ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - LÊ BÁ LONG NHẬT MỘT SỐ VẤN ĐỀ VỀ LÝ THUYẾT ĐỐI NGẪU LIÊN HỢP VÀ LÝ THUYẾT ĐỐI NGẪU LAGRANGE Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Dương Thị Việt An THÁI NGUYÊN - 2020 ▼ö❝ ❧ö❝ ỵ ì♥ ✼ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✽ ✶✳✶ ✶✳✷ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ❧✐➯♥ ❤ñ♣ ✈➔ ởt số t t ữợ ỗ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt số ỵ tt ✤è✐ ♥❣➝✉ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ sỡ ỗ ố t t ữợ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✶ ✶✸ ✶✹ ✶✼ ✶✼ ✷✵ ✷✹ ✷✼ ✸✶ ✸✺ ✶ ❉❛♥❤ ♠ö❝ ỵ R R R+ X x x M ∩N |x| ||x|| B X (0, 1) ✐♥t A inf f (x) x∈K sup f (x) x∈K δΩ (·) epi f dom f x∗ , x tr÷í♥❣ sè t❤ü❝ t➟♣ sè t❤ü❝ s✉② rë♥❣ t➟♣ sè t❤ü❝ ❦❤æ♥❣ ➙♠ ❦❤æ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ✭✤è✐ ♥❣➝✉✮ ❝õ❛ X t➟♣ ré♥❣ ✈ỵ✐ x tỗ t x t ủ M ✈➔ N ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ x ❝❤✉➞♥ ❝õ❛ ✈➨❝tì x ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ tr♦♥❣ X ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ A ✐♥❢✐♠✉♠ ❝õ❛ t➟♣ sè t❤ü❝ {f (x) | x ∈ K} s✉♣r❡♠✉♠ ❝õ❛ t➟♣ sè t❤ü❝ {f (x) | x ∈ K} ❤➔♠ ❝❤➾ ❝õ❛ t tr ỗ t f ỳ ❤✐➺✉ ❝õ❛ ❤➔♠ f ❣✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ x∗ t↕✐ x ✷ ∂f (x) f∗ f ∗∗ l.s.c N (x) val(P ) ữợ ỗ f t↕✐ x ❤➔♠ ❧✐➯♥ ❤ñ♣ ❝õ❛ ❤➔♠ f ❤➔♠ ❧✐➯♥ ủ tự f ỷ tử ữợ ❤å ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x t➟♣ ❝→❝ ❣✐→ trà tố ữ t P ỵ t❤✉②➳t ✤è✐ ♥❣➝✉ ❧➔ ♠ët ❜ë ♣❤➟♥ q✉❛♥ trå♥❣ ❝õ❛ ỵ tt tố ữ ữỡ ự ợ ộ t♦→♥ ◗✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭❝á♥ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❣è❝✮ ❝â ♠ët ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✳ ❇➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝â ♠è✐ ❧✐➯♥ ❤➺ q✉❛ ❧↕✐ ✈ỵ✐ ♥❤❛✉✱ t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❝â t❤➸ ✤÷đ❝ ❦❤↔♦ s→t t❤ỉ♥❣ q✉❛ ❜➔✐ t♦→♥ ❦✐❛✳ ◆❤✐➲✉ q✉② tr➻♥❤ t➼♥❤ t♦→♥ ❤❛② ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❦❤✐ ①❡♠ ①➨t ❝➦♣ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ tr♦♥❣ ♠é✐ q✉❛♥ ❤➺ ❝❤➦t ❝❤➩ ❝õ❛ ❝❤ó♥❣✱ ♠❛♥❣ ❧↕✐ ♥❤ú♥❣ ❧ñ✐ ➼❝❤ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ♣❤→t s✐♥❤ tø t❤ü❝ t➳✳ ❇➔✐ t♦→♥ q✉② ❤♦↕❝❤ t♦→♥ ❤å❝ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ ữủ ự tứ ỳ t trữợ t ✤➛✉ ✈ỵ✐ ♠ỉ ❤➻♥❤ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✈ỉ ❤↕♥ ❝❤✐➲✉✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ tè✐ ÷✉ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✱ ❝â ❝➜✉ tró❝ ♣❤ù❝ t↕♣✱ ♥❤÷ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ✈➔ ❜➔✐ t♦→♥ ❜✐➳♥ ♣❤➙♥ ❝â t❤➸ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t♦→♥ ❤å❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✤➲ t➔✐ ❝ì ❜↔♥ ♥❤➜t ❝õ❛ t r t ỗ ỵ tt ♥➔② ❧↕✐ ❝➔♥❣ trð ♥➯♥ ♣❤♦♥❣ ♣❤ó ♥❤í ♥❤ú♥❣ t➼♥❤ t t t ỗ ỗ ữợ ✈✐ ♣❤➙♥ ❧➔ ❦❤→✐ ✹ ♥✐➺♠ ♠ð rë♥❣ ❝❤♦ ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ❦❤✐ ❤➔♠ ❦❤æ♥❣ ❦❤↔ ✈✐✳ ✣✐➲✉ ♥➔② t trỏ ữợ tr t➼❝❤ ❤✐➺♥ ✤↕✐ ❝ơ♥❣ ❝â t➛♠ q✉❛♥ trå♥❣ ♥❤÷ ✈❛✐ trá ❝õ❛ ✤↕♦ ❤➔♠ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❝ê ✤✐➸♥✳ ❚r♦♥❣ ỵ tt tố ữ õ t ỗ õ r q t t tờ ữợ ỗ tữớ õ trỏ t sù❝ q✉❛♥ trå♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❦❤✐ t❛ ❧➔♠ ✈✐➺❝ ợ t tố ữ õ r ự ỵ tt ố ủ ỵ tt ố r t q ỗ õ t số tr ổ ứ õ ữủ ỗ ố ự q t t tờ ữợ ỗ tữớ ữợ ỳ ❝❤➼♥❤ q✉② t❤➼❝❤ ❤đ♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❞à❝❤ r❛ ❚✐➳♥❣ ❱✐➺t ♠ët sè ♥ë✐ ❞✉♥❣ tø ♠ö❝ ✷✳✺ ❉✉❛❧✐t② ❚❤❡♦r② tr♦♥❣ ❝✉è♥ s→❝❤ ❝❤✉②➯♥ ❦❤↔♦ ✧P❡rt✉r❜❛t✐♦♥ ❆♥❛❧②s✐s ♦❢ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠s✧ ✭❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✵✮ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❏✳ ❋✳ ❇♦♥♥❛♥s ❛♥❞ ❆✳ ❙❤❛♣✐r♦ ❬✸❪✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ t→❝ ❣✐↔ ❝ơ♥❣ t➻♠ ❤✐➸✉✱ tê♥❣ ❤đ♣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥ ✈➔ ❝è ❣➢♥❣ ❞✐➵♥ ✤↕t ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❝→❝ ♠➺♥❤ ✤➲ ỵ ỗ ♣❤➛♥ ❦➳t ❧✉➟♥✱ ❞❛♥❤ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ✈➔ ❤❛✐ ❝❤÷ì♥❣ ❝â ♥ë✐ ❞✉♥❣ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ tự ỡ t ỗ ỗ ❧✐➯♥ ❤đ♣ ❝ị♥❣ ♠ët sè ❦➳t q✉↔ ❜ê trđ ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ð ❝❤÷ì♥❣ s❛✉✳ ✺ ❈❤÷ì♥❣ ✷✿ ▼ët sè ✈➜♥ ✤➲ ✈➲ ỵ tt ố tr t ỵ tt ố ố ủ ố r sỡ ỗ ✤è✐ ♥❣➝✉ ❝ơ♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ð ❝❤÷ì♥❣ ♥➔②✳ ✣➦❝ ❜✐➺t✱ ð ♣❤➛♥ ❝✉è✐ ❝❤÷ì♥❣✱ ♠ët ❦➳t q✉↔ ✈➲ q✉② t t t ữợ tờ ỗ ỷ tử ữợ tữớ t ữủ sỡ ỗ ố ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ữợ sỹ ữợ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ ❚❙✳ ❉÷ì♥❣ ❚❤à ❱✐➺t ❆♥✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❈ỉ ữợ q tr ỳ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❊♠ ❝ô♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ ð tr÷í♥❣✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✻ t❤→♥❣ ✽ ♥➠♠ ✷✵✷✵ ❍å❝ ✈✐➯♥ ▲➯ ❇→ ▲♦♥❣ ◆❤➟t ✼ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè tự ỡ t ỗ ỗ ủ ũ ởt số t q ❜ê trđ ♥❤➡♠ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ s❛✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✸❪✳ ỗ ỗ sỷ X ổ ợ ổ ố tữỡ ự ❧➔ X ∗✱ D ⊂ X, f : D → R = R {} t ủ ữợ epif := {(x, α) ∈ D × R | f (x) ≤ α}, domf := {x ∈ D | f (x) < +}, ữủt ữủ tr ỗ t❤à ✈➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ❤➔♠ f ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ domf = ∅ ✈➔ f (x) > −∞, ∀x ∈ D ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ A X ữủ ỗ x, y ∈ A, ∀λ ∈ (0, 1) ⇒ λx + (1 )y A ữợ t ỗ r ổ ỳ ❤↕♥ ❝❤✐➲✉✱ ♠➦t ♣❤➥♥❣✱ ✤♦↕♥ t❤➥♥❣✱ ✤÷í♥❣ t❤➥♥❣✱ t❛♠ ❣✐→❝✱ t ỗ ❬✶✱ tr❛♥❣ ✹❪✮ ●✐↔ sû Aα ⊂ X(α ∈ I) t ỗ ợ I t số ❜➜t ❦ý✳ ❑❤✐ ✤â A = Aα α∈I ❝ô♥❣ ❧➔ t ỗ tr sû t➟♣ Ai λi ∈ R, 1, m✳ ∈ X t ỗ õ 1A1 + à à à + mAm t ỗ ❬✶✱ tr❛♥❣ ✹❪✮ ●✐↔ sû Xi ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t t Ai Xi ỗ (i = 1, n) ❑❤✐ ✤â✱ t➼❝❤ ✣➲ ❝→❝ A1 × A2 × ì An t ỗ tr X1 ì X2 ì × Xn ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❍➔♠ f : D R ữủ ỗ tr D epif t ỗ tr X ì R ✭①❡♠ ❬✶✱ tr❛♥❣ ✹✵❪✮ ❈❤♦ f : X → (−∞, +] õ f ỗ ♥➳✉ f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), ∀x, y ∈ X, λ ∈ (0, 1) ✭✶✳✶✮ ❈❤ù♥❣ ♠✐♥❤✳ ⇒) ❱➻ f ❧➔ ỗ epif t ỗ õ ợ ♠å✐ (x, r) ∈ epif ✱ (y, s) ∈ epif ✱ λ ∈ (0, 1)✱ t❛ ❝â λ(x, r) + (1 − λ)(y, s) = (λx + (1 − λ)y, λr + (1 − λ)s) ∈ epif ⇔ f (λx + (1 − λ)y) ≤ λr + (1 − λ)s ⇔ f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), (❧➜② r = f (x), s = f (y)) ◆➳✉ x ❤♦➦❝ y ❦❤æ♥❣ t❤✉ë❝ domf t❤➻ f (x) = +∞ ❤♦➦❝ f (y) = +∞✳ ❑❤✐ ✤â ✭✶✳✶✮ ✤ó♥❣✳ ⇐) ◆❣÷đ❝ ❧↕✐ ❣✐↔ sû ✭✶✳✶✮ ✤ó♥❣✳ ▲➜② (x, r) ∈ epif, (y, s) ∈ epif ✱ ✈ỵ✐ ♠å✐ ✾ t❤➻ t❛ ❝â v(u) ≥ u∗ , u − v ∗ (u∗ ) ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✱ v(u) = u∗ , u − v ∗ (u∗ ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u∗ ∈ ∂v(u) ❱➻ v∗(u∗) = ϕ∗ (0, u∗)✱ ✤✐➲✉ ♥➔② s✉② r❛ r➡♥❣ u∗ ❧➔ ♠ët ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ (Du) ♥➳✉ u∗ ∈ ∂v(u) ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤â val (Pu) = val (Du)✱ ✈➔ ♥❣÷đ❝ ❧↕✐✳ ❉♦ ✤â ✭✐✮ ✈➔ ✭✐✐✮ ①↔② r❛✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✱ t❛ ❝â S(Du ) = ∂v ∗∗ (u)✳ ❱➻ v(·) ❧➔ ❦❤↔ ữợ t u tứ t❛ ❝ô♥❣ ❝â ∂v∗∗(u) = ∂v(u)✱ ✈➔ ❞♦ ✤â ✭✐✮ ①↔② r❛✳ ◆➳✉ v∗∗(u) = v(u)✱ t❤➻ ♠ët ❧➛♥ ♥ú❛ ∂v∗∗(u) = ∂v(u)✱ ✈➔ ❦❤✐ ✤â ✭✐✐✮ ①↔② r❛✳ ◆➳✉ val (Pu) = val (Du) ✈➔ ❣✐→ trà ♥➔② ❤ú✉ ❤↕♥✱ t❤➻ rã r➔♥❣ tø ♥❤ú♥❣ ❧➟♣ ❧✉➟♥ tr➯♥✱ x¯ ∈ X ✈➔ u¯∗ ∈ U ∗ ❧➛♥ ❧÷đt ❧➔ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ (Pu ) ✈➔ (Du ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮ ✤ó♥❣✳ ❍ì♥ ♥ú❛✱ t❛ ❝ơ♥❣ t❤➜② r➡♥❣✱ ♥➳✉ ✭✷✳✹✮ ✤ó♥❣ t❤➻ val (Pu) = val (Du)✱ ✈➔ ❞♦ ✤â ✭✐✐✐✮ ①↔② r❛ ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❚❛ ♥â✐ r➡♥❣ ❜➔✐ t♦→♥ (Pu) t➽♥❤ ♥➳✉ val(Pu) ❤ú✉ ❤↕♥ ✈➔ ❤➔♠ ❣✐→ trà tè✐ ÷✉ v(Ã) ữợ t u v(u) = ứ ỵ t õ t q s ▼➺♥❤ ✤➲ ✷✳✺✳ ●✐↔ sû r➡♥❣ val(Pu) ❤ú✉ ❤↕♥✳ ◆➳✉ (Pu) t➽♥❤✳ ❑❤✐ ✤â ❦❤æ♥❣ ❝â ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛ (Pu) ✈➔ (Du)✱ ✈➔ t➟♣ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du) ❦❤→❝ ré♥❣✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛ (Pu ) ✈➔ (Du ) ❜➡♥❣ ❦❤æ♥❣✱ t❤➻ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du ) ❝â ♠ët ♥❣❤✐➺♠ tè✐ ÷✉ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (Pu) t➽♥❤✳ ỵ ỵ ố sỷ r (x, u) ỗ tữớ tr tè✐ ÷✉ v(u) = val(Pu) ❤ú✉ ❤↕♥ ✈➔ ❧✐➯♥ tư❝ t↕✐ u¯ ∈ U ✳ ❑❤✐ ✤â val(Pu¯ ) = val(Du¯ )✱ S(Du¯ ) = ∅✱ ✈➔ ❤ì♥ ♥ú❛ S(Du¯ ) = v( u) ự v(Ã) ỗ tử t u t ỵ t õ v(u) rộ õ t ỵ ✭✐✮✱ t❛ t❤✉ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ v(·) ❧✐➯♥ tư❝ t↕✐ u¯ ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ✤✐➲✉ q õ t t ữợ tữỡ ữỡ v(Ã) ỗ ✈➔ val(Pu¯) ❤ú✉ ❤↕♥ t❤➻ ✤✐➲✉ ❦✐➺♥ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ val(Pu) ❜à ❝❤➦♥ tr➯♥ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ u¯✳ ❍ì♥ ♥ú❛✱ ♥➳✉ U ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ t❤➻ v(·) ❧✐➯♥ tö❝ t↕✐ u¯ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u¯ ∈ int(dom v)✳ ◆➳✉ X, U ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤➻ t❛ ❝â ❦➳t q✉↔ s❛✉✳ ▼➺♥❤ ✤➲ ✷✳✻✳ ❈❤♦ X, U ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ sỷ r tữớ ỗ ỷ tử ữợ v(u) ỳ õ v(Ã) ❧➔ ❧✐➯♥ tö❝ t↕✐ u¯ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u¯ ∈ int(dom v) ϕ(x, u) ❱➼ ❞ö ✷✳✶✳ ❈❤♦ x = (x1, x2) ∈ R2✱ u ∈ R ✈➔ ϕ(x, u) :=   x1 , −x1 + ex2 + u +, 0, tr trữớ ủ ợ u R tt ỵ ✷✳✷ t❤ä❛ ♠➣♥ ✈➔ v(u) = u ✈ỵ✐ ♠å✐ u ∈ R✳ ◆❣♦➔✐ r❛✱ ϕ∗ (0, u∗ ) := v ∗ (u∗ ) =   0, u∗ = 1,  +∞, tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝❀ ❞♦ ✈➟② ❦❤ỉ♥❣ ❝â ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛ (Pu) ✈➔ (Du)❀ ✈➔ S(Du) = {1}✳ ✷✸ ✷✳✸ ✣è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ❚r♦♥❣ ♠ö❝ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝❤ t✐➳♣ ❝➟♥ ✤è✐ ♥❣➝✉ ❞ị♥❣ ❤➔♠ ▲❛❣r❛♥❣❡✳ P❤➛♥ ❝✉è✐ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ❝❤➾ r❛ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✤è✐ ♥❣➝✉ ❧✐➯♥ ❤đ♣ ✭▼ư❝ ✷✳✷✮ ✈➔ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡✳ ❈❤♦ KX ⊂ X, KY ⊂ Y ❧➔ ❝→❝ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣ ❜➜t ❦➻✳ ❚❛ ①➨t ❝➦♣ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ t❤ỉ♥❣ q✉❛ ❤➔♠ ▲ : KX × KY → R✱ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ sup x∈KX y∈KY ▲(x, y), ✭P L✮ ▲(x, y) ✭DL✮ max inf y∈KY x∈KX r tữỡ ự ợ ❜➔✐ t♦→♥ ð tr➯♥✳ ❍✐➺✉ sè ❣✐ú❛ val(P L ) − val(DL ) ✭❦❤✐ val(P L ) ✈➔ val(DL ) ❦❤ỉ♥❣ ❝ị♥❣ ♥❤➟♥ ❣✐→ trà ✈ỉ ❤↕♥✮ ✤÷đ❝ ❣å✐ ❧➔ ố tữỡ ự ợ t ✤è✐ ♥❣➝✉ ð tr➯♥✳ ❚❛ ♥â✐ r➡♥❣ (¯x, y¯) ∈ KX × KY ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ❝õ❛ ❤➔♠ ▲(x, y) ♥➳✉ ▲(¯x, y¯) ∈ R ✈➔✿ ▲(¯x, y) ≤ ▲(¯x, y¯) ≤ ▲(x, y¯), ∀(x, y) ∈ KX ì KY ỵ õ val(DL) ≤ val(P L)✳ ❍ì♥ ♥ú❛ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ val(DL) − val(P L) ✭♥➳✉ ♥â ①→❝ ✤à♥❤✮ ❧➔ ❦❤æ♥❣ ➙♠✳ ✭✐✐✮ ❍➔♠ ▲(x, y) ❝â ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ❜➔✐ t♦→♥ (DL)✈➔ (P L ) ❝â ❝ị♥❣ ❣✐→ trà tè✐ ÷✉ ✈➔ t➟♣ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ♠é✐ ❜➔✐ t♦→♥ ❧➔ ❦❤→❝ ré♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t➟♣ ❤đ♣ ❝→❝ ✤✐➸♠ ②➯♥ ♥❣ü❛ ✤÷đ❝ ❦➼ ❤✐➺✉ S(P L ) × S(DL )✳ ✷✹ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ▲➜② (ˆx, yˆ) ∈ KX × KY ❜➜t ❦➻✳ ❑❤✐ ✤â inf ▲(x, yˆ) ≤ ▲(ˆ x, yˆ) ≤ sup ▲(ˆ x, y) x∈K y∈K X Y ✈➔ sup inf y∈Ky x∈KX ▲(x, y) ≤ x∈K inf sup X y∈KY ▲(x, y) ❚ø ✤â✱ s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ val(DL) ≤ val(P L)✱ ✈➔ ❞♦ ✤â ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ổ sỷ r tỗ t ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ (¯x, y¯)✳ ❑❤✐ ✤â sup y∈KY ▲(¯x, y) ≤ ▲(¯x, y¯) ≤ x∈K inf ▲(x, y¯) X ❚❤ü❝ t➳ t❤➻ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤➼♥❤ ❧➔ ❝→❝ ✤➥♥❣ t❤ù❝✱ ❜ð✐ ✈➻ ❝➟♥ tr➯♥ ✤ó♥❣ ✈➔ ữợ ú ữủt t ữủ ợ y x¯✱ ✈➻ ✈➟② ▲ (¯ x, y) = ▲(¯ x, y¯) = inf ▲(x, y¯) ≤ val(DL ) x∈K y∈K ▼➦t ❦❤→❝✱ ✈➻ val(DL) ≤ val(P L) ♥➯♥ val(DL) = ▲(¯x, y¯) = val(P