O˙’ mu.c tru.´o.c, trong Mˆe.nh d¯ˆe` 2.2.15 ch´ung tˆoi d¯˜a nghiˆen c´u.u d¯u.`o.ng. k´ınh cu˙’a tˆa.p c´ac d¯iˆe˙’m cu. c tiˆ. e˙’u to`an cu.c cu˙’a B`ai to´an quy hoa.ch to`an phu.o.ng lˆ`i ngˇo a.t v´o.i nhiˆe˜u gi´o.i nˆo.i ( ˜P). Trong mu.c n`ay, ch´ung tˆoi nghiˆen c´u.u d¯u.`o.ng k´ınh cu˙’a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c v`a t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P) theo cˆa.n trˆen s cu˙’a h`am nhiˆe˜u p.
Nghiˆen c´u.u c´ac d¯iˆe˙’m infimum to`an cu.c, trong mu.c n`ay ta su˙’ du.ng. h`am bao d¯´ong nu.˙’a liˆen tu.c du.´o.i (xem [54], trang 68-90) lsc ˜f(x) := lim infy→x, y∈Df˜(y) v`a c´o bˆo˙’ d¯ˆ` sau:e
Bˆo˙’ d¯ˆ` 2.3.1.e X´et h`am bi. nhiˆe˜u gi´o.i nˆo.i f˜= f +p. Khi d¯´o (a) V´o.i mo. i x ∈ D th`ılsc ˜f(x) = f(x) + lscp(x),
(b) supx∈D|lscp(x)| ≤ supx∈D|p(x)| ≤ s.
Ch´u.ng minh. (a) Ta c´o lsc ˜f(x) = lim inf y→x, y∈D ˜ f(y) = inf{η : ∃yn →x, yn ∈ D, η = lim n→∞ ˜ f(yn)}.
V`ıf(x) liˆen tu.c nˆen
lsc ˜f(x) = lim inf y→x, y∈D ˜ f(y) = f(x) + inf{η0 : ∃yn → x, yn ∈ D, η0 = lim n→∞p(yn)}.
Do d¯´o, d¯ˆo´i v´o.i h`am to`an phu.o.ng lˆ`i ngˇo a.t v´o.i nhiˆe˜u gi´o.i nˆo.i th`ı lsc ˜f(x) = f(x) + lscp(x). (b) Ta c´o |lscp(x)| = | lim inf y→x, x∈Dp(x)| ≤ lim inf y→x, x∈D|p(x)| ≤ lim inf y→x, x∈Dsup x∈D |p(x)|, nˆen sup x∈D |lscp(x)| ≤ sup x∈D |p(x)| ≤s < +∞.
Bˆo˙’ d¯ˆ` d¯u.o.e . c ch´u.ng minh.
V`ı tˆa.p c´ac d¯iˆe˙’m cu. c tiˆ. e˙’u to`an cu.c l`a tˆa.p con cu˙’a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c, nˆen mˆe.nh d¯ˆe` sau l`a tru.`o.ng ho.. p tˆo˙’ng qu´at cu˙’a Mˆe.nh d¯ˆe` 2.2.15.
Mˆe.nh d¯ˆe` 2.3.16. Nˆe´u x˜∗1, x˜∗2 l`a hai d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a B`ai to´an ( ˜P) th`ı
kx˜∗1 −x˜∗2k ≤ γ∗.
Ch´u.ng minh. V`ı ˜x∗1, x˜∗2 l`a hai d¯iˆe˙’m infimum to`an cu.c bˆa´t k`y cu˙’a B`ai to´an ( ˜P), nˆen theo Mˆe.nh d¯ˆe` 1.3.1 v`a (a) cu˙’a Bˆo˙’ d¯ˆ` 2.3.1 ta suy ra ˜e x∗1, x˜∗2 l`a c´ac d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a lsc ˜f = f + lscp.
