = (1, 9) f(v 1) f(v 2) f(v 3) h(v 1 ) h(v 2 ) h(v 3 ) x = (8, 2) { } 4e 2x 1 5e 3x 2 log(x 1 +x 2 : x ) x 2 3, 2x 1 x 2 4, x 1 0, x 2 0 f(x) = (4e 2x 1 + 5e 3x 2 ) = log(x 1 + x 2 ) + 4 S x 3, x 4

42 x 1 x 2 x 3 x 4 b x x R 1 = {(1, 4)} W 1 = {(1, 2)} d 1 = (1, 0) a 2,2 (1, 2) W 1 x 1 x 2 x 3 x 4 b x /3 1/3 7/3 x /3 1/3 2/3 R 1 = {(1, 4), (1, 2)} W 1 = d 2 = (1/3, 2/3) v 1 = (3, 0) v 2 = (7/3, 2/3) t(1, 0) + z(1, 2) t, z > 0 t z 4e 2(t+z) 5e 6z = 0 (t,z) log(t+3z)+4 t z 4e 2(t+z) 5e 6z 9 log(t+3z)+4 log(t+3z)+4 0 t,z S f(v 1 ) f(v 2 ) x = (3, 0)

43 f/h f h f/h f h h f S f S F (q) q > 0 f h f(x) q f(x) q {f(x) q : x S} q > 0 q f/h f(x) = x 3 [ 1, 1] = 1 x 2 [ 1, 1] z(x) = x 3 x [ 1, 1] x 1 = 0 x 2 = 1 z(0) = z(1) = 1 x λ (x 1 : x 2 ) z(x λ ) < f(x) = x 3 3 = 1 x 2 3 z(x) = x 3 x 2 + 1

44 f f(x) = 1 = e (x 2)2 e x2 z(x) = f(x) = e (x 2)2 x 1 = 1 x 2 = 3 x λ (x 1 : x 2 ) z(x λ ) < {z(x 1 ), z(x 2 ))} f(x) = 1 2 = e (x 2)2 2 z(x) = e (x 2)2 X R n, f : X R, g : X R m, l : X R k {f(x) : g(x) 0, l(x) = 0, x X} {θ(u, v) = x X {f(x) + ug(x) + vl(x)} : u Rm, u 0, v R k } x (u, v) f(x) θ(u, v)

45 X f, g l {f(x) : g(x) 0, l(x) = 0, x X} = {θ(u, v) : u R m, u 0, v R k } x (u, v) ug(x) = 0. θ X f, g, l X { 1 x { u x : x > 0, 1 x 2} { 1 x + u 1(1 x) + u 2 (x 2) : x > 0 } : u 1, u 2 0} x = 2 1/2 { f(x) } : g(x) 0, x X X R n f, h : X R g : X R m > 0 x X z 0 = 1 z = xz 0 {z 0 f(z/z 0 ) : g(z/z0) 0, z 0 h(z/z 0 ) = 1, (z 0, z) W } X f, h, g X {f(x) : x X} < 0 h

46 {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) = 1, (z 0, z) W } máx (u,v) 0 { ínf {z 0 f(z/z 0 ) + uz 0 g(z/z 0 ) + v(z 0 h(z/z 0 ) 1)} (z 0,z) W } θ(u) = ínf x X { máx ínf u 0 x X { } f(x)+ug(x) { }} f(x) + ug(x) u R m, u 0 ug(x) { 1 x { u x : x > 0, 1 x 2} { } } 1+u1 (1 x)+u 2 (x 2) : x > 0 u x 1, u 2 0 = { { = 1 (1 + u u x x 1 2u 2 ) + u 2 u 1 : x > 0 }, u 1, u 2 0} = { = 1 (1 + u u x 1 2u 2 ) + u 2 u 1 : 1 (1 + u x 2 1 2u 2 ) = 0, x > 0, u 1, u 2 0 } u 2 = 1 2 u {u 2 u 1 : u 1, u 2 0} = { 1 2 u : u 1 0 } = 1 2 u = (0, 1/2) { } f(x) f(y)+ug(y) : y X h(y) { f(y)+ug(y) h(y) } : y X f(y)+ug(y) h(y) x u = f(y) h(y) + u g(y) h(y) f(y) h(y) y X {g(y) 0} { } f(x) : g(x) 0, x X {θ(u) : u 0}

