Khi b i to¡n (3.1) ch¿ phö thuëc tham sè µ, Quincampoix v Zlateva [14] ¢ thi¸t lªp mët sè i·u ki»n õ cho t½nh Lipschitz cõa tªp iºm y¶n ngüa.
Trong möc n y chóng ta thi¸t lªp c¡c i·u ki»n õ cho t½nh Lipschitz cõa tªp iºm y¶n ngüa trong b i to¡n minimax câ tham sè M (µ, λ, θ)
inf
x∈K(λ) sup
y∈L(θ)
ð â f (x, y, µ) = 1 2x TA(µ)x+xTB(µ)y − 1 2y TC(µ)y+ a(µ)Tx+b(µ)Ty, K : Λ ⊂ Rp →2Rn, L : Θ ⊂Rq →2Rm, Λ,Θ l c¡c tªp lçi, K(λ) = {x ∈ Rn|Dx ≥λ, x ≥ 0}, L(θ) = {y ∈ Rm|Ey ≥ θ, y ≥0}, ð â D ∈ Rp×n, E ∈ Rq×m, A : Rs → Rn×n, µ 7→ A(µ) ∈ Rn×n B : Rs → Rn×m, µ 7→B(µ) ∈ Rn×m C : Rs →Rm×m, µ 7→C(µ) ∈ Rm×m a : Rs → Rn, µ 7→ a(µ) ∈ Rn×1 b : Rs → Rm, µ 7→ b(µ) ∈ Rm×1.
Chu©n trong khæng gian c¡c ma trªn c§p m×n ÷ñc x¡c ành bði
kMkm×n = n X j=1 m X i=1 |rij|2 !1/2 , ð â M = (rij)m×n.
°t e(µ) l gi¡ trà ri¶ng nhä nh§t cõa A(µ), d(µ) l gi¡ trà ri¶ng nhä nh§t cõa C(µ).
°t G(x, y, µ) = (F1(x, y, µ),−F2(x, y, µ)), ð â
F1(x, y, µ) := A(µ)x+B(µ)y+ a(µ), F2(x, y, µ) := B(µ)Tx−C(µ)y +b(µ).
ành lþ 3.1. X²t b i to¡n (3.2). L§y µ,¯ λ,¯ θ¯ ∈ Rp×Rq×Rs v (¯x,y¯) ∈
(i) G l Lipschitz àa ph÷ìng trong l¥n cªn cõa (¯x,y,¯ µ¯),
(ii) tçn t¤i l¥n cªn U¯ (¯µ) cõa µ¯ v α > 0 sao cho
min{e(µ), d(µ)}> α ∀µ ∈ U¯(¯µ).
Khi â ∃kµ¯ > 0,∃kλ,¯θ¯ > 0,∃U (¯µ), V λ¯, W θ¯ sao cho
a) Vîi méi (µ, λ, θ) ∈ U (¯µ) ×V ¯λ× W θ¯ b i to¡n (3.2) câ duy nh§t iºm y¶n ngüa, kþ hi»u iºm y¶n ngüa l (x(µ, λ), y(µ, θ)).
b) Vîi måi (µ0, λ0, θ0),(µ, λ, θ) ∈ U (¯µ)×V ¯λ×W θ¯ ta câ k(x(µ0, λ0), y(µ0, θ0))−(x(µ, λ), y(µ, θ))k ≤ kµ¯kµ0 −µk+kλ,¯θ¯k(λ0, θ0)−(λ, θ)k. Chùng minh. °t Γ (λ, θ) = (x, y) ∈ Rn ×Rm| H x y ≥ λ θ , x y ≥ 0 , ð â H(p+q)×(n+m) := Dp×n Op×m Oq×n Eq×m , O l ma trªn khæng. Ta kh¯ng ành r¬ng hG(x0, y0, µ)−G(x, y, µ),(x0, y0)−(x, y)i ≥ α kx0 −xk2 +ky0 −yk2 ∀µ∈ U (¯µ), ∀(x0, y0),(x, y) ∈ Rn×Rm.
Thªt vªy, hG(x0, y0, µ)−G(x, y, µ),(x0, y0)−(x, y)i = hA(µ) (x0 −x), x0 −xi+hC (µ) (y0 −y), y0 −yi ≥ e(µ) n X i=1 (xi0 −xi)2 +d(µ) m X i=1 (yi0 −yi)2 ≥ α kx0−xk2 +ky0−yk2 ∀µ ∈ U(¯µ),∀(x0, y0),(x, y) ∈ Rn×m. V¼ min{e(µ), d(µ)} > α > 0 n¶n A(µ) v C(µ) l c¡c ma trªn x¡c ành d÷ìng trong l¥n cªn U¯(¯µ). Do â vîi méi µ ∈ U¯ (¯µ), λ ∈ Λ, θ ∈ Θ
ta câ f(., y, µ) l lçi tr¶n K(λ) v f(x, ., µ) l lãm tr¶n L(θ) vîi måi
(x, y) ∈ K(λ)×L(θ). Ta suy ra tø ành lþ 1.1 r¬ng (x, y) l iºm y¶n ngüa cõa b i to¡n (3.2) khi v ch¿ khi (x, y) l nghi»m cõa b i to¡n b§t ¯ng thùc bi¸n ph¥n:
T¼m (x, y) ∈ K(λ)×L(θ) sao cho hG(x, y, µ),(x0, y0)−(x, y)i ≥ 0
∀(x0, y0) ∈ K(λ)×L(θ).
Tø [24, Theorem 3.1] suy ra kh¯ng ành cõa ành lþ.
K¸t luªn
Luªn v«n tr¼nh b y mët sè k¸t qu£ ban ¦u v· b§t ¯ng thùc bi¸n ph¥n minimax trong [9] v ¡p döng nghi¶n cùu t½nh ch§t Lipschitz cõa tªp iºm y¶n ngüa trong c¡c b i to¡n minimax câ tham sè; cö thº:
1. Giîi thi»u v· b§t ¯ng thùc bi¸n ph¥n minimax.
2. Ph¥n lo¤i c¡c b§t ¯ng thùc bi¸n ph¥n minimax ìn i»u.
3. Tr¼nh b y v· sü tçn t¤i v duy nh§t nghi»m cõa c¡c b§t ¯ng thùc bi¸n ph¥n minimax trong mët sè khæng gian: b§t ¯ng thùc bi¸n ph¥n minimax khæng ìn i»u trong khæng gian Euclide, b§t ¯ng thùc bi¸n ph¥n minimax gi£ ìn i»u trong khæng gian Banach ph£n x¤, b§t ¯ng thùc bi¸n ph¥n minimax ìn i»u m¤nh trong khæng gian Hilbert.
4. Giîi thi»u b i to¡n minimax câ tham sè v mët k¸t qu£ mîi v· t½nh Lipschitz cõa tªp iºm y¶n ngüa trong b i to¡n minimax câ tham sè.
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