Mˇa.t kh´ac, t`u. (b) cu˙’a Bˆo˙’ d¯ˆe` ta c´o 2 r 2 sup x∈D |lscp(x)|/λmin ≤ 2 r 2 sup x∈D |p(x)|/λmin ≤2p2s/λmin = γ∗,
nˆen theo Mˆe.nh d¯ˆe` 2.1.11 ta suy ra lsc ˜f = f + lscp l`a γ-lˆ`i ngo`o ai ngˇa.t khi
γ > γ∗. Ap du.ng Mˆe.nh d¯ˆe´ ` 2.2.15 cho h`am lsc ˜f = f + lscp ta nhˆa.n d¯u.o..c
kx˜∗1 −x˜∗2k ≤ γ∗.
V´ı du. 2.3.6. X´et c´ac h`am f(x) = x2, p(x) = ( −0.5 nˆe´u |x| ≥ 1 0.5 nˆe´u |x| < 1. Khi d¯´o ˜ f(x) = f(x) + p(x) = ( x2 −0.5 nˆe´u |x| ≥1 x2 + 0.5 nˆe´u |x|< 1,
c´o ba d¯iˆe˙’m infimum to`an cu.c l`a x1 = −1, x2 = 0, x3 = 1, s = supx∈IR|p(x)| = 0.5, λmin = 1 v`a γ∗ = 2p2s/λmin = 2√
2×0.5 = 2. Theo mˆe.nh d¯ˆe` trˆen th`ı
2 = max{kxi −xjk | i, j = 1,2,3} ≤ 2 = 2p2s/λmin,
suy ra
max{kxi −xjk | i, j = 1,2,3} = γ∗.
Biˆe˙’u th´u.c cuˆo´i cho ph´ep kˆe´t luˆa.n d¯u.`o.ng k´ınh cu˙’a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa.t v´o.i nhiˆe˜u n´oi chung khˆong nho˙’ ho.n γ∗.
Khi x´et l´o.p h`am to`an phu.o.ng lˆ`i ngˇo a.t v´o.i nhiˆe˜u gi´o.i nˆo.i, cˆau ho˙’i d¯u.o..c d¯ˇa.t ra l`a: C´ac d¯iˆe˙’m infimum to`an cu.c cu˙’a h`am n`ay thay d¯ˆo˙’i nhu. thˆe´ n`ao so v´o.i d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t x∗ cu˙’a h`am to`an phu.o.ng (nˆe´u tˆ`no ta.i)? c´o thˆe˙’ d¯´anh gi´a d¯u.o..c khoa˙’ng c´ach gi˜u.a ch´ung hay khˆong? D-i.nh l´y sau s˜e tra˙’ l`o.i cho ta cˆau ho˙’i n`ay.
D- i.nh l´y 2.3.13. Nˆe´u x∗ ∈ D l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P), x˜∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a B`ai to´an ( ˜P), th`ı
kx˜∗ −x∗k ≤ γ∗/2.
i) ˜x∗ l`a cu.. c tiˆe˙’u to`an cu.c cu˙’a ˜f = f +p. D- ˇa.t ϕ(t) : = f(x∗ +t(˜x∗ −x∗))−f(x∗) = hAx∗, x∗i +hb, x∗i+h2Ax∗ +b,x˜∗ −x∗it +hA(˜x∗ −x∗),x˜∗ −x∗it2 − hAx∗, x∗i+hb, x∗i = h2Ax∗ +b,x˜∗ −x∗it+hA(˜x∗ −x∗),x˜∗ −x∗it2. Do d¯´o ϕ0(t) = h2Ax∗ +b,x˜∗ −x∗i+ 2hA(˜x∗ −x∗),x˜∗ −x∗it (2.3.8) v`a ϕ00(t) = 2hA(˜x∗ −x∗),x˜∗ −x∗i. (2.3.9) Mˇa.t kh´ac, v`ıx∗ l`a cu.. c tiˆe˙’u to`an cu.c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa.t f trˆen
D, nˆen ϕ(t) ≥ 0 v´o.i mo.i t∈ [0,1]. Ta c´o ϕ0(0) = lim t&0 ϕ(t)−ϕ(0) t = limt&0 ϕ(t) t ≥ 0.
