d¯˜a d¯u.o.. c x´et d¯ˆe´n trong Chu.o.ng 2. Hai d¯i.nh l´y du.´o.i d¯ˆay l`a nh˜u.ng kˆe´t qua˙’ co. ba˙’n cu˙’a mu.c n`ay.
D- i.nh l´y 3.3.18. Nˆe´u x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P), x˜∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P). Khi d¯´o
˜ x∗ ∈ x∗ + 0.5M(2s). (3.3.23) Ch´u.ng minh. V`ıf(˜x∗) ≥ f(x∗) nˆen f(˜x∗)−f(x∗) ≥ 1 2f(˜x ∗) + 1 2f(x ∗)−f(x∗) = 1 2f(˜x ∗)− 1 2f(x ∗) = 1 2 hAx˜∗,x˜∗i+hb,x˜∗i − hAx∗, x∗i − hb, x∗i = 1 2 hA(x∗ + ˜x∗ −x∗), x∗ + ˜x∗ −x∗i +hb, x∗ + ˜x∗ −x∗i −hAx∗, x∗i − hb, x∗i = 1 2 hAx∗, x∗i+ 2hAx∗,x˜∗ −x∗i+hAx˜∗ −x∗,x˜∗ −x∗i +hb,x˜∗ −x∗i+hb, x∗i − hAx∗, x∗i − hb, x∗i .
R´ut go.n biˆe˙’u th´u.c trˆen ta d¯u.o..c 1
2 hA(˜x∗ −x∗),x˜∗ −x∗i+h2Ax∗ +b,x˜∗ −x∗i
≤ f(˜x∗)−f(x∗).
Theo (2.3.10) Chu.o.ng 2 th`ıh2Ax∗+b,x˜∗−x∗i ≥ 0 nˆen t`u. biˆe˙’u th´u.c trˆen ta suy ra
1
2hA(˜x∗ −x∗),x˜∗ −x∗i ≤ f(˜x∗)−f(x∗). (3.3.24) Nˆe´u ˜x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a ˜f th`ı ˜f(˜x∗)−f˜(x∗) ≤ 0 v`a c´o thˆe˙’ biˆe´n d¯ˆo˙’i
f(˜x∗)−f(x∗) = f˜(˜x∗)−f˜(x∗)−p(˜x∗) +p(x∗)
≤ f˜(˜x∗)−f˜(x∗) + 2s
≤ 2s.
Do d¯´o kˆe´t ho.. p v´o.i (3.3.24) ta d¯u.o.. c
Nˆe´u ˜x∗ chı˙’ l`a d¯iˆe˙’m infimum to`an cu.c cu˙’a ˜f th`ı n´o l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a lsc ˜f = f + lscp nˆen lsc ˜f(˜x∗)− lsc ˜f(x∗) ≤ 0. Kˆe´t ho.. p v´o.i supx∈D|lsc p(x)| ≤s ta suy ra
f(˜x∗)−f(x∗) = lsc ˜f(˜x∗)−lsc ˜f(x∗)−lscp(˜x∗) + lscp(x∗)
≤ lsc ˜f(˜x∗)−lsc ˜f(x∗)
≤ 2s.
Thay bˆa´t d¯ˇa˙’ng th´u.c trˆen v`ao (3.3.24) ta la.i nhˆa.n d¯u.o..c (3.3.25). T`u. (3.3.25) v`a d¯i.nh ngh˜ıa tˆa.p M(γ) ta suy ra
˜
x∗ −x∗ ∈ 0.5M(2s),
t´u.c l`a ˜x∗ ∈ x∗ + 0.5M(2s).
Nhˆa.n x´et 3.3.7. D- i.nh l´y 3.3.18 ma.nh ho.n D- i.nh l´y 2.3.13 o.˙’ Mu.c 2.3 Chu.o.ng 2.
V´ı du. sau d¯ˆay cho ta thˆa´y tˆa.p 0.5M(2s) l`a nho˙’ nhˆa´t trong c´ac tˆa.p ch´u.a ˜x∗ −x∗.
V´ı du. 3.3.8. X´et h`am to`an phu.o.ng lˆ`i ngˇo a.t
f(x) =ξ12 + 2ξ22 trˆen IR2 v`a h`am nhiˆe˜u p(x) = ( s−f(x) nˆe´u x = (ξ1, ξ2) ∈ 0.5M(2s) 0 nˆe´u x = (ξ1, ξ2) ∈/ 0.5M(2s) . Khi d¯´o ˜ f(x) = ( s nˆe´u x = (ξ1, ξ2) ∈ 0.5M(2s) f(x) nˆe´u x = (ξ1, ξ2) ∈ IR2 \0.5M(2s).
