BỘ GIÁO DỤC VÀ ĐÀO TẠO BỘ QUỐC PHÒNG HỌC VIỆN KỸ THUẬT QUÂN SỰ Bài toán quy hoạch toàn phương lồi ngặt với nhiễu giới nội Võ Minh Phổ Chuyên ngành: Toán học Mã số: 62 46 30 01 Người hướng dẫn khoa học: GS TSKH Hoàng Xuân P PGS TS Phan Thành Anh 2011 `.I CAM D - OAN LO Tˆoi xin cam d¯oan nh˜ u.ng kˆe´t qua˙’ d¯u.o c tr`ınh b`ay luˆa.n a´n l`a m´o.i, d¯a˜ d¯u.o c cˆong bˆo´ trˆen c´ac ta.p ch´ı To´an ho.c quˆo´c tˆe´ C´ac kˆe´t qua˙’ viˆe´t u v`a PGS TS Phan Th`anh An d¯a˜ chung v´o.i GS TSKH Ho`ang Xuˆan Ph´ d¯u.o c su d¯`oˆng y ´ cu˙’a c´ac d¯`ˆong t´ac gia˙’ d¯u.a v`ao luˆa.n ´an C´ac kˆe´t qua˙’ nˆeu luˆa.n a´n l`a trung thu c v`a chu.a t` u.ng d¯u.o c cˆong bˆo´ bˆa´t k` y cˆong tr`ınh n`ao kh´ac tru.o´.c d¯o´ Nghiˆen c´ u.u sinh ˙’ M O.N `.I CA LO Luˆa.n ´an d¯u.o c ho`an th`anh du.o´.i su hu.o´.ng dˆa˜n, chı˙’ ba˙’o cu˙’a GS TSKH Ho`ang Xuˆan Ph´ u v`a PGS TS Phan Thanh An T´ac gia˙’ chˆan th`anh ca˙’m `ay d¯˜a d`anh cho T´ac gia˙’ b`ay to˙’ l`ong o.n su gi´ up d¯o˜ mo.i mˇa.t m`a c´ac Thˆ `ay d¯a˜ u, Thˆ biˆe´t o.n sˆau sˇa´c v`a chˆan th`anh t´o.i GS TSKH Ho`ang Xuˆan Ph´ `eu kiˆe.n d¯ˆe˙’ t´ac quan tˆam, hu.o´.ng dˆa˜n tˆa.n t`ınh, nghiˆem khˇa´c v`a ta.o mo.i d¯iˆ gia˙’ c´o thˆe˙’ ho`an th`anh nh˜ u.ng mu.c tiˆeu d¯ˇa.t cho luˆa.n a´n T´ac gia˙’ xin - oˆng Yˆen, PGS TS Ta Duy b`ay to˙’ l`ong biˆe´t o.n d¯ˆe´n GS TSKH Nguyˆ˜en D Phu.o ng, PGS TS Nguyˆ˜en Nˇang Tˆam v`a c´ac d¯`ˆong nghiˆe.p thuˆo.c Ph`ong Gia˙’i t´ıch sˆo´ v`a T´ınh to´an Khoa ho.c Viˆe.n To´an ho.c v`ı d¯a˜ c´o nh˜ u.ng y ´ kiˆe´n qu´ y b´au cho t´ac gia˙’ qu´a tr`ınh nghiˆen c´ u.u T´ac gia˙’ xin d¯u.o c b`ay to˙’ l`ong ca˙’m o.n d¯ˆe´n Ban chu˙’ nhiˆe.m Khoa Cˆong Nghˆe thˆong tin, Ph`ong Sau d¯a.i ho.c v`a Ban Gi´am d¯oˆ´c Ho.c viˆe.n K˜ y thuˆa.t `eu kiˆe.n thuˆa.n lo i d¯ˆe˙’ t´ac gia˙’ c´o nhiˆ `eu th`o.i gian thu c Quˆan su d¯a˜ ta.o mo.i d¯iˆ hiˆe.n luˆa.n ´an - a`o Thanh T˜ınh, T´ac gia˙’ c˜ ung b`ay to˙’ l`ong biˆe´t o.n d¯ˆe´n PGS TS D -u PGS TS Nguyˆ˜en D ´.c Hiˆe´u, PGS TS Nguyˆ˜en Thiˆe.n Luˆa.n, PGS TS `ong, TS Nguyˆ˜en H˜ Tˆo Vˇan Ban, TS Nguyˆ˜en Nam Hˆ u.u Mˆo.ng, TS V˜ u Thanh H`a, TS Nguyˆ˜en Ma.nh H` ung, TS Nguyˆ˜en Tro.ng To`an, TS Ngˆo -u - `ınh So.n, TS Trˆ `an Nguyˆen Ngo.c H˜ u.u Ph´ uc, TS Tˆo´ng Minh D ´.c, TS Lˆe D v`a tˆa´t ca˙’ c´ac d¯`oˆng nghiˆe.p Khoa Cˆong Nghˆe thˆong tin, HVKTQS, d¯a˜ d¯ˆo.ng viˆen, kh´ıch lˆe v`a c´o nh˜ u.ng trao d¯oˆ˙’i h˜ u.u ´ıch suˆo´t th`o.i gian nghiˆen c´ u.u v`a cˆong t´ac T´ac gia˙’ ca˙’m o.n sˆau sˇa´c GS TSKH Pha.m Thˆe´ Long, Gi´am d¯ˆo´c Ho.c `eu kiˆe.n vˆ `e mˇa.t thu˙’ tu.c c˜ Viˆe.n KTQS, ngu.o`.i d¯a˜ ta.o mo.i d¯iˆ ung nhu chuyˆen mˆon d¯ˆe˙’ t´ac gia˙’ c´o thˆe˙’ ho`an th`anh luˆa.n ´an n`ay Cuˆo´i c` ung t´ac gia˙’ gu˙’.i l`o.i c´am o.n t´o.i vo v`a c´ac con, nh˜ u.ng ngu.o`.i d¯a˜ `eu kiˆe.n cho t´ac gia˙’ qu´a tr`ınh l`am d¯oˆ ng viˆen, chˇam s´oc v`a ta.o mo.i d¯iˆ luˆa.n ´an Mu.c lu.c L` o.i cam d ¯oan L` o.i ca˙’m o.n Danh mu.c c´ ac k´ y hiˆ e.u thu.