Kˆe´t ho.. p biˆe˙’u th´u.c (2.3.8) khi t = 0 v´o.i bˆa´t d¯ˇa˙’ng th´u.c trˆen ta suy ra
h2Ax∗ +b,x˜∗ −x∗i ≥ 0. (2.3.10)
Theo cˆong th´u.c Tay lo th`ı
ϕ(t) = ϕ(0) +ϕ0(0)t+ ϕ
00(0)
2 t
2
nˆen, v´o.i t= 12 ta suy ra
ϕ(1 2) = ϕ 0(0)1 2 +ϕ 00(0)1 8.
Thay c´ac gi´a tri. cu˙’a ϕ0(0), ϕ00(0) theo c´ac cˆong th´u.c (2.3.8) v`a (2.3.9) ta nhˆa.n d¯u.o..c
f(x˜ ∗ +x∗ 2 ) = f(x ∗)+1 2h2Ax∗+b,x˜∗−x∗i+1 4hA(˜x∗−x∗),x˜∗−x∗i. (2.3.11) Theo biˆe˙’u th´u.c (2.1.2) th`ı
1 4 D A(˜x∗ −x∗),x˜∗ −x∗ E = 1 2f(˜x ∗) + 1 2f(x ∗)−f(x˜ ∗ +x∗ 2 ), (2.3.12)
nˆen thay f(x˜∗+2x∗) o.˙’ (2.3.11) v`ao (2.3.12), chuyˆe˙’n vˆe´ v`a r´ut go.n ta d¯u.o..c
D
A(˜x∗ −x∗),x˜∗ −x∗
E
= f(˜x∗)−f(x∗)− h2Ax∗ +b,x˜∗ −x∗i.
Kˆe´t ho.. p v´o.i (2.3.10) suy ra
A(˜x∗ −x∗),x˜∗ −x∗ ≤ f(˜x∗)−f(x∗). (2.3.13) D- ˇa.t η := kx˜∗ −x∗k, t`u. (2.1.3) v`a (2.3.13) suy ra
λminη2 ≤ f(˜x∗)−f(x∗). (2.3.14)
Mˇa.t kh´ac, v`ı ˜x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a ˜f = f + p nˆen
f(˜x∗) + p(˜x∗) ≤f(x∗) +p(x∗).
Kˆe´t ho.. p v´o.i supx∈D|p(x)| ≤s, ta suy ra
0≤ f(˜x∗)−f(x∗) ≤2s. (2.3.15)
Thay (2.3.15) v`ao (2.3.14) ta nhˆa.n d¯u.o..c
λminη2 ≤ 2s
tu.o.ng d¯u.o.ng v´o.i
η ≤ p2s/λmin.
ii) ˜x∗ l`a d¯iˆe˙’m infimum to`an cu.c (khˆong l`a d¯iˆe˙’m cu. c tiˆ. e˙’u to`an cu.c) cu˙’a ˜
f = f + p. Khi d¯´o ˜x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a lsc ˜f = f + lscp. Su.˙’ du.ng i) cho h`am lsc ˜f ta c˜ung nhˆa.n d¯u.o..c
f(˜x∗)−f(x∗) ≤lscp(˜x∗)−lscp(x∗).
D- ˇa.t η := kx˜∗ −x∗k. Theo Bˆo˙’ d¯ˆ` 2.3.1 th`ıe
lscp(˜x∗)−lscp(x∗) ≤ 2s,
nˆen
Kˆe´t ho.. p (2.3.16) v´o.i (2.3.13), ta suy ra λminη2 ≤ 2s, v`ı vˆa.y η ≤ p2s/λmin, t´u.c l`a kx˜∗ −x∗k ≤ γ∗/2.
D- i.nh l´y d¯˜a d¯u.o..c ch´u.ng minh.
D- i.nh l´y trˆen d¯˜a d¯u.o..c H. X. Phu ch´u.ng minh rˆa´t go.n trong [51].