Ta thˆa´y, x∗ = 0 l`a d¯iˆe˙’m cu.. c tiˆe˙’u duy nhˆa´t cu˙’a f trˆen IR2 v`a ˜f d¯a.t cu. c. tiˆe˙’u ta.i mo.i ˜x∗ ∈ 0.5M(s). Do d¯´o ta suy ra 0.5M(2s) l`a tˆa.p nho˙’ nhˆa´t ch´u.a ˜
Go.i S0 l`a tˆa.p c´ac d¯iˆe˙’m cu. c tiˆ. e˙’u cu˙’a B`ai to´an (P) v`a Ss l`a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P). Khoa˙’ng c´ach Hausdorff l`a d¯a.i lu.o.. ng: dH(S0, Ss) = max{sup x∈S0 inf y∈Sskx−yk,sup y∈Ss inf x∈S0kx−yk}.
Trong [50] H. X. Phu d¯˜a ph´at biˆe˙’u v`a ch´u.ng minh d¯i.nh l´y sau:
D- i.nh l´y 3.3.19. Gia˙’ su.˙’ B`ai to´an (P) c´o d¯iˆe˙’m cu.. c tiˆe˙’u x∗ v`a (x∗ + ¯B(0, r))∩D l`a d¯´ong v´o.i gi´a tri. r > 0 n`ao d¯´o .
Nˆe´u sup x∈D |p(x)| ≤ s ≤ 1 2r 2λmin, th`ı tˆa. p Ss l`a kh´ac rˆo˜ng v`a dH({x∗}, Ss) ≤p2s/λmin.
3.4. Du.´o.i vi phˆan suy rˆo.ng thˆo v`a d¯iˆe` u kiˆe.n tˆo´i u.u Trong mu.c 2.4 Chu.o.ng 2, ch´ung tˆoi d¯˜a tr`ınh b`ay t´ınh chˆa´t tu..a v`a d¯iˆ` u kiˆe.n tˆo´i u.u cu˙’a B`ai to´an ( ˜e P). O˙’ mu.c n`ay ta x´et la.i c´ac t´ınh chˆa´t trˆen. v`a nhˆa.n d¯u.o..c c´ac kˆe´t qua˙’ tˆo˙’ng qu´at ho.n c´ac kˆe´t qua˙’ tru.´o.c d¯´o.
D- i.nh ngh˜ıa 3.4.10. ([50]) Cho tˆa. p cˆan Γ ta n´oi ξ l`a du.´o.i vi phˆan suy rˆo. ng thˆo cu˙’a h`am g : D → IR ta. i d¯iˆe˙’m x∗ ∈ D nˆe´u
inf
x0∈(x∗+Γ)∩D g(x0) +hξ, x0i
≤g(x) +hξ, xi v´o.i mo. i x ∈ D.
Khi g = ˜f = f +p ta c´o d¯i.nh l´y sau:
D- i.nh l´y 3.4.20. Gia˙’ su.˙’ 0 < supx∈D|p(x)| ≤ s < +∞, f(x) =
hAx, xi − hb, xi. Khi d¯´o, v´o.i x∗ ∈ D n`ao d¯´o th`ı
inf
x0∈(x∗+0.5M(2s))∩D
˜
Trong tru.`o.ng ho.. p d¯ˇa. c biˆe.t, nˆe´u D d¯´ong v`a p l`a nu.˙’a liˆen tu.c du.´o.i, th`ı v´o.i mˆo˜i x∗ ∈ D tˆ`n ta.io ˜ x∗ ∈ x∗ + 0.5M(2s)∩D sao cho ˜ f(˜x∗)− h2Ax∗ +b,x˜∗i = min x0∈(x∗+0.5M(2s))∩D ˜ f(x0)− h2Ax∗ +b, x0i v`a ˜ f(˜x∗)− h2Ax∗ + b,x˜∗i ≤ f˜(x)− h2Ax∗ +b, xi v´o.i mo. i x ∈ D,
hoˇa. c tu.o.ng d¯u.o.ng l`a ˜
f(x) ≥ f˜(˜x∗) +h2Ax∗ +b, x−x˜∗i v´o.i mo. i x ∈ D.