47 f(x) = θ(u) x, u { } f(x) : g(x) 0, x X = θ(u) = u 0 {θ(u) : u 0} = + X f, h, g X{f(x) : x X} < 0 h { } f(x) : g(x) 0, x X = {θ(u) : u 0} x u ug(x) = 0 {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) = 1, (z 0, z) W } γ = {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) = 1, (z 0, z) W } γ < γ = {θ(u) : u 0} = γ z 0 f(z/z 0 ) γ < 0 z 0 g(z/z 0 ) 0 z 0 h(z/z 0 ) 1 = 0 (z 0, z) W γ (u 0, u, v) (u 0, u) 0 (u 0, u) > 0 u 0 (z 0 f(z/z 0 ) γ) + uz 0 g(z/z 0 ) + v(z 0 h(z/z 0 ) 1) 0 (z 0, z) W. u 0 > 0 u 0 = 0 (ẑ 0, ẑ) W ẑ 0 g(ẑ/ẑ 0 ) < 0 ẑ 0 h(ẑ/ẑ 0 ) = 1 u ẑ 0 g(ẑ/ẑ 0 ) }{{} +v (ẑ 0h(ẑ/ẑ 0 ) 1) }{{} 0 uẑ 0g(ẑ/ẑ 0 ) 0 u = 0 < 0 = 0 u 0 = u = 0 u 0 > 0 u 0 z 0 f(z/z 0 ) + uz 0 g(z/z 0 ) + v(z 0 h(z/z 0 ) 1) γ (z 0, z) W u = u u 0 v = v u 0

48 f(x) + ug(x) γ {θ(u) : u 0} θ(u) γ x X θ(u) γ θ(u) = γ u x g(x) 0, x X f(x) = γ x f(x) + u g(x) = f(x) ug(x) = 0 X f, h, g X {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) = 1, z 0 0, (z 0, z) W } h z 0 h(z/z 0 ) = 1 h {f(x) : x X} < 0 {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) 1, z 0 0, (z 0, z) W } {f(x) : x X} 0 {z 0 f(z/z 0 ) : z 0 g(z/z 0 ) 0, z 0 h(z/z 0 ) 1, z 0 0, (z 0, z) W } u,v,w 0 { inf {z 0 f(z/z 0 ) + z 0 ug(z/z 0 ) + v( z 0 h(z/z 0 ) + 1) + w( z 0 )} (z 0,z) W z 0 f(z/z 0 ) + ug(z/z 0 ) vh(z/z 0 ) z z 0 ( f(z/z 0 ) + u g(z/z 0 ) v h(z/z 0 )) w = 0 }

49 z f(z/z 0 ) + u g(z/z 0 ) v h(z/z 0 ) = 0 f(z/z 0 ) + ug(z/z 0 ) vh(z/z 0 ) = w 0 f(z/z 0 ) + u g(z/z 0 ) v h(z/z 0 ) = 0 v f(z/z 0 ) + u g(z/z 0 ) v h(z/z 0 ) = 0 f(z/z 0 ) + ug(z/z 0 ) vh(z/z 0 ) 0 u, v 0 v f(x) + u g(x) v = 0 f(x) + ug(x) v 0 u, v 0 {f(x) : x X} < 0 v 0 v x (u, v) (x, u, v) v = f(x) { } (x1 3) 2 +(x 2 2) 2 x 1 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 v 2(x 1 3) v + 2u 1 x 1 + u 2 = 0 x u 1 x 2 + u 2 = 0 (x 1 3) 2 + (x 2 2) 2 v(x 1 + 1) + u 1 (x x 2 2 5) + u 2 (x 1 + 2x 2 4) 0 x 1, x 2, u 1, u 2, v 0 v = 2/3, x = (2, 1), u = (11/20, 9/20)