2.4. T´ınh chˆa´t tu.. a v`a d¯iˆ` u kiˆe e.n tˆo´i u.u
Trong mu.c n`ay, ta nghiˆen c´u.u t´ınh chˆa´t tu..a cu˙’a h`am ˜f = f + p v`a su.. tˆ`n ta.i c´ac d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜o P), cu. thˆe˙’ l`a D- i.nh l´y Kuhn-Tucker suy rˆo.ng cho B`ai to´an ( ˜P) khi D l`a tˆa.p lˆo`i tho˙’a m˜an mˆo.t trong hai tru.`o.ng ho.. p sau:
D = {x ∈ S | gi(x) ≤ 0, i = 1, . . . , m}, (2.4.17)
trong d¯´o gi : IRn → IR, i = 1, . . . , m, l`a c´ac h`am lˆ`i v`o a S ⊂ IRn l`a tˆa.p lˆo`i d¯´ong, hoˇa.c
D = {x ∈ IRn | hci, xi ≤ di, i= 1, . . . , m}. (2.4.18)
Mˆo.t t´ınh chˆa´t d¯ˇa.c biˆe.t cu˙’a h`am lˆo`i bˆa´t k`y g : IRn →IR l`a v´o.i x∗ ∈ IRn
n`ao d¯´o, tˆ`n ta.io ξ ∈ IRn go.i l`a du.´o.i vi phˆan sao cho
g(x) ≥ g(x∗) +hξ, x−x∗i, v´o.i mo.i x ∈ IRn,
t´u.c l`a ta.i mo.i d¯iˆe˙’m x∗ ∈ IRn h`am lˆ`io g tu.. a trˆen mˆo.t h`am tuyˆe´n t´ınh
g(x∗) + hξ, x − x∗i v`a ta go.i l`a t´ınh chˆa´t tu. a cu˙’a. g. Trong tru.`o.ng ho.. p
g = f th`ıξ = 2Ax∗ + b. V`ı h`am nhiˆe˜u p chı˙’ gia˙’ thiˆe´t gi´o.i nˆo.i nˆen khˆong hy vo.ng h`am bi. nhiˆe˜u ˜f = f +p c´o t´ınh chˆa´t tu.. a nhu. trˆen. Tuy nhiˆen, ta
s˜e chı˙’ ra h`am lˆ`i thˆo o ˜f s˜e c˜ung c´o t´ınh chˆa´t tu.. a thˆo. Muˆo´n vˆa.y, ta viˆe´t la.i t´ınh chˆa´t tu.. a nhu. sau:
g(x∗) +hξ, x∗i ≤ g(x) +hξ, xi, v´o.i mo.i x ∈ IRn. (2.4.19) Thay thˆe´ vˆe´ tr´ai cu˙’a (2.4.19) bo.˙’ i
inf
x0∈B(x∗,r)
˜
f(x0)− hξ, x0i hoˇa.c min
x0∈B¯(x∗,r)
˜
f(x0)− hξ, x0i
v´o.i mˆo.t r > 0 ho.. p l´y n`ao d¯´o v`a vˆe´ pha˙’i cu˙’a (2.4.19) bo.˙’ i ˜f(x) − hξ, xi ta d¯u.o.. c inf x0∈B(x∗,r) ˜ f(x0)− hξ, x0i ≤f˜(x)− hξ, xi v´o.i mo.i x ∈ IRn, hoˇa.c min x0∈B¯(x∗,r) ˜ f(x0)− hξ, x0i ≤f˜(x)− hξ, xi v´o.i mo.i x ∈ IRn.
Nh˜u.ng cˆong th´u.c trˆen mˆo ta˙’ t´ınh chˆa´t tu.. a thˆo cu˙’a h`am ˜f . T´ınh chˆa´t n`ay d¯˜a d¯u.o.. c H. X. Phu chı˙’ ra khi nghiˆen c´u.u c´ac h`am γ-lˆ`i ngo`o ai tˆo˙’ng qu´at [44]. Bˇa`ng c´ach su.˙’ du.ng D- i.nh l´y 2.3.13 cho h`am to`an phu.o.ng lˆo`i ngˇa.t v´o.i nhiˆe˜u gi´o.i nˆo.i ˜f = f + p, mˆe.nh d¯ˆe` du.´o.i d¯ˆay cho ta kˆe´t qua˙’ tˆo´t ho.n, t´u.c l`a chı˙’ ra r = γ∗/2 v`a ξ = 2Ax∗ +b.