D- i.nh l´y n`ay do H. X. Phu ph´at biˆe˙’u v`a ch´u.ng minh trong [50].
Nghiˆen c´u.u su.. tˆ`n ta.i d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜o P), ta c´o D- i.nh l´y Kuhn-Tucker suy rˆo.ng nhu. sau:
D- i.nh l´y 3.4.21. X´et B`ai to´an ( ˜P) v´o.i miˆ` ne D d¯u.o.. c cho bo˙’ i. (2.4.17). Khi d¯´o
(a) Nˆe´u x˜∗ l`a d¯iˆe˙’m infimum to`an cu. c, th`ı tˆ`n ta.i duy nhˆa´to x∗ ∈ (˜x∗ + 0.5M(2s))∩ D v`a c´ac nhˆan tu.˙’ Lagrange µ0 ≥ 0, µ1 ≥ 0, . . . , µm ≥ 0,
sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an d¯iˆe` u kiˆe.n Kuhn-Tucker
L(x∗, µ0, . . . , µm) = min
x∈S L(x, µ0, . . . , µm), (3.4.26) d¯iˆ` u kiˆe.n b`ue
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m. (3.4.27) Nˆe´u d¯iˆ` u kiˆe.n Slatere (1.1.7) tho˙’a m˜an th`ıµ0 6= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) Nˆe´u tˆ`n ta.io x∗ tho˙’a m˜an (3.4.26) v`a (3.4.27) v´o.i µ0 = 1 th`ı tˆ`n ta.io ˜
Ch´u.ng minh. (a) Gia˙’ thiˆe´t (a) cu˙’a d¯i.nh l´y c˜ung l`a gia˙’ thiˆe´t (a) cu˙’a D- i.nh l´y 2.4.14 Chu.o.ng 2, nˆen suy ra tˆ`n ta.i duy nhˆa´t d¯iˆe˙’mo x∗ ∈ D v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an d¯iˆ` u kiˆe.n Kuhn-Tuckere
L(x∗, µ0, . . . , µm) = min
x∈S L(x, µ0, . . . , µm) v`a
µigi(x∗) = 0 v´o.i mo.i i = 1, . . . , m.
Nˆe´u d¯iˆ` u kiˆe.n Slater (1.1.7) tho˙’a m˜an, th`ıe µ0 6= 0 v`a c´o thˆe˙’ coi µ0 = 1.
Ngo`ai ra, x∗ l`a cu.. c tiˆe˙’u to`an cu.c cu˙’a B`ai to´an (P) trˆen D, nˆen theo D- i.nh l´y 3.3.18 ta suy ra x∗ ∈ (˜x∗ + 0.5M(2s)) ∩D.
(b) V`ı c´ac gia˙’ thiˆe´t cu˙’a d¯i.nh l´y c˜ung l`a c´ac gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.14 Chu.o.ng 2, nˆen tˆ`n ta.i ˜o x∗ l`a d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P). Mˇa.t kh´ac, theo D- i.nh l´y 1.1.5 th`ıx∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a B`ai to´an (P) trˆen D. Do d¯´o, ´ap du.ng D- i.nh l´y 3.3.18 ta suy ra
˜
x∗ ∈ (x∗+ ∈ 0.5M(2s))∩D.
D- i.nh l´y d¯u.o..c ch´u.ng minh.
Mo.˙’ rˆo.ng D- i.nh l´y 2.4.15 l`a d¯i.nh l´y sau:
D- i.nh l´y 3.4.22. Gia˙’ su.˙’ D d¯u.o.. c cho bo˙’ i cˆ. ong th´u.c (2.4.17), gi : IRn →
IR, i = 1, . . . , m, l`a c´ac h`am lˆ`i, c`o ung liˆen tu. c ´ıt nhˆa´t ta. i mˆo. t d¯iˆe˙’m cu˙’a tˆa. p lˆ`i, d¯´o ong S ⊂ IRn. Khi d¯´o
(a) Nˆe´u x˜∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P) th`ı tˆ`n ta.i duyo nhˆa´t x∗ ∈ (˜x∗+ 0.5M(2s))∩D v`a c´ac nhˆan tu.˙’ Lagrange µ0 ≥ 0, µ1 ≥
0, . . . , µm ≥ 0, ch´ung khˆong c`ung triˆe.t tiˆeu tho˙’a m˜an 0 ∈ µ0(2Ax∗ + b) + m X i=1 µi∂gi(x∗) +N(x∗|S) (3.4.28) v`a µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m, (3.4.29)
trong d¯´o N(x∗|S) l`a n´on ph´ap tuyˆe´n cu˙’a S ta. i x∗.