` o.ng d` ung `au Mo˙’ d ¯ˆ `oi `oi, quy hoa.ch to` a h` am lˆ B` to´ an quy hoa.ch lˆ an phu.o.ng v` thˆ o `oi, quy hoa.ch to`an phu.o.ng 1.1 B`ai to´an quy hoa.ch lˆ `oi suy rˆo.ng thˆo 1.2 H`am lˆ 12 `oi ngo`ai 1.3 H`am γ-lˆ 13 `oi ngo`ai 1.4 H`am Γ-lˆ 15 `oi 1.5 H`am γ-lˆ 17 - iˆ D e˙’m infimum to` an cu.c cu˙’a B` to´ an (P˜ ) `oi ngo`ai cu˙’a h`am bi nhiˆ˜eu 2.1 T´ınh γ-lˆ - iˆe˙’m cu c tiˆe˙’u to`an cu.c v`a d¯iˆe˙’m infimum to`an cu.c 2.2 D 2.3 C´ac t´ınh chˆa´t cu˙’a d¯iˆe˙’m infimum to`an cu.c `eu kiˆe.n tˆo´i u.u 2.4 T´ınh chˆa´t tu a v`a d¯iˆ 20 20 27 28 33 ˜ `oi ngo` T´ınh Γ-lˆ cu˙’a h` am bi nhiˆ e u v` a d ¯iˆ e˙’m infimum to` an cu.c cu˙’a B` to´ an (P˜ ) `oi ngo`ai cu˙’a h`am bi nhiˆ˜eu 3.1 T´ınh Γ-lˆ - iˆe˙’m infimum to`an cu.c cu˙’a b`ai to´an nhiˆ˜eu 3.2 D 43 3.3 T´ınh oˆ˙’n d¯.inh cu˙’a tˆa.p c´ac d¯iˆe˙’m infimum to`an cu.c `eu kiˆe.n tˆo´i u.u 3.4 Du.o´.i vi phˆan suy rˆo.ng thˆo v`a d¯iˆ 55 ˜ - iˆ D e˙’m supremum cu˙’a B` to´ an (Q) `oi cu˙’a h`am bi nhiˆ˜eu 4.1 T´ınh γ-lˆ - iˆe˙’m supremum to`an cu.c cu˙’a h`am bi nhiˆ˜eu 4.2 D 4.3 T´ınh chˆa´t cu˙’a tˆa.p c´ac d¯iˆe˙’m supremum to`an cu.c 4.4 T´ınh chˆa´t cu˙’a tˆa.p c´ac d¯iˆe˙’m supremum d¯.ia phu.o.ng 43 52 58 64 64 66 73 86 Kˆ e´t luˆ a.n chung 94 Danh mu.c cˆ ong tr`ınh cu˙’a t´ ac gia˙’ liˆ en quan d ¯ˆ e´n luˆ a.n ´ an 96 T` liˆ e.u tham kha˙’o 97 `.NG DUNG ´ KY ´ HIE ˆ U THU.O ` DANH MU C CAC `eu • IRn : Khˆong gian Euclide n chiˆ • · : Chuˆa˙’n Euclide IRn • x, y : T´ıch vˆo hu.o´.ng cu˙’a v´ec to x, y `au mo˙’ b´an k´ınh r tˆam x • B(x, r) := {y | y − x < r} : H`ınh cˆ ¯ r) := {y | y − x ≤ r} : H`ınh cˆ `au d¯o´ng b´an k´ınh r tˆam x • B(x, • A ∈ IRn×n , A : Ma trˆa.n d¯oˆ´i x´ u.ng x´ac d¯.inh du.o.ng • AT : Ma trˆa.n chuyˆe˙’n vi cu˙’a ma trˆa.n A • λmin , (λmax ) : Gi´a tri riˆeng nho˙’ nhˆa´t (l´o.n nhˆa´t) cu˙’a ma trˆa.n A • λ(A) : Tˆa.p c´ac gi´a tri riˆeng cu˙’a ma trˆa.n A √ • A = { max λ | λ ∈ λ(AT A)} : Chuˆa˙’n cu˙’a ma trˆa.n A IRn×n `oi ngˇa.t • f (x) = Ax, x + b, x : H`am to`an phu.o.ng lˆ • p(x), supx∈D |p(x)| ≤ s v´o.i s ∈ [0, +∞[ : H`am nhiˆ˜eu gi´o.i nˆo.i `oi ngˇa.t v´o.i nhiˆ˜eu gi´o.i nˆo.i • f˜ = f + p : H`am to`an phu.o.ng lˆ • f (x) := Ax, x + b, x → inf, x ∈ D : B`ai to´an quy hoa.ch to`an phu.o.ng (P ) • f (x) := Ax, x + b, x → sup, x ∈ D : B`ai to´an quy hoa.ch to`an phu.o.ng (Q) • f (x) := Ax, x + b, x + p(x) → inf, x ∈ D : B`ai to´an quy hoa.ch `oi ngˇa.t v´o.i nhiˆ˜eu (P˜ ) to`an phu.o.ng lˆ • f (x) := Ax, x + b, x + p(x) → sup, x ∈ D : B`ai to´an quy hoa.ch ˜ `oi ngˇa.t v´o.i nhiˆ˜eu (Q) to`an phu.o.ng lˆ • ∂g(x∗ ) : Du.o´.i vi phˆan cu˙’a g ta.i d¯iˆe˙’m x∗ • L(x, µ0 , , µm ) := m i=0 µi gi (x) : H`am Lagrange • T´ınh chˆa´t (Mγ ) : Mˆo˜i d¯iˆe˙’m γ-cu c tiˆe˙’u x∗ cu˙’a f l`a d¯iˆe˙’m cu c tiˆe˙’u to`an cu.c • T´ınh chˆa´t (Iγ ) : Mˆo˜i d¯iˆe˙’m γ-infimum x∗ cu˙’a f l`a d¯iˆe˙’m infimum to`an cu.c • Lα (f˜) := {x | x ∈ D, f˜(x) ≤ α}, α ∈ IR : Tˆa.p m´ u.c du.o´.i cu˙’a h`am f˜ = f + p (f (x0 ) • h1 (γ) := inf x0 , x1 ∈D, x0 −x1 =γ • h2 (γ) := inf x0 , x1 ∈D, x0 −x1 =γ,−x0 +2x1 ∈D + f (x1 )) − f ( 12 (x0 + x1 )) f (x0 )−2f (x1 )+f (−x0 +2x1 ) • aff D : Bao aphin cu˙’a tˆa.p D `oi d¯a diˆe.n D • ext D : Tˆa.p c´ac d¯iˆe˙’m cu c biˆen cu˙’a tˆa.p lˆ • JD (x∗ ) := ext D \ {x∗ }, x∗ ∈ ext D • d(x, D) := inf y∈D x − y : Khoa˙’ng c´ach t` u x d¯ˆe´n D `oi cu˙’a tˆa.p D • conv D : Bao lˆ • dD := minx∗ ∈ext D {d x∗ , conv JD (x∗ ) } • D(x∗ , β) := {x ∈ D | x = (1 − α)x∗ + αy, y ∈ D, ≤ α ≤ − β}, x∗ ∈ ext D, β ∈ [0, 1] • C (D) := {p : D → IR | p C0 := supx∈D |p(x)| < +∞} ¯C (0, r) : H`ınh cˆ `au d¯o´ng b´an k´ınh r tˆam C (D) • B ˙’ D ˆU -` MO A `en thˆo´ng c´o da.ng B`ai to´an quy hoa.ch to`an phu.o.ng truyˆ f (x) := Ax, x + b, x → inf, x∈D d¯o´ A ∈ IRn×n l`a ma trˆa.n vuˆong, b ∈ IRn l`a v´ec to v`a D ⊂ IRn l`a tˆa.p `oi lˆ `oi, b`ai to´an quy hoa.ch to`an phu.o.ng C` ung v´o.i b`ai to´an quy hoa.ch lˆ `eu nh`a to´an ho.c Viˆe.t nam v`a quˆo´c tˆe´ nghiˆen c´ d¯u.o c nhiˆ u.u, v´ı du nhu H W Kuhn v`a A W Tucker [22], B Bank v`a R Hasel [5], E Blum v`a W Oettli [7], B C Eaves [12], M Frank v`a P Wolfe [13], O L Magasarian [26], G M Lee, N N Tam v`a N D Yen [31], H X Phu [45], H X Phu v`a N D Yen [53], M Schweighofer [57], H Tuy [63], [64], [72], H H Vui v`a P T Son [66] u.u c´ac b`ai to´an C´ac kˆe´t qua˙’ quan tro.ng d¯a˜ thu d¯u.o c nghiˆen c´ `on ta.i nghiˆe.m tˆo´i `e su tˆ quy hoa.ch to`an phu.o.ng cu˙’a c´ac nh`a to´an ho.c l`a vˆ `eu kiˆe.n cˆ `an tˆo´i u.u, d¯iˆ `eu kiˆe.n d¯u˙’ tˆo´i u.u, thuˆa.t to´an t`ım nghiˆe.m tˆo´i u.u, d¯iˆ u.u, t´ınh oˆ˙’n d¯.inh cu˙’a nghiˆe.m tˆo´i u.u c´ac b`ai to´an trˆen bi t´ac d¯ˆo.ng bo˙’.i `eu kˆe´t qua˙’ nghiˆen c´ `e b`ai to´an trˆen d¯a˜ d¯u.o c u nhiˆ˜eu Nhiˆ u.u vˆ ´.ng du.ng d¯ˆe˙’ gia˙’i c´ac b`ai to´an kinh tˆe´ v`a k˜ y thuˆa.t, nhu b`ai to´an lu a cho.n d¯`ˆau tu (portfolio selection) ([27], [28]), b`ai to´an ph´at d¯iˆe.n tˆo´i u.u (economic power dispatch) ([6], [11], [69]), b`ai to´an kinh tˆe´ d¯oˆ´i s´anh (matching economic), ([17]), b`ai to´an m´ay hˆo˜ tro v´ec to (support vector machine) ([29]) Khi A l`a nu˙’.a x´ac d¯i.nh du.o.ng hoˇa.c nu˙’.a x´ac d¯i.nh ˆam th`ı b`ai to´an trˆen c´o thˆe˙’ phˆan r˜a th`anh hai b`ai to´an kh´ac sau: f (x) := Ax, x + b, x → inf, x∈D (P ) f (x) := Ax, x + b, x → sup, x ∈ D (Q) v`a `oi ngˇa.t Luˆa.n ´an n`ay nghiˆen c´ u.u c´ac b`ai to´an quy hoa.ch to`an phu.o.ng lˆ v´o.i nhiˆ˜eu gi´o.i nˆo.i sau: f˜(x) := Ax, x + b, x + p(x) → inf, x∈D (P˜ ) f˜(x) := Ax, x + b, x + p(x) → sup, x ∈ D, ˜ (Q) v`a `eu kiˆe.n supx∈D |p(x)| ≤ s v´o.i gi´a tri d¯o´ p : D → IR tho˙’a m˜an d¯iˆ ˜ d¯u.o c gia˙’ thiˆe´t l`a s ∈ [0, +∞[ v`a A c´ac b`ai to´an (P ), (Q), (P˜ ) v`a (Q) ma trˆa.n d¯oˆ´i x´ u.ng x´ac d¯.inh du.o.ng V`ı c´ac b`ai to´an trˆen d¯u.o c cho.n d¯ˆe˙’ nghiˆen c´ u.u? R˜o r`ang, s = ˜ ch´ınh l`a c´ac b`ai to´an (P ) v`a (Q), hay n´oi c´ach th`ı c´ac b`ai to´an (P˜ ) v`a (Q) kh´ac c´ac b`ai to´an (P ) v`a (Q) l`a c´ac tru.o`.ng ho p riˆeng cu˙’a c´ac b`ai to´an (P˜ ) ˜ D - aˆy l`a l´ v`a (Q) y d¯ˆe˙’ tiˆe´n h`anh nghiˆen c´ u.u c´ac b`ai to´an trˆen, tˆo´i thiˆe˙’u y thuyˆe´t Tuy nhiˆen, c`on mˆo.t sˆo´ l´ y thu c tˆe´ kh´ac du.o´.i t` u quan d¯iˆe˙’m l´ ˜ l`a thu c su cˆ `an d¯aˆy, cho thˆa´y viˆe.c nghiˆen c´ u.u c´ac b`ai to´an (P˜ ), (Q) L´ y th´ u nhˆa´t: f (x) = Ax, x + b, x l`a h`am mu.c tiˆeu ban d¯`aˆu v`a `om c´ac t´ac d¯oˆ ng bˆo˙’ sung p l`a h`am nhiˆ˜eu n`ao d¯o´ H`am nhiˆ˜eu p c´o thˆe˙’ bao gˆ (tˆa´t d¯.inh hoˇa.c ngˆa˜u nhiˆen) lˆen h`am mu.c tiˆeu v`a c´ac lˆo˜i gˆay qu´a - iˆe˙’m d¯aˇ c biˆe.t l`a o˙’ chˆo˜, ch´ ung tr`ınh mˆo h`ınh h´oa, d¯o d¯a.c, t´ınh to´an D ta ha.n chˆe´ chı˙’ x´et nhiˆ˜eu gi´o.i nˆo.i Ha.n chˆe´ n`ay l`a khˆong qu´a ngˇa.t, c´o thˆe˙’ `eu b`ai to´an thu c tˆe´, chˇa˙’ng ha.n nhu hai v´ı d¯u.o c tho˙’a m˜an nhiˆ du minh ho.a sau d¯ˆay Mˆo.t nh˜ u.ng u ´.ng du.ng nˆo˙’i bˆa.t cu˙’a quy hoa.ch to`an phu.o.ng l`a b`ai to´an lu a cho.n d¯`ˆau tu (H M Markowitz [27], [28]) B`ai to´an ph´at biˆe˙’u nhu sau: Phˆan phˆo´i vˆo´n qua n ch´ u.ng kho´an (asset) c´o sˇa˜n d¯ˆe˙’ c´o thˆe˙’ gia˙’m thiˆe˙’u ru˙’i ro v`a tˆo´i d¯a lo i nhuˆa.n, t´ u.c l`a t`ım v´ec to tı˙’ lˆe x ∈ D, D := {x = (x1 , x2 , , xn ) | nj=1 xj = 1} d¯ˆe˙’ f (x) = ωxT Σx − ρT x d¯a.t gi´a tri nho˙’ nhˆa´t, d¯´o xj , j = 1, , n, l`a ty˙’ lˆe ch´ u.ng kho´an th´ u j danh mu.c d¯`ˆau tu., ω l`a tham sˆo´ ru˙’i ro, Σ ∈ IRn×n l`a ma trˆa.n hiˆe.p phu.o.ng sai, ρ ∈ IRn l`a v´ec to lo i nhuˆa.n k` y vo.ng V`ı Σ v`a ρ thu.o`.ng ˜ v`a ρ˜, d¯o´ ch´ khˆong d¯u.o c x´ac d¯i.nh ch´ınh x´ac m`a chı˙’ xˆa´p xı˙’ bo˙’.i Σ ung ˜ − ρ˜T x = f (x) + p(x), d¯´o ta pha˙’i cu c tiˆe˙’u h´oa h`am f˜(x) = ωxT Σx ˜ − Σ)x − (˜ p(x) = ωxT (Σ ρ − ρ)T x Khi quy d¯.inh, khˆong d¯u.o c b´an khˆo´ng, t´ u.c l`a xj ≥ 0, j = 1, , n, th`ı tˆa.p chˆa´p nhˆa.n d¯u.o c D l`a gi´o.i nˆo.i V`ı vˆa.y nhiˆ˜eu p c˜ ung gi´o.i nˆo.i trˆen D N´oi mˆo.t c´ach tˆo˙’ng qu´at, t´ınh gi´o.i nˆo.i cu˙’a nhiˆ˜eu luˆon d¯u.o c d¯a˙’m ba˙’o D gi´o.i nˆo.i v`a p liˆen tu.c trˆen D Gia˙’ thiˆe´t `eu b`ai to´an thu c tˆe´ n`ay c˜ ung ph` u ho p v´o.i nhiˆ Mˆo.t v´ı du n˜ u.a cho thˆa´y l`a nhiˆ˜eu gi´o.i nˆo.i luˆon xuˆa´t hiˆe.n gia˙’i mˆo.t `an l´o.n c´ac sˆo´ b`ai to´an tˆo´i u.u (P ) hoˇa.c (Q) n`ao d¯´o bˇa` ng m´ay t´ınh Do phˆ `au hˆe´t thu c khˆong thˆe˙’ biˆe˙’u diˆ˜en ch´ınh x´ac bˇa` ng m´ay t´ınh, nˆen d¯ˆo´i v´o.i hˆ x ∈ D ta khˆong thˆe˙’ t´ınh ch´ınh x´ac d¯a.i lu.o ng f (x) = Ax, x + b, x m`a chı˙’ c´o thˆe˙’ xˆa´p xı˙’ f (x) bo˙’.i mˆo.t sˆo´ dˆa´u chˆa´m d¯oˆ ng f˜(x) n`ao d¯´o H`am f˜ `oi, khˆong to`an phu.o.ng v`a thˆa.m ch´ı l`a khˆong liˆen tu.c trˆen D Khi khˆong lˆ d¯o´ h`am p := f˜− f mˆo ta˙’ c´ac lˆo˜i t´ınh to´an C´ac lˆo˜i d¯´o bi chˇa.n bo˙’.i mˆo.t cˆa.n trˆen s ∈ [0, +∞[ n`ao d¯o´ c´o thˆe˙’ u.o´.c lu.o ng d¯u.o c, t´ u.c l`a supx∈D |p(x)| ≤ s Ngo`ai ra, bˇa` ng c´ach su˙’ du.ng c´ac sˆo´ dˆa´u chˆa´m d¯oˆ ng d`ai ho.n v`a/hoˇa.c c´ac thuˆa.t to´an tˆo´t ho.n, ta c´o thˆe˙’ gia˙’m cˆa.n trˆen s L´ y th´ u hai: f˜ l`a h`am mu.c tiˆeu d¯´ıch thu c v`a f l`a h`am mu.c tiˆeu `eu d¯u.o c l´ y tu.o˙’.ng h´oa hoˇa.c l`a h`am mu.c tiˆeu thay thˆe´ Trong thu c tˆe´, nhiˆ `oi, hoˇa.c to`an h`am thˆe˙’ hiˆe.n mˆo.t sˆo´ mu.c tiˆeu thu c tiˆ˜en d¯u.o c gia˙’ d¯.inh l`a lˆ phu.o.ng, hoˇa.c c´o mˆo.t sˆo´ t´ınh chˆa´t thuˆa.n tiˆe.n d¯a˜ d¯u.o c nghiˆen c´ u.u k˜ y, hoˇa.c - iˆ `eu n`ay d¯˜a d¯u.o c dˆ˜e nghiˆen c´ u.u, nhu.ng thu c th`ı khˆong pha˙’i l`a nhu vˆa.y D H X Phu, H G Bock v`a S Pickenhain d¯`ˆe cˆa.p d¯ˆe´n [48] Trong bˆo´i ca˙’nh d¯´o, p = f˜ − f l`a h`am hiˆe.u chı˙’nh C´o thˆe˙’ gia˙’ thiˆe´t p l`a gi´o.i nˆo.i (tˆo´i thiˆe˙’u trˆen tˆa.p chˆa´p nhˆa.n d¯u.o c) bo˙’.i mˆo.t sˆo´ du.o.ng kh´a b´e s, v`ı nˆe´u |p(x)| qu´a l´o.n th`ı su thay thˆe´ khˆong