Mˆe.nh d¯ˆe` 2.4.17. ([51]) Cho D = IRn. Khi d¯´o v´o.i x∗ ∈ IRn v`a > 0 th`ı inf
x0∈B(x∗,γ∗/2+)
˜
f(x0)− hξ, x0i
≤ f˜(x)− hξ, xi v´o.i mo. i x ∈ IRn,
D- ˇa.c biˆe.t, nˆe´u p l`a nu.˙’a liˆen tu.c du.´o.i th`ı min
x0∈B¯(x∗,γ∗)
˜
f(x0)− hξ, x0i
≤f˜(x)− hξ, xi v´o.i mo. i x ∈ IRn.
Mˆe.nh d¯ˆe` trˆen d¯u.o.. c H. X. Phu chı˙’ ra. Ch´u.ng minh chi tiˆe´t c´o thˆe˙’ xem trong [51]
Trong qu´a tr`ınh kha˙’o s´at d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜P), ch´ung tˆoi su.˙’ du.ng bˆo˙’ d¯ˆe` sau:
Bˆo˙’ d¯ˆ` 2.4.2.e Cho h`am g : D ⊂ IRn → IR l`a nu.˙’a liˆen tu.c du.´o.i, bi. chˇa.n du.´o.i, D l`a tˆa. p d¯´ong v`a limkxk→+∞, x∈Dg(x) = +∞. Khi d¯´o tˆ`n ta.io x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu. c cu˙’a g trˆen D.
Ch´u.ng minh. Bˆo˙’ d¯ˆ` n`e ay c´o thˆe˙’ ch´u.ng minh nhu. l`a hˆe. qua˙’ cu˙’a D- i.nh l´y 8.2 (xem [76], trang 119-121). Tuy nhiˆen c´o thˆe˙’ ch´u.ng minh tru.. c tiˆe´p nhu. sau:
Cˆo´ d¯i.nh d¯iˆe˙’m x0 ∈ D. V`ı limkxk→∞, x∈Dg(x) = +∞ nˆen
∃r ∈ [kx0k,+∞[ : x ∈ D, kxk > r ⇒g(x) > g(x0).
Do d¯´o, d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c nˆe´u c´o th`ı s˜e chı˙’ nˇa`m trong miˆe` nB(0, r)∩D.
V`ıg bi. chˇa.n du.´o.i nˆen
inf
x∈B(0,r)∩D
g(x) > −∞.
Mˇa.t kh´ac, tˆo`n ta.i d˜ay (xi) ⊂ B(0, r)∩ D sao cho lim
i→∞g(xi) = inf
x∈B(0,r)∩D g(x).
Tˆa.p B(0, r)∩D l`a d¯´ong, gi´o.i nˆo.i trong IRn nˆen l`a tˆa.p compact, v`ı vˆa.y t`u. d˜ay (xi) ⊂B(0, r)∩D c´o thˆe˙’ tr´ıch d˜ay con hˆo.i tu.. Khˆong gia˙’m tˆo˙’ng qu´at, ta coi ch´ınh d˜ay d¯´o hˆo.i tu., t´u.c l`a limi→∞xi = x∗ v`a x∗ ∈ B(0, r)∩D. V`ı
g l`a nu.˙’a liˆen tu.c du.´o.i nˆen
g(x∗) ≤ lim
i→∞g(xi) = inf
x∈B(0,r)∩D g(x).
Do d¯´o x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u trˆen B(0, r)∩D. V`ı d¯iˆe˙’m cu.. c tiˆe˙’u cu˙’a g trˆen
B(0, r) ∩ D l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c, nˆen x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a g trˆen D.
Tru.´o.c khi ph´at biˆe˙’u v`a ch´u.ng minh D- i.nh l´y Kuhn-Tucker suy rˆo.ng cho B`ai to´an ( ˜P), ta nhˇa´c la.i h`am Lagrange cho B`ai to´an ( ˜P) d¯u.o.. c d¯i.nh ngh˜ıa theo cˆong th´u.c (1.1.4), t´u.c l`a
L(x, µ0, . . . , µm) =µ0f(x) +
m
X
i=1
D- i.nh l´y 2.4.14. Gia˙’ su.˙’ D d¯u.o.. c cho bo˙’ i cˆ. ong th´u.c (2.4.17).