Nˆe´u d¯iˆ` u kiˆe.n Slatere (1.1.7) tho˙’a m˜an th`ıµ0 6= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) Nˆe´u c´o x∗ ∈ D tho˙’a m˜an (3.4.28) v`a (3.4.29) v´o.i µ0 = 1 th`ı tˆ`n ta.io ˜
x∗ ∈ (x∗+ 0.5M(2s))∩D l`a d¯iˆe˙’m infimum to`an cu. c duy nhˆa´t cu˙’a B`ai to´an ( ˜P).
Ch´u.ng minh. (a) V`ı gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.15 c˜ung l`a gia˙’ thiˆe´t cu˙’a d¯i.nh l´y n`ay, nˆen tˆ`n ta.io x∗ ∈ D v`a c´ac nhˆan tu.˙’ Lagrange µi, i = 0, . . . , m, tho˙’a m˜an c´ac d¯iˆ` u kiˆe.ne 0 ∈ µ0(2Ax∗ +b) + m X i=1 µi∂gi(x∗) +N(x∗|S) v`a µigi(x∗) = 0 v´o.i mo.i i = 1, . . . , m,
trong d¯´o N(x∗|S) l`a n´on ph´ap tuyˆe´n cu˙’a S ta.i x∗. D- ˆo`ng th`o.i, theo D- i.nh l´y Kuhn-Tucker cho B`ai to´an (P) ta c˜ung suy ra x∗ l`a cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t cu˙’a B`ai to´an (P). ´Ap du.ng D- i.nh l´y 3.3.18 cho c´ac d¯iˆe˙’m ˜x∗ v`a
x∗ ta nhˆa.n d¯u.o..c
x∗ ∈ (˜x∗ + 0.5M(2s)) ∩D.
Ta c˜ung suy ra, nˆe´u d¯iˆ` u kiˆe.n Slater (1.1.7) tho˙’a m˜an th`ıe µ0 6= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) C´ac d¯iˆ` u kiˆe.n (3.4.28) v`a (3.4.29) tho˙’a m˜an (b) cu˙’a De - i.nh l´y 2.4.15, nˆen tˆ`n ta.i ˜o x∗ l`a d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P). Mˇa.t kh´ac c´ac d¯iˆ` u kiˆe.n (3.4.28) v`a (3.4.29) tho˙’a m˜an De - i.nh l´y Kuhn-Tucker cho B`ai to´an (P) nˆen x∗ l`a cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa.t
f. Ap du.ng D´ - i.nh l´y 3.3.18 cho c´ac d¯iˆe˙’m ˜x∗ v`a x∗ ta suy ra ˜
x∗ ∈ (x∗ + 0.5M(2s)) ∩D.
Nhˆa.n x´et 3.4.8. (a) Nˆe´u S = IRn th`ı khi d¯´o N(x∗|S) = {0}, nˆen biˆe˙’u th´u.c (3.4.28) d¯u.o.. c thay bo˙’ i.
0 ∈ µ0(2Ax∗ + b) +
m
X
i=1
µi∂gi(x∗).
(b) Nˆe´u gi, i= 1, . . . , m lˆ`i, kha˙’ vi liˆen tu.c th`ı biˆe˙’u th´u.c trˆen c´o da.ngo 0 = µ0(2Ax∗ + b) +
m
X
i=1
µi5gi(x∗).
Khi D l`a tˆa.p lˆo`i d¯a diˆe.n ta c´o d¯i.nh l´y sau:
D- i.nh l´y 3.4.23. Gia˙’ su.˙’ D d¯u.o.. c cho bo˙’ i. (2.4.18). Khi d¯´o
(a) Nˆe´u x˜∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P) th`ı tˆ`n ta.i duyo nhˆa´t x∗ ∈ (˜x∗ + 0.5M(2s))∩D v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 1, . . . , m, sao cho (2Ax∗ +b) + m X i=1 µici = 0, (3.4.30) µi(hci, x∗i −di) = 0 v´o.i mo. i i = 1, . . . , m. (3.4.31) Ho.n thˆe´ n˜u.a ta c´o
2Ax˜∗ +b+
m
X
i=1
µici ∈ AM(2s).