c`on ph` u ho p n˜ u.a - ˆe˙’ gia˙’i th´ıch d¯iˆ `eu n`ay, ta d¯`ˆe cˆa.p d¯ˆe´n vˆa´n d¯`ˆe thu.o`.ng d¯u.o c nghiˆen c´ D u.u cu˙’a ph´at d¯iˆe.n tˆo´i u.u, t´ u.c l`a b`ai to´an phˆan bˆo´ lu.o ng d¯iˆe.n nˇang cho t` u.ng tˆo˙’ m´ay ph´at nhiˆe.t d¯iˆe.n cho tˆo˙’ng chi ph´ı (gi´a th`anh) l`a cu c tiˆe˙’u, d¯`oˆng `au lu.o ng d¯iˆe.n nˇang v`a thoa˙’ m˜an r`ang buˆo.c th`o.i vˆa˜n d¯a´p u ´.ng d¯u.o c nhu cˆ 90 Ch´ u.ng minh Nˆe´u s = th`ı ξ(s) = nˆen kˆe´t luˆa.n cu˙’a bˆo˙’ d¯`ˆe l`a hiˆe˙’n nhiˆen Nˆe´u s > 0, lˆa´y bˆa´t k` y x∗ ∈ Slocal (0) V´o.i mo.i x ∈ D, x = x∗ ta c´o x − x∗ f (x) = f (x ) + 2Ax + b, x − x∗ nˆen, v´o.i mo.i x ∈ D, x = x∗ th`ı ∗ x − x∗ + A(x − x∗ ), x − x∗ , ∗ x − x∗ f (x) = 2Ax + b, x − x∗ x − x∗ + A(x − x∗ ), x − x∗ + 3s + f (x∗ ) − 3s ∗ (4.4.27) X´et biˆe˙’u th´ u.c 2Ax∗ + b, x − x∗ x − x∗ x − x∗ + A(x − x∗ ), x − x∗ + 3s V`ı A(x − x∗ ), x − x∗ ≤ λmax x − x∗ tho˙’a m˜an x − x∗ = ξ(s), ta d¯u.o c v`a ξ(s) > 0, nˆen v´o.i mo.i x ∈ D x − x∗ x − x∗ + A(x − x∗ ), x − x∗ + 3s ∗ x−x ≤ λmax ξ(s) + η0 ξ(s) + 3s 2Ax∗ + b, V`ı s ≤ s0 v`a ξ(s) ≤ ξ(s0 ) nˆen λmax ξ(s)2 + η0 ξ(s) + 3s ≤ λmax ξ(s0 )2 + η0 ξ(s0 ) + 3s0 Thay s0 , ξ(s) d¯u.o c x´ac d¯.inh theo (4.4.25) v`a (4.4.26) ta suy λmax ξ(s0 )2 + η0 ξ(s0 ) + 3s0 = Do d¯o´ x − x∗ 2Ax + b, x − x∗ ∗ x − x∗ + A(x − x∗ ), x − x∗ + 3s ≤ Kˆe´t ho p biˆe˙’u th´ u.c n`ay v´o.i (4.4.27) ta suy kˆe´t luˆa.n cu˙’a bˆo˙’ d¯`ˆe 91 ˜ Khi d¯´o - i.nh l´ D y 4.4.25 X´et B` to´an (Q) ¯C (0, s0 ) : ∀p ∈ B max x∗ ∈Slocal (0) d x∗ , Slocal (p) ≤ ξ( p C ) `eu khˇa˙’ng d¯.inh l`a hiˆe˙’n nhiˆen Ch´ u.ng minh Nˆe´u p ≡ th`ı d¯iˆ - aˇ t s := p C Do D l`a Lˆa´y bˆa´t k` y p ∈ C (D) cho < p C ≤ s0 D ¯ x∗ , ξ(s) ∩D l`a tˆa.p compact V`ı h`am to`an phu.o.ng lˆ `oi `oi d¯a diˆe.n nˆen B tˆa.p lˆ ¯ x∗ , ξ(s) ∩D, nˆen ngˇa.t bi nhiˆ˜eu gi´o.i nˆo.i f˜ = f +p bi chˇa.n (trˆen) trˆen tˆa.p B ¯ x∗ , ξ(s) ∩ D `on ta.i (˜ supx∈B(x a.t kh´ac, tˆ xi ) B ¯ ∗ , ξ(s))∩D f˜(x) < +∞ Mˇ cho limi→∞ x˜i = x˜∗ v`a limi→∞ f˜(˜ xi ) = supx∈B(x ¯ ∗ , ξ(s))∩D f˜(x) Ta khˇa˙’ng d¯i.nh x˜∗ ∈ B(x∗ , ξ(s)) ∩ D Thˆa.t vˆa.y gia˙’ su˙’ kˆe´t luˆa.n trˆen l`a ¯ x∗ , ξ(s) \ B(x∗ , ξ(s) ∩ D Ta c´o sai, d¯´o x˜∗ ∈ B sup ¯ ∗ , ξ(s))∩D x∈B(x f˜(x) = lim f˜(˜ xi ) i→∞ = lim f (˜ xi ) + p(˜ xi ) i→∞ (4.4.28) `on ta.i limi→∞ f˜(˜ `oi V`ı tˆ xi ) = limi→∞ f (˜ xi ) + p(˜ xi ) , m`a h`am to`an phu.o.ng lˆ `on ta.i limi→∞ p(˜ ngˇa.t f liˆen tu.c ta.i mo.i d¯iˆe˙’m thuˆo.c D nˆen tˆ xi ) Do d¯o´ thay limi→∞ f˜(˜ xi ) + p(˜ xi ) = limi→∞ f (˜ xi ) + limi→∞ p(˜ xi ) v`ao (4.4.28) v`a ch´ u y ´ rˇ`a ng f (˜ x∗ ) ≤ f (x∗ ) − 3s theo Bˆo˙’ d¯`ˆe 4.4.8, ta d¯u.o c sup ¯ ∗ , ξ(s))∩D x∈B(x f˜(x) = lim f (˜ xi ) + lim p(˜ xi ) i→∞ ∗ i→∞ ≤ f (˜ x )+s ≤ f (x∗ ) − 3s + s ≤ f (x∗ ) + p(x∗ ) − s = f˜(x∗ ) − s Suy sup ¯ ∗ , ξ(s))∩D x∈B(x f˜(x) ≤ sup f˜(x) − s ¯ ∗ , ξ(s))∩D x∈B(x Biˆe˙’u th´ u.c nhˆa.n d¯u.o c l`a vˆo l´ y, nˆen x˜∗ ∈ B(x∗ , ξ(s)) ∩ D, d¯o´ suy x˜∗ ∈ D l`a d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a f˜ = f + p trˆen D Ngo`ai v`ı 92 ¯ ∗ , ξ(s)) nˆen x˜∗ ∈ B(x max x∗ ∈Slocal (0) d x∗ , Slocal (p) = ≤ ≤ = max x∗ ∈Slocal (0) max x∗ ∈Slocal (0) inf y˜∗ ∈Slocal (p) x˜∗ − x∗ max ξ(s) max ξ( p x∗ ∈Slocal (0) x∗ ∈Slocal (0) y˜∗ − x∗ C ) T´om la.i ta nhˆa.n d¯u.o c ¯C (0, s0 ) : ∀p ∈ B max x∗ ∈Slocal (0) d x∗ , Slocal (p) ≤ ξ( p C ) - i.nh l´ D y d¯a˜ d¯u.o c ch´ u.ng minh `e 4.4.30 H` Mˆ e.nh d ¯ˆ am d¯a tri Slocal (p) l`a nu˙’.a liˆen tu.c du.