L)✳ val(P L ) ≤ sup X Y ❍ì♥ ♥ú❛✱ ▲ val(P L ) = (¯ x, y¯) = sup ✈➔ y∈KY ▲ val(DL ) = (¯ x, y¯) = inf x∈KX ▲(¯x, y) ▲(¯x, y) ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝ x¯ ∈ S(P L) ✈➔ y¯ ∈ S(DL) t÷ì♥❣ ù♥❣✳ ❇➙② ❣✐í t❛ s➩ ❝❤➾ r❛ r➡♥❣ ❣✐→ trà tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❧➔ ❜➡♥❣ ♥❤❛✉✱ ✈➔ ♥➳✉ x¯ ∈ S(P L) ✈➔ y¯ ∈ S(DL)✱ t❤➻ (¯x, y¯) ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ❝õ❛ ▲✳ ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② r➡♥❣ ✤✐➲✉ ❦✐➺♥ ♥➔② ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ✱ ✈➔ tø ✤â t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❧➔ S(P L) × S(DL)✳ ❚❤➟t ✈➟②✱ ✈➻ val(DL ) = inf x∈KX ▲(x, y¯) ≤ ▲(¯x, y¯) ≤ y∈K sup ▲(¯ x, y) = val(P L ), Y ✷✺ ✈➔ ❣✐→ trà tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❧➔ ❜➡♥❣ ♥❤❛✉✱ t❛ ❝â ▲(¯x, y¯) = x∈K inf ▲(x, y¯) ≤ ▲(x, y¯), ∀x ∈ KX X ❚÷ì♥❣ tü✱ t❛ ❝ơ♥❣ ❝â ▲(¯x, y¯) = y∈K sup ▲(x, y¯) ≥ ▲(¯ x, y), ∀y ∈ KY Y ❑❤✐ ✤â (¯x, y¯) ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛✳ ✣➸ t❤➜② ✤÷đ❝ q✉❛♥ ❤➺ ❣✐ú❛ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ✈➔ ✤è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣✱ t❛ ①➨t ❤➔♠ s❛✉✱ ❤➔♠ ♥➔② ❝â t❤➸ ✤÷đ❝ ①❡♠ ❧➔ ❤➔♠ ▲❛❣r❛♥❣❡ ❝õ❛ ❝➦♣ ❜➔✐ t♦→♥ ❣è❝ (Pu) ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du) t÷ì♥❣ ù♥❣✱ L(x, u∗ , u) := u∗ , u − ϕ∗u (x, u∗ ) , ð ✤â✱ ✈ỵ✐ x trữợ u ủ t ❜✐➳♥ u✿ ϕ∗u (x, u∗ ) = sup { u∗ , u − ϕ(x, u )} u ∈U ❑❤✐ ✤â t❛ ❝â inf L(x, u∗ , u) = u∗ , u − x∈X sup { u∗ , u − ϕ(x, u )} , (x,u )∈X×U ✈➔ ❞♦ ✤â inf L(x, u∗ , u) = u∗ , u − ϕ∗ (0, u∗ ) x∈X ❱➟② ♥➯♥✱ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du) tữỡ ữỡ ợ max u U inf L(x, u∗ , u) x∈X ▼➦t ❦❤→❝✱ t❛ ❝â sup L(x, u∗ , u) = sup { u∗ , u − ϕ∗u (x, u∗ )} = ϕ∗∗ u (x, u) u U u U ú ỵ r➡♥❣ ϕ∗∗u ϕ✱ ✈➻ ✈➟② val(Du ) = sup inf L(x, u∗ , u) u∗ ∈U ∗ x∈X = inf ϕ∗∗ u (x, u) x∈X inf sup L(x, u∗ , u) x∈X u∗ ∈U ∗ val(Pu ) ◆❣♦➔✐ r❛✱ ♥➳✉ (x, Ã) ỗ õ ợ x X t r ỵ t❛ ❝â sup L(x, u∗ , u) = ϕ(x, u), u∗ ∈U ∗ ✈➻ ✈➟② ❜➔✐ t♦→♥ ❣è❝ (Pu) ❝â t t ữợ xX sup L(x, u , u) u∗ ∈U ∗ ❙✉② r❛ r➡♥❣ ♥➳✉ ϕ(x, Ã) ởt ỗ ỷ tử ữợ t❤÷í♥❣ t❤➻ ❜➔✐ t♦→♥ (Pu) ✈➔ (Du) ❝â ✤÷đ❝ ♥❤í t❤❛② ✤ê✐ t❤ù tü tr♦♥❣ ✤â ❝→❝ t♦→♥ tû ✧♠❛①✧ ✈➔ ✧♠✐♥✧ ✤÷đ❝ →♣ ❞ư♥❣ ❝❤♦ ❤➔♠ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ L(x, u∗, u)✳ ❚ù❝ ❧➔✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ✤è✐ ♥❣➝✉ ❧✐➯♥ ❤đ♣ trị♥❣ ✈ỵ✐ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡✱ ✈➔ ♥➳✉ x¯, u¯∗ ❧➛♥ ❧÷đt ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ✤è✐ ♥❣➝✉✱ ✈➔ ❝→❝ ❣✐→ trà ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ♥➔② ❜➡♥❣ ♥❤❛✉✱ t❤➻ (¯x, u¯∗) ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ❝õ❛ L(·, ·, u)✱ tù❝ ❧➔ sup L(¯ x, u∗ , u) = L(¯ x, u¯∗ , u) = inf L(x, u¯∗ , u) x∈X u∗ U sỡ ỗ ✤è✐ ♥❣➝✉ ❈❤♦ X, X ∗ ✈➔ Y, Y ∗ ổ tỡ tổổ ỗ ữỡ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉✳ (P ) min{f (x) + F (G(x))}, ✷✼ ✭✷✳✺✮ tr♦♥❣ ✤â f : X → R✱ F : Y → R ❧➔ ❝→❝ ❤➔♠ ❝❤➼♥❤ t❤÷í♥❣✱ ✈➔ G : X → Y ✳ ❚➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ (P ) ❧➔ Φ := {x ∈ dom f | G(x) ∈ dom F } ú ỵ r F (Ã) = K (Ã) ❤➔♠ ❝❤➾ ❝õ❛ t➟♣ ❦❤æ♥❣ ré♥❣ K ⊂ Y ✱ ❦❤✐ ✤â ❜➔✐ t♦→♥ (P ) ❝â ❞↕♥❣ (P ) f (x) x∈X s❛♦ ❝❤♦ G(x) ∈ K ✭✷✳✻✮ ❚❛ ①➨t ❤å ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â t❤❛♠ sè min{f (x) + F (G(x) + y)}, x∈X ✭Py ✮ tr♦♥❣ ✤â y ∈ Y ❧➔ t❤❛♠ sè✳ ❍✐➸♥ ♥❤✐➯♥ ❦❤✐ y = ❜➔✐ t♦→♥ (P0) trị♥❣ ✈ỵ✐ ❜➔✐ t♦→♥ (P )✳ ✣➦t ϕ(x, y) = f (x) + F (G(x) + y) inf ϕ(x, y) ❱➻ ❍➔♠ ❣✐→ trà tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ v(y) = val(Py ) ❤❛② v(y) = x∈X ❤➔♠ f ✈➔ F ❧➔ tữớ tỗ t x dom f y ∈ dom F ✳ ❑❤✐ ✤â ϕ(x, y − G(x)) = f (x) + F (y) < +∞✱ s✉② r❛ (x, y − G(x)) ∈ dom ϕ✳ ❍ì♥ ♥ú❛ ϕ(x, y) ≥ −∞✱ ✈ỵ✐ ♠å✐ (x, y) ∈ X × Y ✱ ❞♦ ✤â ϕ ❧➔ ❤➔♠ ❝❤➼♥❤ t❤÷í♥❣✳ f F ỷ tử ữợ t ụ ỷ tử ữợ rữớ ủ r F (·) = δK (·) ❧➔ ❤➔♠ ❝❤➾ ❝õ❛ t➟♣ K õ F ỷ tử ữợ ✈➔ ❝❤➾ ❦❤✐ K ✤â♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ❚❛ ♥â✐ r➡♥❣ ❜➔✐ t♦→♥ (P ) ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ỗ F (Ã) ỷ tử ữợ f (x) (x, y) = F (G(x) + y) ỗ õ r➡♥❣ ❜➔✐ t♦→♥ (P ) ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ỗ f (x) ỗ t K ỗ õ G(x) ỗ tữỡ ự ợ t (K) (x, y) := K (G(x) + y) ỗ r t (P ) ❧➔ L(x, y ∗ ) := f (x) + y ∗ , G(x) ▼➺♥❤ ✤➲ ✷✳✼✳ ❈❤♦ ❤➔♠ ϕ(x, y) = f (x) + F (G(x) + y)✳ ❑❤✐ ✤â ❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ♥❤➜t ϕ∗ ✈➔ t❤ù ❤❛✐ ϕ∗∗❝õ❛ ϕ ❧➛♥ ❧÷đt ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ϕ∗ (x∗ , y ∗ ) = sup{ x∗ , x − L(x, y ∗ )} + F ∗ (y ∗ ) x∈X ϕ∗∗ (x, y) = sup { y ∗ , y + inf L(x, y ∗ ) − F ∗ (y ∗ )} x∈X y ∗ ∈Y ∗ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❤➔♠ ❧✐➯♥ ❤đ♣✱ t❛ ❝â ϕ∗ (x∗ , y ∗ ) = { x∗ , x + y ∗ , y − ϕ(x, y)} sup (x,y)∈X×Y = { x∗ , x + y ∗ , y − f (x) − F (G(x) + y)} sup (x,y)∈X×Y = { x∗ , x + y ∗ , G(x) + y sup (x,y)∈X×Y − y ∗ , G(x) − f (x) − F (G(x) + y)} = sup{ x∗ , x − f (x) − y ∗ , G(x) x∈X + sup[ y ∗ , G(x) + y − F (G(x) + y)]} y∈Y ❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ G(x) + y −→ y t❛ ✤÷đ❝ ϕ∗ (x∗ , y ∗ ) = sup{ x∗ , x − L(x, y ∗ )} + F ∗ (y ∗ ) x∈X ❇➡♥❣ ❝→❝❤ ❜✐➳♥ ✤ê✐ t÷ì♥❣ tü t❛ ❝ơ♥❣ t➼♥❤ ✤÷đ❝ ϕ∗∗ (x, y) = sup { y ∗ , y + inf L(x, y ∗ ) − F ∗ (y ∗ )} x∈X y ∗ ∈Y ∗ ✷✾ ❇➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Dy ) ❝õ❛ ❜➔✐ t♦→♥ (Py ) ❝â ❞↕♥❣ ✭Dy ✮ { y ∗ , y + inf L(x, y ∗ ) − F ∗ (y ∗ )} max ∗ ∗ y ∈Y x∈X ❚r÷í♥❣ ❤đ♣ y = 0✱ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ (P ) ❧➔ ✭D ✮ max { inf L(x, y ∗ ) − F ∗ (y ∗ )} y ∗ ∈Y ∗ x∈X ❚❛ ❧✉æ♥ ❝â val(P ) ≥ val(D ) ✈➻ v(u) ≥ v∗∗(u)✳ ◆➳✉ ✈ỵ✐ ♠ët ✈➔✐ x0 ∈ X, y¯∗ ∈ Y ∗ ♠➔ ✭✷✳✼✮ f (x0 ) + F (G(x0 )) = inf L(x, y¯∗ ) − F ∗ (¯ y ∗ ), x∈X t❤➻ val(P ) = val(D ) t ỵ val(P ) = val(D ) ❤ú✉ ❤↕♥ t❤➻ x0 ∈ X ✈➔ y¯∗ ∈ Y ∗ t÷ì♥❣ ù♥❣ ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (P ) ✈➔ (D )✳ ✣✐➲✉ ❦✐➺♥ ✭✷✳✼✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ = f (x0 ) + F (G(x0 )) − inf L(x, y¯∗ ) − F ∗ (¯ y∗) x∈X ⇔ = L(x0 , y¯∗ ) − y ∗ , G(x0 ) + F (G(x0 )) − inf L(x, y¯∗ ) + F ∗ (¯ y ∗ ) x∈X ❍❛② L(x0 , y¯∗ )− inf L(x, y¯∗ ) + F (G(x0 )) + F ∗ (¯ y ∗ ) − y ∗ , G(x0 ) x∈X = ✭✷✳✽✮ ◆❤➟♥ ①➨t ✷✳✷✳ ✭✐✮ L(x0, y¯∗) ≥ x∈X inf L(x, y¯∗ ) ❤❛② L(x0 , y¯∗ )− inf L(x, y¯∗ ) ≥ x∈X ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x0 ∈ arg L(x, y¯∗ ) x∈X ∗ ∗ ∗ ✭✐✐✮ F (G(x0)) ≥ y , G(x0) − F (¯y ) ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ y¯∗ ∈ ∂F (G(x0)) ❚ø ❝→❝ ♥❤➟♥ ①➨t tr➯♥ t❛ ❝â ✤✐➲✉ ❦✐➺♥ tữỡ ữỡ ợ x0 arg L(x, y ) ✈➔ y¯∗ ∈ ∂F (G(x0 )) x∈X ✸✵ ✭✷✳✾✮ ỵ val(P ) = val(D ) x0 ∈ X, ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (P ) ✈➔ (D ) t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â ✤✐➲✉ ❦✐➺♥ ✭✷✳✾✮ t❤ä❛ ♠➣♥✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭✷✳✾✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠ët ✈➔✐ x0, y¯∗✱ ❦❤✐ ✤â x0 ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (P )✱ y¯∗ ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (D ) ✈➔ y¯∗ ∈ Y ∗ val(P ) = val(D ) ❇➙② ❣✐í t❛ ①➨t ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✻✳ ✣✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② v(y) < +∞ ✈ỵ✐ ♠å✐ y t❤✉ë❝ ❧➙♥ ❝➟♥ ❝õ❛ 0✱ ✤✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ∈ int(dom v)✳ ▼➦t ❦❤→❝✱ t❛ ❝â v(y) < + tỗ t x dom f s❛♦ ❝❤♦ G(x) + y ∈ K ✳ ❚ù❝ ❧➔ dom v = K − G(dom f ) ❱➻ ✈➟② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② int(dom v) ữủ t ữợ int(G(dom f ) − K) ✷✳✺ ⑩♣ ❞ö♥❣ t➼♥❤ t♦→♥ ữợ X, Y ổ ❇❛♥❛❝❤✱ f : X → R ✈➔ g : Y R ỗ tữớ A : X → Y ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤✳ ❳➨t ❤➔♠ F (x) := f (x) + g(Ax), ✈ỵ✐ ♠✐➲♥ ❤ú✉ ❤✐➺✉ dom F := {x ∈ dom f | Ax ∈ dom g} ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ (P ) min{f (x) + g(Ax)} x∈X ✸✶ ✭✷✳✶✵✮ ✭❇➔✐ t♦→♥ ♥➔② trữớ ủ r t ợ g ≡ F, A ≡ G.✮ ❍➔♠ ▲❛❣r❛♥❣❡ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ L(x, y ∗ ) = f (x) + y ∗ , Ax = f (x) + A∗ y ∗ , x , s✉② r❛ inf L(x, y ∗ ) = − sup{−f (x) + −A∗ y ∗ , x } = −f ∗ (−A∗ y ∗ ) x∈X x∈X ❱➻ ✈➟② ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✵✮ ❧➔ (D ) {−f ∗ (−A∗ y ∗ ) − g ∗ (y ∗ )} max ∗ ∗ y Y tr tố ữ tữỡ ự ợ t♦→♥ ✭✷✳✶✵✮ ❧➔ v = inf {f (x) + g(Ax)} x∈X ❚❛ ❝â dom v = {x ∈ dom f | Ax ∈ dom g} = dom g − A(dom f ) ✣✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ∈ int(dom v) ✤÷đ❝ ✈✐➳t ❧↕✐ t❤➔♥❤ ∈ int{A(dom f ) − dom g} ỵ X, Y ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❝→❝ ❤➔♠ f : X → R ✈➔ g : Y → R ❧➔ ❝→❝ ❤➔♠ ỗ tữớ s A : X Y t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✈➔ F (x) = f (x) + g(Ax)✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✭✷✳✶✶✮ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❜➜t ❦➻ x0 ∈ dom F ✱ t❛ ❝â ∂F (x0 ) = ∂f (x0 ) + A∗ [∂g(Ax0 )] ❚r♦♥❣ tr÷í♥❣ ❤đ♣ X = Y ✱ →♥❤ ①↕ t✉②➳♥ t➼♥❤ A ❧➔ ỗ t t õ t q s ỵ X ổ f, g : X R ỗ ❧✳s✳❝✳✱ ❝❤➼♥❤ t❤÷í♥❣✳ ◆➳✉ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✭✷✳✶✷✮ ∈ int{dom f − dom g} ✤÷đ❝ t❤ä❛ ♠➣♥ t❤➻ ✈ỵ✐ ❜➜t ❦➻ x0 ∈ (dom f ) ∩ dom g✱ t❛ ❝â ∂(f + g)(x0 ) = ∂f (x0 ) + g(x0 ) ỵ q t t t ữợ tờ ỗ ỷ tử ữợ tữớ ✤➙② t❛ ①➨t ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✤➸ t trỏ tt tr ỵ ✷✳✻✳ ✣➛✉ t✐➯♥ ❧➔ ♠ët ✈➼ ❞ö ❝❤➾ r❛ sü ❝➛♥ t❤✐➳t ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✭✷✳✶✷✮✳ ❱➼ ❞ö ✷✳✷✳ ▲➜② X = Y = R✱ f ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ f (x) = ♥➳✉ x = ✈➔ f (x) = +∞ ♥➳✉ x = 0✳ ❈❤♦ g ✤÷đ❝ ❝❤♦ ❜ð✐ g(y) = −√y ♥➳✉ y ≥ ✈➔ g(y) = +∞ ♥➳✉ y < 0✳ ❑❤✐ ✤â g(Ax) = g(x) = ❚❛ ❝â A(dom f ) = dom f = {0},   √  − x ✐❢ x ≥ 0,   +∞ ✐❢ x < dom g = [0, +∞)✳ ❙✉② r❛ 0∈ / int(A(dom f ) − dom g) ❍ì♥ ♥ú❛ F (x) = f (x) + g(Ax) =    0 ✐❢ x = 0,   +∞ ✐❢ x = ❈❤å♥ x¯ := ∈ dom F ✱ t❛ ❝â ∂F (¯x) = R tr♦♥❣ ❦❤✐ ✤â ∂f (¯ x) + A∗ (∂g(A¯ x)) = ∅ ✸✸ ❚✐➳♣ t❤❡♦✱ ✈➼ ❞ö s❛✉ ✤➙② ❝❤ù♥❣ tä r➡♥❣ ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❧✳s✳❝✳ ❝õ❛ f ✈➔ g ❦❤æ♥❣ t❤➸ ❜ä q tr ỵ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❑❤✐ ✤â ❧✉æ♥ tỗ t t t ổ tử f : X → R✳ ✣➦t g := −f ✱ t❛ ❝â dom f = dom g = X ✱ ✈➻ ✈➟② ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✭✷✳✶✷✮ t❤ä❛ ♠➣♥✳ ❱➻ f ✈➔ g ❧➔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ❧✐➯♥ tư❝✱ ♥➯♥ ❝❤ó♥❣ ❦❤ỉ♥❣ ❧✳s✳❝✳ ▼ët ♠➦t t❛ ❝â✱ ∂f (x) = ∂g(x) = ∅ ✈ỵ✐ ❜➜t ❦➻ x ∈ X ✳ ▼➦t ❦❤→❝✱ ✈➻ f (x) + g(x) ≡ 0✱ t❛ ❝â ∂(f + g)(x) = {0}✳ ❱➻ ✈➟②✱ ✭✷✳✶✸✮ ❦❤ỉ♥❣ ✤ó♥❣✳ ✸✹ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ú tổ ự q t t t ữợ tờ ỗ tữớ ỷ tử ữợ ữủ ỗ ố ♥❣➝✉✳ ❈ư t❤➸✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ỵ tt ố ự ổ tự t ữợ tờ ỗ tữớ ỷ tử ữợ sỷ ỳ ❝❤➼♥❤ q✉② t❤➼❝❤ ❤đ♣✳ ❈→❝ ❦➳t q✉↔ ð ✤➙② ✤÷đ❝ ①➨t tr➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❞à❝❤✱ tê♥❣ ❤đ♣ ✈➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t t❤❡♦ ❝→❝ ♥ë✐ ❞✉♥❣ t÷ì♥❣ ù♥❣ ❝õ❛ ♠ư❝ ✷✳✺ ❉✉❛❧✐t② ❚❤❡♦r② tr♦♥❣ ❝✉è♥ s→❝❤ ❝❤✉②➯♥ ❦❤↔♦ ❬✸❪✳ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ✣é ❱➠♥ ▲÷✉✱ P❤❛♥ t ỗ t ❑ÿ t❤✉➟t✱ ❍➔ ◆ë✐ ✭✷✵✵✵✮ ❬✷❪ ❍✉ý♥❤ ❚❤➳ P❤ị♥❣✱ ❈ì s t ỗ t t ◆❛♠✱ ✣➔ ◆➤♥❣ ✭✷✵✶✷✮ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❏✳ ❋✳ ❇♦♥♥❛♥s ❛♥❞ ❆✳ ❙❤❛♣✐r♦✱ P❡rt✉r❜❛t✐♦♥ ❆♥❛❧②s✐s ♦❢ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦ ✭✷✵✵✵✮ ✸✻ ... ▲í✐ ❝↔♠ ì♥ ✼ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✽ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ❧✐➯♥ ủ ởt số t t ữợ ỗ ởt số t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ▼ët sè ✈➜♥ ✤➲ ✈➲ ỵ tt ố Pt ❜✐➸✉ ❜➔✐ t♦→♥... ❤➳t sù❝ q✉❛♥ trå♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❦❤✐ t❛ ợ t tố ữ õ r ự ỵ tt ố ủ ỵ tt ố r t q ỗ õ t số tr ổ ứ õ ữủ ỗ ố ự q t t tờ ữợ ỗ tữớ ữợ ỳ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② t❤➼❝❤ ❤ñ♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛... ❝â ♥ë✐ ❞✉♥❣ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ tự ỡ t ỗ ỗ ủ ũ ởt số t q trđ ♥❤➡♠ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ð ❝❤÷ì♥❣ s❛✉✳ ✺ ❈❤÷ì♥❣ ✷✿ ▼ët sè ✈➜♥ ỵ