(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 d¯iˆe˙’m x∗ ∈ D sao cho
kx˜∗ −x∗k ≤ γ∗/2
v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, 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) (2.4.20) v`a d¯iˆ` u kiˆe.n b`ue
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m. (2.4.21)
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∗ ∈ D tho˙’a m˜an (2.4.20), (2.4.21) v´o.i µ0 = 1 th`ı tˆ`n ta.io ˜
x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P) trˆenD, tho˙’a m˜an
kx˜∗ −x∗k ≤ γ∗/2
v`a khˆong c´o d¯iˆe˙’m infimum to`an cu. c n`ao cu˙’a B`ai to´an ( ˜P) nˇa`m ngo`ai h`ınh cˆ` ua B(x∗, γ∗/2).
Ch´u.ng minh. (a) X´et tˆa.p D, v`ıS l`a lˆ`i d¯´o ong, gi(x), i = 1, . . . , m, l`a c´ac h`am lˆ`i nˆeno D = {x ∈ S |gi(x) ≤ 0, i = 1, . . . m} c˜ung l`a lˆ`i d¯´o ong. Khi d¯´o i) Nˆe´u D gi´o.i nˆo.i, th`ı v`ıf(x) = hAx, xi + hb, xi l`a lˆ`i ngˇo a.t, liˆen tu.c nˆen tˆ`n ta.i duy nhˆa´t d¯iˆe˙’m cu..c tiˆe˙’u to`an cu.co x∗ ∈ D.
ii) Nˆe´u D khˆong gi´o.i nˆo.i, th`ı
f(x) = hAx, xi+hb, xi ≥ λminkxk2 − kbkkxk,
nˆen
lim
H`am to`an phu.o.ng lˆ`i ngˇo a.t f trˆen tˆa.p lˆo`i D tho˙’a m˜an c´ac d¯iˆ` u kiˆe.n cu˙’ae Bˆo˙’ d¯ˆ` 2.4.2, do d¯´e o tˆ`n ta.i d¯iˆe˙’m cu..c tiˆe˙’u to`an cu.c v`a duy nhˆa´to x∗ trˆen D.
V`ı ˜x∗ l`a d¯iˆe˙’m infimum to`an cu.c cu˙’a ˜f = f +p, nˆen theo D- i.nh l´y 2.3.13, ta nhˆa.n d¯u.o..c
kx˜∗ −x∗k ≤ γ∗/2.
Mˇa.t kh´ac, x∗ l`a cu.. c tiˆe˙’u to`an cu.c cu˙’a f trˆen D, t´u.c l`a x∗ l`a nghiˆe.m cu˙’a B`ai to´an (P),do d¯´o theo (a) cu˙’a D- i.nh l´y Kuhn-Tucker 1.1.1 suy ra tˆo`n ta.i µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an
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, nˆen c´o thˆe˙’ cho.n
µ0 = 1.
(b) V`ıx∗ ∈ D tho˙’a m˜an d¯iˆ` u kiˆe.n (2.4.20), (2.4.21) v´o.ie µ0 = 1 nˆen x∗
tho˙’a m˜an (b) cu˙’a D- i.nh l´y Kuhn-Tucker 1.1.1 cho B`ai to´an (P). Do d¯´o x∗
l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c cu˙’a f trˆen D v`a v`ıf l`a lˆ`i ngˇo a.t nˆen x∗ l`a d¯iˆe˙’m cu.. c tiˆe˙’u to`an cu.c duy nhˆa´t cu˙’a f trˆen D.
Ta ch´u.ng minh tˆ`n ta.i d¯iˆe˙’m infimum to`an cu.c cu˙’a B`ai to´an ( ˜o P). Thˆa.t vˆa.y, x´et h`am lsc ˜f = f + lscp trˆen D, xa˙’y ra c´ac tru.`o.ng ho.. p sau:
i) D gi´o.i nˆo.i, khi d¯´o tˆo`n ta.i M > 0 sao cho kxk ≤ M v´o.i mo.i x ∈ D.
V`ı vˆa.y lsc ˜f(x) ≥ hAx, xi − kbkkxk −sup x∈D |p(x)| ≥ −kbkM −sup x∈D |p(x)|,
suy ra h`am bi. chˇa.n du.´o.i trˆen D. V`ı h`am lsc ˜f = f + lscp nu.˙’a liˆen tu.c