´o.i ta.i d¯iˆe˙’m Ch´ u.ng minh Theo d¯.inh ngh˜ıa h`am nu˙’.a liˆen tu.c du.o´.i, nˆe´u Slocal (0) = ∅ th`ı Slocal (p) nu˙’.a liˆen tu.c du.o´.i ta.i l`a hiˆe˙’n nhiˆen Ta x´et tru.o`.ng ho p Slocal (0) = ∅ Lˆa´y tˆa.p mo˙’ bˆa´t k` y V ⊂ IRn tho˙’a m˜an Slocal (0) ∩ V = ∅ Khi d¯o´ ∃ x∗ ∈ Slocal (0) : x∗ ∈ V ¯ ∗ , δ) ⊂ V Mˇa.t kh´ac, `on ta.i δ > cho B(x V`ı V l`a tˆa.p mo˙’., ta suy tˆ ξ(s) d¯u.o c x´ac d¯.inh bo˙’.i biˆe˙’u th´ u.c (4.4.26) nˆen lim ξ(s) = lim − η0 − s→0 s→0 = η02 − 12λmax s /(2λmax ) − η0 − (−η0 ) /(2λmax ) = Biˆe˙’u th´ u.c n`ay cho ph´ep ta cho.n sˆo´ du.o.ng s1 ≤ s0 cho v´o.i mo.i s ∈ ]0, s1 ] th`ı ξ(s) ≤ δ, v`ı vˆa.y ¯ x∗ , ξ(s) ∩ D ⊂ B(x ¯ ∗ , δ) ∩ D ⊂ V B (4.4.29) - i.nh l´ Mˇa.t kh´ac, theo D y 4.4.25, v´o.i mo.i p ∈ C (D) tho˙’a m˜an p C ≤ s0 ¯ x∗ , ξ( p C ) ∩ D l`a d¯iˆe˙’m `on ta.i x˜∗ ∈ B (d¯u.o c x´ac d¯i.nh theo (4.4.25)) tˆ 93 supremum d¯.ia phu.o.ng cu˙’a f˜ = f + p, t´ u.c l`a x˜∗ ∈ Slocal (p) V`ı s1 ≤ s0 nˆen kˆe´t ho p v´o.i biˆe˙’u th´ u.c (4.4.29) ta nhˆa.n d¯u.o c ¯C (0, s1 ) : V ∩ Slocal (p) = ∅, ∀p ∈ B Do d¯o´ Slocal (p) l`a nu˙’.a liˆen tu.c du.o´.i ta.i d¯iˆe˙’m T` u d¯.inh ngh˜ıa 4.3.14 ta suy h`am Slocal (p) khˆong nu˙’.a liˆen tu.c trˆen `on ta.i lˆan cˆa.n V tho˙’a m˜an Slocal (0) ⊆ V v`a d˜ay (pi ), i = 1, ta.i nˆe´u tˆ `e C (D) cho Sglobal (pi ) \ V = ∅ v´o.i mo.i i = 1, 2, hˆo.i tu vˆ V´ı du sau d¯aˆy chı˙’ Slocal (p) khˆong nu˙’.a liˆen tu.c trˆen ta.i V´ı du 4.4.14 Cho f (x) = x2 , x ∈ [ 0, ] 1/i − 2x2 nˆe´u x ∈ [ 0, 1/i ] pi (x) = nˆe´u x ∈ [ 0, ] \ [ 0, 1/i ] i = 1, Ta t´ınh d¯u.o c pi 0C (D) = 1/i, Slocal (0)={2} v`a Slocal (pi ) = {0, 2}, i = 1, 2, Lˆa´y tˆa.p mo˙’ V = ]1.5, 2.1[ ta c´o {2} = Slocal (0) ⊂ V = ]1.5, 2.1[ Trong d¯o´ v´o.i mo.i i th`ı ∈ Slocal (pi ) nhu.ng ∈ / V, nˆen suy Slocal (pi ) \ V = ∅ Do d¯o´ Slocal (p) khˆong nu˙’.a liˆen tu.c trˆen ta.i Kˆ e´t luˆ a.n: C´ac kˆe´t qua˙’ d¯a.t d¯u.o c, co ba˙’n d¯u.o c tr`ınh b`ay c´ac Mu.c `om: mˆo.t sˆo´ d¯iˆ `eu kiˆe.n d¯u˙’ d¯ˆe˙’ h`am to`an phu.o.ng lˆ `oi 4.1–4.4, ch´ ung bao gˆ `oi (Mˆe.nh d¯`ˆe 4.1.24); c´ac t´ınh chˆa´t cu˙’a ngˇa.t bi nhiˆ˜eu gi´o.i nˆo.i l`a γ-lˆ ˜ (c´ac mˆe.nh d¯`ˆe c´ac d¯iˆe˙’m cu c d¯a.i v`a supremum to`an cu.c cu˙’a B`ai to´an (Q) 4.2.25, 4.2.26, 4.2.27, 4.2.28); t´ınh ˆo˙’n d¯i.nh, nu˙’.a liˆen tu.c trˆen cu˙’a h`am tˆa.p ˜ (D - i.nh l´ c´ac d¯iˆe˙’m supremum to`an cu.c cu˙’a B`ai to´an (Q) y 4.3.24, Mˆe.nh d¯`ˆe 4.3.29); t´ınh oˆ˙’n d¯.inh, nu˙’.a liˆen tu.c du.o´.i cu˙’a h`am tˆa.p c´ac d¯iˆe˙’m supremum ˜ (D - i.nh l´ d¯.ia phu.o.ng cu˙’a B`ai to´an (Q) y 4.4.25, Mˆe.nh d¯`ˆe 4.4.30) 94 ˆ´T LUA ˆ N CHUNG KE `e: ac vˆ a´n d ¯ˆ Luˆ a.n ´ an d ¯˜ a gia˙’i quyˆ e´t d ¯u.o c c´ `oi ngo`ai v´o.i mo.i γ ≥ γ ∗ , d¯o´ • Chı˙’ h`am bi nhiˆ˜eu f˜ = f + p l`a γ-lˆ γ ∗ = 2s/λmin ; d¯iˆe˙’m γ ∗ -cu c tiˆe˙’u cu˙’a f˜ l`a d¯iˆe˙’m cu c tiˆe˙’u to`an cu.c; d¯u.o`.ng k´ınh cu˙’a tˆa.p c´ac d¯iˆe˙’m cu c tiˆe˙’u to`an cu.c cu˙’a B`ai to´an (P˜ ) nho˙’ ho.n hoˇa.c bˇa` ng γ ∗ ; khoa˙’ng c´ach 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 cu c tiˆe˙’u to`an cu.c cu˙’a h`am f nho˙’ ho.n hoˇa.c bˇa` ng `eu kiˆe.n tˆo´i u.u suy rˆo.ng γ ∗ Ngo`ai t´ınh chˆa´t tu a thˆo v`a mˆo.t sˆo´ d¯iˆ cu˙’a h`am f˜ c˜ ung d¯u.o c tr`ınh b`ay C´ac kˆe´t qua˙’ trˆen d¯˜a d¯u.o c cˆong bˆo´ b`ai b´ao “Global infimum of strictly convex quadratic functions with bounded perturbation” (xem Danh mu.c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa.n ´an) `oi ngo`ai v´o.i tˆa.p cˆan d¯aˇ c biˆe.t Γ ⊂ IRn ; • Ch´ u.ng minh d¯u.o c, h`am f˜ l`a Γ-lˆ d¯iˆe˙’m Γ-tˆo´i u.u d¯.ia phu.o.ng cu˙’a B`ai to´an (P˜ ) l`a d¯iˆe˙’m tˆo´i u.u to`an cu.c; hiˆe.u cu˙’a hai nghiˆe.m tˆo´i u.u bˆa´t k` y cu˙’a B`ai to´an (P˜ ) n`am tˆa.p Γ; x∗ − x˜∗ ∈ 12 Γ nˆe´u x∗ l`a nghiˆe.m cu c tiˆe˙’u to`an cu.c cu˙’a f trˆen D v`a x˜∗ l`a nghiˆe.m tˆo´i u.u to`an cu.c bˆa´t k` y cu˙’a B`ai to´an (P˜ ); tˆa.p nghiˆe.m tˆo´i - i.nh u.u Ss cu˙’a (P˜ ) l`a oˆ˙’n d¯i.nh theo khoa˙’ng c´ach Hausdorff dH (.,.) D l´ y Kuhn-Tucker suy rˆo.ng cho B`ai to´an (P˜ ) c˜ ung d¯u.o c ch´ u.ng minh C´ac kˆe´t qua˙’ trˆen d¯˜a d¯u.o c d¯ˇang ta˙’i b`ai b´ao “ Some properties of boundedly disturbed strictly convex quadratic functions” (xem Danh mu.c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa.n ´an) `oi v´o.i γ ≥ (2/λmin ) v`a γ-lˆ `oi ngˇa.t v´o.i • Chı˙’ h`am f˜ l`a γ-lˆ 1 γ > (2/λmin ) ; D bi chˇa.n v`a γ = (2/λmin ) , mo.i d¯iˆe˙’m supremum ˜ chı˙’ c´o thˆe˙’ l`a d¯iˆe˙’m γ- cu c biˆen cu˙’a D v`a c´o to`an cu.c cu˙’a B`ai to´an (Q) ´ıt nhˆa´t mˆo.t d¯iˆe˙’m l`a γ-cu c biˆen ngˇa.t Mˆo.t sˆo´ t´ınh chˆa´t quan tro.ng cu˙’a tˆa.p c´ac d¯iˆe˙’m supremum to`an cu.c Sglobal (p) v`a tˆa.p c´ac d¯iˆe˙’m supremum ˜ nhu t´ınh oˆ˙’n d¯i.nh v`a t´ınh nu˙’.a d¯.ia phu.o.ng Slocal (p) cu˙’a B`ai to´an Q 95 `an l´o.n c´ac kˆe´t qua˙’ d¯u.o c liˆe.t kˆe o˙’ trˆen liˆen tu.c c˜ ung d¯u.o c chı˙’ Phˆ d¯a˜ d¯u.o c cˆong bˆo´ b`ai b´ao “Maximizing strictly convex quadratic functions with bounded perturbation” (xem Danh mu.c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa.n ´an) `an tiˆ a´n cˆ e´p tu.c nghiˆ en c´ u.u: Nh˜ u.ng vˆ `e l´ Luˆa.n ´an chı˙’ m´o.i d¯`ˆe cˆa.p d¯ˆe´n mˆo.t sˆo´ vˆa´n d¯`ˆe vˆ y thuyˆe´t cu˙’a B`ai to´an `oi ngˇa.t v´o.i nhiˆ˜eu gi´o.i nˆo.i Do d¯´o ch´ quy hoa.ch to`an phu.o.ng lˆ ung tˆoi c`on u.ng vˆa´n d¯`ˆe sau d¯aˆy tiˆe´p tu.c nghiˆen c´ u.u nh˜ • Xˆay du ng thuˆa.t to´an t´ınh to´an t`ım l`o.i gia˙’i tˆo´i u.u cu˙’a c´ac b`ai to´an ˜ (P˜ ) v`a (Q) ´ du.ng thuˆa.t to´an t´ınh to´an t`ım l`o.i gia˙’i tˆo´i u.u cu˙’a c´ac b`ai to´an (P˜ ) • Ap ˜ v`ao c´ac b`ai to´an thu c tˆe´ nhu b`ai to´an ph´at d¯iˆe.n tˆo´i u.u, kinh v`a (Q) tˆe´ d¯oˆ´i s´anh, Tuy nhiˆen, v`ı th`o.i gian ha.n he.p nˆen ch´ ung tˆoi c˜ ung chu.a tra˙’ l`o.i d¯u.o c c´ac vˆa´n d¯`ˆe trˆen Ch´ ung tˆoi hy vo.ng rˇ`a ng c´ac vˆa´n d¯`ˆe n`ay s˜e s´o.m d¯u.o c gia˙’i quyˆe´t 96 ˆ DANH MU TR`INH C CONG ´ GIA˙’ LIEN ˆ QUAN D ˆ´N LUA ˆ N AN ´ -E CU˙’A TAC C´ac b`ai b´ao 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Problems, (Tiˆe´ng Nga), Nauka, Moscow, 1974 [75] B N Pshenhishnui, Convex Analysis and Extremal Problems, (Tiˆe´ng Nga), Nauka, Moscow, 1980 [76] M M Vainberg, Variational Method and Method Monotone Operators, (Tiˆe´ng Nga), Nauka, Moscow, 1972 [77] G E Silov, Finite-dimentional Linear Spaces, (Tiˆe´ng Nga), Nauka, Moscow, 1969 [...]