(b) Nˆe´u c´o x∗ ∈ D tho˙’a m˜an (3.4.30), (3.4.31) th`ı tˆ`n ta.io x˜∗ ∈ (x∗ + 0.5M(2s))∩D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P).
Ch´u.ng minh. (a) Gia˙’ thiˆe´t cu˙’a d¯i.nh l´y c˜ung l`a gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.16, do d¯´o tˆ`n ta.i duy nhˆa´to x∗ c`ung c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m,
sao cho
(2Ax∗ +b) +X
i=1
µici = 0
v`a
Mˇa.t kh´ac x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t cu˙’a B`ai to´an (P) nˆen theo D- i.nh l´y 3.3.18 suy ra
x∗ ∈ (˜x∗ + 0.5M(2s)) ∩D. Ta la.i c´o 2Ax˜∗ +b+ m X i=1 µici = 2Ax∗ +b+ m X i=1 µici + 2Ax˜∗ −2Ax∗ = 2A(˜x∗ −x∗) ∈ AM(2s). (3.4.32)
(b) C´ac d¯iˆ` u kiˆe.n (3.4.30) v`a (3.4.31) tho˙’a m˜an (b) cu˙’a De - i.nh l´y 2.4.16, do d¯´o tˆ`n ta.i ˜o x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P). V`ı x∗ l`a cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t cu˙’a B`ai to´an (P) nˆen theo D- i.nh l´y 3.3.18 suy ra mo.i d¯iˆe˙’m infimum to`an cu.c ˜x∗ ∈ (x∗ + 0.5M(2s)) ∩ D. D- i.nh l´y d¯u.o..c ch´u.ng minh.
Kˆe´t luˆa.n: Trong chu.o.ng n`ay ch´ung tˆoi d¯˜a gia˙’i quyˆe´t d¯u.o.. c nh˜u.ng vˆa´n d¯ˆ`e co. ba˙’n d¯u.o.. c d¯ˇa.t ra o˙’ d¯ˆ. ` u chu.o.ng l`a a: chı˙’ ra c´ac d¯iˆ` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am bi.e nhiˆe˜u gi´o.i nˆo.i ˜f = f+pl`a Γ-lˆ`i ngo`o ai (c´ac Mˆe.nh d¯ˆe` 3.1.18 – 3.1.19); ch´u.ng minh d¯iˆe˙’m Γ-cu.. c tiˆe˙’u (Γ-infimum) l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c (infimum to`an cu.c) khi Γ = M(2s) (c´ac Mˆe.nh d¯ˆe` 3.2.21 – 3.2.22); x´ac lˆa.p d¯u.o..c quan hˆe. gi˜u.a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a B`ai to´an (P) v`a d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P), d¯ˆ`ng th`o o.i ch´u.ng minh d¯u.o.. c t´ınh ˆo˙’n d¯i.nh nghiˆe.m theo khoa˙’ng c´ach Hausdorff (c´ac d¯i.nh l´y 3.3.18, 3.4.20); tr`ınh b`ay du.´o.i vi phˆan suy rˆo.ng thˆo cu˙’a h`am ˜f = f + p v`a chı˙’ ra c´ac d¯iˆ` u kiˆe.n tˆo´i u.u cu˙’a B`aie to´an ( ˜P) (c´ac d¯i.nh l´y 3.4.21 – 3.4.23).
CHUONG 4
D- IˆE˙’M SUPREMUM CU˙’ A B `AI TO ´AN ( ˜Q)
B`ai to´an d¯u.o.. c x´et trong chu.o.ng n`ay l`a ˜
f(x) := f(x) +p(x) →sup, x ∈ D, ( ˜Q)
trong d¯´o D l`a tˆa.p lˆo`i, f tho˙’a m˜an cˆong th´u.c (1.0.1), t´u.c l`a f =
hAx, xi + hb, xi, A ∈ IRn×n l`a ma trˆa.n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng, p l`a nhiˆe˜u gi´o.i nˆo.i, t´u.c l`a
sup
x∈D
|p(x)| ≤s < +∞.