50

51

52 { f(x) } : g(x) 0, x X X R n, f, h : X R, g : X R m > 0 X S = {x R n : g(x) 0, x X} S f, h (P P ) q F (q) = {f(x) q : x S} F : R R S, f, h { } x S, x x f(x) f(x) : x S x F 0 ( ) { } f(x) = 0 f(x) f(x) : x S = { f(x) } f(x) : x S F (q) x 1 S q 1 := f(x 1) h(x 1 ) F (q k ) = {f(x) q k : x S} x k+1 F (q k ) = 0 x k q k+1 = f(x k+1) h(x k+1 k = k + 1 ) {x k } S

53 q k = f(x k) h(x k ) φ k (x) = f(x) q k φ k (x) < 0 x S f(x) < f(x k) h(x k ) φ k (x) < 0 f(x) f(x k) f(x) h(x k < 0 < f(x k) ) h(x k ) x k S x k+1 F (q k ) x k F (q k ) f(x k+1) < f(x k) h(x k+1 ) h(x k ) x k F (q k ) x k+1 F (q k ) f(x k+1) h(x k+1 ) < f(x k) h(x k ) φ k (x k+1 ) < φ k (x k ) = 0 x k F (q k ) = 0 x k x k+1 A (x k+1 A(x k )) (x k+1 = A(x k )) Ω A x 1 S {x k } Ω S 1, S 2 A : S 1 S 2 x S 1

54 {x k } S 1 x {y k } A(x k ) y A(x) S R n Ω S A : S S {x k } S α α(y) < α(x) x / Ω y A(x) A Ω Ω {x k } {x k } Ω {α(x k )} α(x) x Ω D(x k ) = { { }} x S : x f(x) f(x k) : x S h(x k ) f/h {x k } Ω D S Ω {x k } {y k } x k S k x k = x S Ω y k D(x k ) k y k = y y D(x) y k S S y S Γ(x, y) = f(y) f(x) h(y) ŷ D(x) ŷ { } f(x) f(x) : x S f(ŷ) f(x) f(x) h(ŷ) f(y) h(y) Γ(x, ŷ) Γ(x, y) { y k D(x k ) y k f(x) } f(x) : x S x

55 Γ(x k, y k ) Γ(x k, ŷ) Γ(x, y) S S f, h (x, y) Γ(x k, y k ) Γ(x k, ŷ) k k Γ(x, y) Γ(x, ŷ) Γ(x, y) = Γ(x, ŷ) y D(x) F (q) F (q) f h S G(q, x) = f(x) q F (q) F (q) F (q) { } (x1 3) 2 +(x 2 2) 2 x 1 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0 S = {(x 1, x 2 ) R 2 : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0} f, h (0, 0) q 1 = f(0,0) h(0,0) = 13 F (13) = {(x 1 3) 2 + (x 2 2) 2 13(x 1 + 1) : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2

56 0} = (2.1881, ) q 2 = f(2.1881, ) h(2.1881, ) = F ( ) = {(x 1 3) 2 + (x 2 2) (x 1 + 1) : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0} = (2, 1) q 3 = f(2,1) h(2,1) = 2/3 F (2/3) = {(x 1 3) 2 + (x 2 2) (x 1 + 1) : x x 2 2 5, x 1 + 2x 2 4, x 1 0, x 2 0} = e 16 (2, 1) 2/3 f, h, g X f, h, g X v f(x) + u g(x) v = 0 f(x) + ug(x) v 0 u 0 {f(x) : g(x) 0, x X} < 0 {f(x) : g(x) 0, x X} 0 v 0 v 0 ( f(x), ) = 0 v v = f(x) x { f(x) } : Ax = b, x 0

57 S = {x : Ax = b, x 0} f/h p, q R n, A µ m n (R), α, β R { f(x) = p x + α q x + β } : Ax = b, x 0 f(x) = p x+α S q x+β q x + β 0 S f S q x + β > 0 S x 1, x 2 S q x 1 + β < 0 q x 2 + β > 0 S x λ S q x λ + β = 0 f x 1, x 2 S f(x 1 ) (x 2 x 1 ) 0 f(x 2 ) f(x 1 ) f(x 1 ) = (q x 1 +β)p (p x 1 +α)q (q x 1 +β) 2 f(x 1 ) (x 2 x 1 ) 0 (q x 1 + β) 2 > 0 [p (q x 1 + β) q (p x 1 + α)] (x 2 x 1 ) 0 p (q x 1 + β)x 2 q (p x 1 + α)x 2 p (q x 1 + β)x 1 + q (p x 1 + α)x 1 0 (p x 2 + α)(q x 1 + β) (q x 2 + β)(p x 1 + α) 0 (p x 2 + α)(q x 1 + β) (q x 2 + β)(p x 1 + α)