... gian t´o.i s˜e c´o thˆem mˆo.t sˆo´ kˆe´t qua˙’ m´o.i CHU O NG 1 `ˆ I, ` TOAN ´ QUY HOA BAI CH LO ` HAM ` ˆ I THO ˆ ` PHU.O.NG VA L` O QUY HOA CH TOAN - i.nh l´ Trong chu.o.ng n`ay, ch´ ung tˆoi nhˇa´c la.i D y Kuhn-Tucker cho b`ai - i.nh l´ `oi, D `e d¯iˆ `eu kiˆe.n cˆ `an cu c tri cho b`ai to´an quy hoa.ch to´an quy hoa.ch lˆ y vˆ - `oˆng th`o.i ch´ ung tˆoi c˜ ung tr`ınh b`ay la.i mˆo.t sˆo´... luˆa.n ´an l`a chı˙’ ra d¯iˆ `om 4 chu.o.ng Luˆa.n ´an gˆ `oi, to`an phu.o.ng v`a h`am Chu.o.ng 1 v´o.i tiˆeu d¯`ˆe “B`ai to´an quy hoa.ch lˆ - i.nh l´ - i.nh `oi thˆ `oi, D lˆ o” tr`ınh b`ay D y Kuhn-Tucker cu˙’a b`ai to´an quy hoa.ch lˆ `e d¯iˆ `eu kiˆe.n cu c tri cu˙’a b`ai to´an quy hoa.ch to`an phu.o.ng v`a mˆo.t sˆo´ loa.i l´ y vˆ `oi thˆo nhu γ-lˆ `oi ngo`ai, Γ-lˆ `oi ngo`ai, γ-lˆ `oi trong c` h`am... • Ta c˜ ung k´ y hiˆe.u λmin , λmax v`a λ(A) lˆ nhˆa´t, l´o.n nhˆa´t v`a tˆa.p c´ac gi´a tri riˆeng cu˙’a ma trˆa.n A 1.1 `oi, quy hoa.ch to` B` ai to´ an quy hoa.ch lˆ an phu.o.ng - i.nh l´ Trong mu.c n`ay, ch´ ung tˆoi tr`ınh b`ay D y Kuhn-Tucker cho b`ai to´an `oi sau: quy hoa.ch lˆ g0 (x) → inf, x∈D D = {x ∈ S | gi (x) ≤ 0, i = 1, , m}, (L1 ) `oi, S ⊂ IRn l`a tˆa.p trong d¯o´ gi : IRn → IR,... 1.1.1 Nˆe´u S = IRn th`ı khi d¯´o N (x∗ |S) = {0}, nˆen biˆe˙’u th´ u.c (1.1.8) d¯u.o c thay bo˙’.i m µi ∂gi (x∗ ) 0∈ (1.1.10) i=0 - oˆ´i v´o.i b`ai to´an quy hoa.ch to`an phu.o.ng ta c´o d¯.inh l´ D y sau: - i.nh l´ D y 1.1.3 (Xem [31]) X´et b`ai to´an quy hoa.ch to`an phu.o.ng M x, x + b, x → inf, x∈D D = {x ∈ IRn | ci , x ≤ di , i = 1, , m}, (L2 ) trong d¯´ o M ∈ IRn×n l` a ma trˆa.n d¯ˆo´i x´ u.ng,... nghiˆen c´ u.u t` u rˆa´t s´o.m, mˆo.t trong nh˜ u.ng kˆe´t qua˙’ quan tro.ng l`a d¯.inh l´ y Kuhn-Tucker do W H Kuhn v`a A W Tucker `oi d¯u.a ra v`ao nˇam 1951 trong [22] cˆong tr`ınh khai ph´a cu˙’a Quy hoa.ch lˆ Trong B`ai to´an (L1 ) h`am Lagrange d¯u.o c d¯.inh ngh˜ıa nhu sau: m L(x, µ0 , , µm ) := µi gi (x), i=0 (1.1.3) 10 trong d¯o´ µi , i = 0, 1, , m, nhˆa.n c´ac gi´a tri thu c, x ∈ D... a´t cu˙’a d ¯iˆ e˙’m infimum to` an cu.c 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 tiˆe˙’u to`an cu.c cu˙’a B`ai to´an quy hoa.ch to`an `oi ngˇa.t v´o.i nhiˆ˜eu gi´o.i nˆo.i (P˜ ) Trong mu.c n`ay, ch´ ung tˆoi nghiˆen phu.o.ng lˆ 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 oˆ˙’n d¯.inh ... o.ng d` ung `au Mo˙’ d ¯ˆ `oi `oi, quy hoa.ch to` a h` am lˆ B` to´ an quy hoa.ch lˆ an phu.o.ng v` thˆ o `oi, quy hoa.ch to`an phu.o.ng 1.1 B`ai to´an quy hoa.ch lˆ `oi suy rˆo.ng thˆo... B`ai to´an quy hoa.ch to`an phu.o.ng (P ) • f (x) := Ax, x + b, x → sup, x ∈ D : B`ai to´an quy hoa.ch to`an phu.o.ng (Q) • f (x) := Ax, x + b, x + p(x) → inf, x ∈ D : B`ai to´an quy hoa.ch `oi... d¯`ˆe “B`ai to´an quy hoa.ch lˆ - i.nh l´ - i.nh `oi thˆ `oi, D lˆ o” tr`ınh b`ay D y Kuhn-Tucker cu˙’a b`ai to´an quy hoa.ch lˆ `e d¯iˆ `eu kiˆe.n cu c tri cu˙’a b`ai to´an quy hoa.ch to`an phu.o.ng