Trong chu.o.ng n`ay ch´ung tˆoi nghiˆen c´u.u t´ınh γ-lˆ`i trong cu˙’a h`o am bi. nhiˆe˜u ˜
f = f + p; mˆo.t sˆo´ t´ınh chˆa´t cu˙’a d¯iˆe˙’m supremum to`an cu.c cu˙’a B`ai to´an ( ˜Q); t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa.p c´ac d¯iˆe˙’m supremum to`an cu.c v`a t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa.p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q) theo nhiˆe˜u p.
4.1. T´ınh γ-lˆ` i trong cu˙’a h`o am bi. nhiˆe˜u
Trong mu.c n`ay, ta tr`ınh b`ay mˆo.t sˆo´ d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am to`an phu.o.ng lˆ`i ngˇo a.t bi. nhiˆe˜u gi´o.i nˆo.i ˜f = f +p l`a γ-lˆ`i trong. Do - ´o l`a c´ac mˆe.nh d¯ˆe` sau:
Mˆe.nh d¯ˆe` 4.1.23. Cho γ > 0, f, p x´ac d¯i.nh theo cˆong th´u.c (1.0.1) v`a (1.0.2), tu.o.ng ´u.ng. Khi d¯´o
(a) Nˆe´u supx∈D|p(x)| ≤ λminγ2/2, th`ı f˜= f +p l`a γ-lˆ`i trong.o
(a) Nˆe´u supx∈D|p(x)| < λminγ2/2, th`ı f˜= f +p l`a γ-lˆ`i trong ngˇo a. t.
Ch´u.ng minh. X´et bˆa´t k`y x0, x1 ∈ D tho˙’a m˜an kx0−x1k = γ,−x0+ 2x1 ∈ D, ta c´o f(x0)−2f(x1) +f(−x0 + 2x1) = hAx0, x0i+hb, x0i −2hAx1, x1i −2hb, x1i +hA(−x0 + 2x1),(−x0 + 2x1)i+hb,−x0 + 2x1i = 2hAx0, x0i −4hAx0, x1i + 2hAx1, x1i = 2hA(x0 −x1), x0 −x1i.
Khˆong gia˙’m t´ınh tˆo˙’ng qu´at ta go.i λi, i = 1, . . . , n, l`a c´ac gi´a tri. riˆeng cu˙’a ma trˆa.n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng A (c´o thˆe˙’ c´o mˆo.t sˆo´ gi´a tri. tr`ung nhau),
{ei | i = 1,2, . . . , n} l`a co. so.˙’ tru.. c chuˆa˙’n trong IRn v`a d¯ˆ`ng th`o o.i ei l`a v´ec to. riˆeng ´u.ng v´o.i gi´a tri. riˆeng λi, i= 1,2. . . , n. Khi d¯´o
x0 = n X i=1 ζ0iei, x1 = n X i=1 ζ1iei v`a 2hA(x0 −x1), x0 −x1i = 2h n X i=1 λi(ζ0i −ζ1i)ei, n X j=1 (ζ0j −ζ1j)eji = 2 n X i=1 n X j=1 λi(ζ0i −ζ1i)(ζ0j −ζ1j)hei, eji = 2 n X i=1 λi(ζ0i −ζ1i)2 ≥ 2λminkx0 −x1k2.
T`u. biˆe˙’u th´u.c cuˆo´i v`a d¯i.nh ngh˜ıa h`am h2(γ) trong Mˆe.nh d¯ˆe` 1.5.7 ta nhˆa.n d¯u.o..c
h2(γ) := inf
x0,x1∈D,kx0−x1k=γ,−x0+2x1∈D f(x0)−2f(x1)+f(−x0+2x1) ≥ 2λminγ2.
(4.1.1) Theo gia˙’ thiˆe´t (a) v`a biˆe˙’u th´u.c (4.1.1) th`ı
|p(x)| ≤ h2(γ)/4 v´o.i mo.i x ∈ D,
Tu.o.ng tu.. theo gia˙’ thiˆe´t (b) v`a biˆe˙’u th´u.c (4.1.1) ta c˜ung c´o
|p(x)| < h2(γ)/4 v´o.i mo.i x ∈ D.
´
Ap du.ng Mˆe.nh d¯ˆe` 1.5.7 ta suy ra ˜f l`a γ-lˆ`i trong ngˇo a.t. Vˆa.y (a), (b) d¯u.o..c ch´u.ng minh.
Mˆe.nh d¯ˆe` 4.1.24. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1), p tho˙’a m˜an