58 (q x 1 + α)(q x 2 + β) > 0 p x 2 +α p x 1 +α q x 2 +β q x 1 +β x 1, x 2 S f(x 1 ) (x 2 x 1 ) 0 f(x 2 ) f(x 1 ) f S x B x B = B 1 b > 0, x N = 0 r = B f(x) B 1 A f(x) r N = Bf(x) B 1 N N f(x) r N 0 (x, r) x r j = {r i : r i 0, i N} x j f(x)d < 0 f(x) = p x+α S q x+β q x + β 0 S x S, d 0 f(x) d < 0 f(x + λd) λ d f x f(x + λd) < f(x), λ > 0

59 r n { } f(x) = p x+α : Ax = b, x 0 q x+β Ax = b, x 0} S q x + β > 0 S S = {x R n : x 1 B A = (B, N) r N = Bf(x) B 1 N N f(x) r N 0 x k r j = {r i : r i 0, i N} x j x i b i y ij { = b i y ij } : y ij > 0, i B b = B 1 b, y j = B 1 a j a j j A ( ) B a j y ij 0 i d = S e j { } 2x 1 +x 2 +2 : x x 1 +3x x 2 4, 2x 1 + 2x 2 14, x 2 6, x 1, x 2 0

60 f/h ( f(x 1, x 2 ) = x 3, x 4, x 5 x 1 + x 2 + x 3 = 4 2x 1 + 2x 2 + x 4 = 14 7x 2 10 (x 1 +3x 2, +4) 2 ) 7x 1 2 (x 1 +3x 2 +4) 2 x 2 + x 5 = 6 x 1 = (0, 0, 4, 14, 6) f(x 1 ) 10/16 2/ x 1 x 2 x 3 x 4 x 5 b 0 x x x r j 10/16 2/ r j = {10/16, 2/16} = 10/16 x 1 b i y i1 = {14/2} x 4 f(x 2 ) 10/121 47/ x 1 x 2 x 3 x 4 x 5 b 0 x 3 0 3/2 1 1/2 0 10/121 x 1 1 1/2 0 1/2 0 0 x r j 0 47/ /121 0 r j 0 x = (7, 0)

61 x 1 + 3x = 0 { } f(x) = p x+α : Ax = b, x 0 q x+β S = {x R n : Ax b, x 0} q x + β > 0 S z 0 = 1 q x+β, z = xz 0 {p z + αz 0 : Az bz 0 0, q z + βz 0 = 1, z 0, z 0} S q z+βz 0 = 1 (0, 0) (z, z 0 ) z 0 > 0 x = z/z 0

62 { } 2x 1 +x 2 +2 : x x 1 +3x x 2 4, 2x 1 + 2x 2 14, x 2 6, x 1, x 2 0 2z 1 + z 2 + 2z 0 z 1 + z 2 4z 0 0 2z 1 + 2z 2 14z 0 0 z 2 6z 0 0 z 1 + 3z 2 + 4z 0 = 1 z 1, z 2, z 0 0 z 1 = 63/99, z 2 = 0 z 0 = 9/99 x = (z 1 /z 0, z 2 /z 0 ) = (7, 0)

63 F (q)

64

65 n c i x 2 i + n x i x i+2 n x i x i+2 i=1 i=1 i %2=1 i=1 i %2=0 log(1+ 20 t i x i ) i=1 n x i = 40 n x 2 i = 240 n x i 15 i=1 i=1 i=1 i %2=0 n i=1 i %2=1 x i 15 0 x i 100 i

66 c i t i [0, 1)

67

68 f h f/h f h n c i x 2 i + n x i x i+2 + n x i x i+2 i=1 i=1 i %2=1 i=1 i %2=0 log(1+ 20 t i x i ) i=1 n x i = 40 n x 2 i = 240 n x i 15 i=1 i=1 i=1 i %2=0 n i=1 i %2=1 x i 15 0 x i 100 i

69

70 f/h f h f h

71

72

73