非線形最適化の基礎〜射影・錐・凸関数〜

射影の非拡大性，錐と極錐，Farkasの補題，凸関数に関するスライド

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1. ඇઢܗ࠷దԽͷجૅ – Farkas’ lemma – miruca Graduate School of Informatics, Kyoto University March 15, 2019
2. ͜ͷεϥΠυͷ໨త ʰඇઢܗ࠷దԽͷجૅʱ(෱ౡ, 2001) ʹؔͯ͠ • ࣹӨͷඇ֦େੑʹ͍ͭͯཧղ͢Δ • ਲ਼ͱۃਲ਼ʹ͍ͭͯཧղ͢Δ • Farkas ͷิ୊Λཧղ͢Δ • ತؔ਺ʹ͍ͭͯཧղ͢Δ • ತؔ਺Ͱ͋ΔͨΊͷඞཁे෼৚݅Λཧղ͢Δ ˞஫ҙ • ຊεϥΠυͷఆཧ౳ͷ൪߸͸ʰඇઢܗ࠷దԽͷجૅʱʹ४ͣΔ • ਤ͸ͳ͍ͷͰదٓखΛಈ͔͠ͳ͕Βཧղ͢Δ͜ͱΛਪ঑ • ఆཧͷূ໌͸ॏཁͳ΋ͷʹ͍ͭͯͷΈߦ͏
3. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) Today’s Topic 1. ࣹӨ (projection) 2. Farkas ͷิ୊ (Farkas’ lemma) 3. ತؔ਺ (convex function) 3 / 20
4. 1. ࣹӨ (projection) 2. Farkas ͷิ୊ (Farkas’ lemma) 3. ತؔ਺ (convex function)
5. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ࣹӨͷఆٛ ఆٛ: ࣹӨ (projection) ඇۭͷดತू߹ *1)S ⫅ Rn Λߟ͑Δɽ೚ҙͷ఺ x ∈ Rn ʹର͠ɼ ∥x − P s(x)∥ = min {∥x − z | z ∈ S} (1) Λຬͨ͢఺ P s(x) ∈ S ͕ଘࡏ͢Δ *2)ɽ͜ͷࣸ૾ P s : Rn → S, x → P s(x) Λ S ΁ͷࣹӨ(projection) ͱݺͿɽ • ࣜ (1) Λຬͨ͢఺ P s(x) ∈ S ͸།ҰͰ͋Δɽ • x ∈ S ͳΒ͹ x = P s(x) Ͱ͋Δɽ • ೚ҙͷ఺ x ∈ Rn ʹରͯ͠ɼ͕࣍ࣜ੒Γཱͭɿ ⟨x − P s(x), z − P s(z)⟩ ≦ 0 (∀z ∈ S). (2) *1)ดू߹͔ͭತू߹Ͱ͋ΔΑ͏ͳू߹Λดತू߹ͱ͍͏ɽ *2)P s(x) ͸ S ͷதͰ x Λ࠷ྑʹۙࣅ͢Δ఺ͱͳ͍ͬͯΔɽ 5 / 20
6. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ࣹӨʹؔ͢Δఆཧ ఆཧ 2.6 S ⫅ Rn ΛۭͰͳ͍ดತू߹ͱ͢Δɽͦͷͱ͖ɼ೚ҙͷ఺ x ∈ Rn ʹରͯ͠ɼx ͷ S ΁ͷࣹӨ P s(x) ͕Ұҙతʹଘࡏ͠ɼෆ౳ࣜ ⟨x − P s(x), z − P s(x)⟩ ≦ 0 (∀z ∈ S) (2) ͕੒Γཱͭɽ • ಉ͜͡ͱ͕ͩɼ೚ҙͷ x⋆ ∈ S ʹରͯ͠ɼ x⋆ = P s(x) ⇔ ⟨x − x⋆, z − x⋆⟩ ≦ 0 (∀z ∈ S) (3) ͕੒Γཱͭɽ • ࣜ (2) ͸ʮx⋆ Λ࢝఺ͱ͢Δ 2 ͭͷϕΫτϧ x − x⋆ ͱ z − x⋆ ͷͳ͕֯͢Ӷ֯ʹͳΒͳ͍ʯͱ͍͏زԿֶతͳಛ௃෇͚͕Ͱ ͖Δɽ 6 / 20
7. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ࣹӨͷඇ֦େੑ ຊఆཧ͸ม෼ෆ౳ࣜ (VIP) ʹର͢Δஞ࣍ۙࣅΞϧΰϦζϜͷղੳͳ ͲͰॏཁͳ໾ׂΛՌͨ͢ॏཁͳఆཧͰ͋Δɽ ఆཧ 2.7 S ⫅ Rn ΛۭͰͳ͍ดತू߹ͱ͢Δɽ͜ͷͱ͖ ∥P s(x) − P s(y)∥ ≦ ∥x − y∥ (∀x, y ∈ Rn) (4) ͕੒Γཱͭɽ • ఆཧ 2.7 ͸ɼ೚ҙͷ 2 ఺ؒͷڑ཭͸ดತू߹΁ͷࣹӨʹΑͬͯ ֦͕Βͳ͍͜ͱΛ͍ࣔͯ͠Δɽ • ࣹӨ P s Λ Rn ͔Β Rn ΁ͷࣸ૾ͱΈͳ͢ͱɼఆཧ 2.7 ͸ࣸ૾ P s ͷඇ֦େੑΛ͍ࣔͯ͠Δ *3)ɽ *3)Ұൠʹࣸ૾ T : Rn → Rn ͕৚݅ ∥T(x) − T(y)∥ ≦ ∥x − y∥ (∀x, y ∈ Rn) Λຬͨ ͢ͱ͖ɼT ͸ඇ֦େࣸ૾ (nonexpansive map) Ͱ͋Δͱ͍͏ɽ 7 / 20
8. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ࣹӨͷඇ֦େੑ ূ໌. ఆཧ 2.6 ΑΓɼ೚ҙͷ఺ x, y ∈ Rn ʹରͯ͠ɼ ⟨x − P s(x), z − P s(x)⟩ ≦ 0 (∀z ∈ S) (5) ⟨y − P s(y), z − P s(y)⟩ ≦ 0 (∀z ∈ S) (6) ͕੒Γཱͭɽࣜ (5)(6) ͸೚ҙͷ z ∈ S ʹରͯ͠੒ΓཱͭͷͰɼ ⟨x − P s(x), P s(y) − P s(x)⟩ ≦ 0 (7) ⟨y − P s(y), P s(x) − P s(y)⟩ ≦ 0 (8) ΋੒Γཱͭ ( (∵)P s(x) ∈ S, P s(y) ∈ S )ɽ ࣜ (7)(8) ΛลʑՃ͑ɼίʔγʔɾγϡϫϧπͷෆ౳ࣜΛద༻͢Δͱ ∥P s(x) − P s(y)∥2 ≦ ⟨P s(x) − P s(y), x − y⟩ (9) ≦ ∥P s(x) − P s(y)∥ · ∥x − y∥ (10) ͕ಘΒΕ (෇࿥ A)ɼลʑ ∥P s(x) − P s(y)∥ ͰׂΔͱࣜ (4) ͕ಘΒ ΕΔɽ(ূ໌ऴ) 8 / 20
9. 1. ࣹӨ (projection) 2. Farkas ͷิ୊ (Farkas’ lemma) 3. ತؔ਺ (convex function)
10. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ਲ਼ͷఆٛ ఆٛ: ਲ਼ ࣍ͷ৚݅Λຬ଍͢Δू߹ C ∈ Rn Λਲ਼(cone) ͱ͍͏ɽ x ∈ C, α ∈ [0, ∞) ⇒ αx ∈ C ಛʹɼਲ਼͕ತू߹Ͱ͋Δͱ͖ತਲ਼ɼดू߹Ͱ͋Δͱ͖ดਲ਼ͱ͍͍ɼ ดू߹Ͱ͋ΔΑ͏ͳತਲ਼Λดತਲ਼ͱ͍͏ɽ ྫ 2.7 ࣍ͷू߹ Ci (i = 1, 2, 3, 4) ͸͍ͣΕ΋ดತਲ਼Ͱ͋Δɽ • C1 = { x ∈ Rn | x = ∑ m i=1 αiai, αi ≧ 0 (i = 1, . . . , m) } • C2 = { x ∈ Rn | ⟨ ai, x ⟩ ≦ 0 (i = 1, . . . , m) } • C3 = { x ∈ Rn | x2 1 ≧ x2 2 + · · · + x2 n , x1 ≧ 0 } 10 / 20
11. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ۃਲ਼ͷఆٛ ఆٛ: ۃਲ਼ ೚ҙͷਲ਼ C ⫅ Rn ʹରͯ͠ C⋆ := {y ∈ Rn | ⟨y, x⟩ ≦ 0 (∀x ∈ C)} Ͱఆٛ͞ΕΔू߹ C⋆ ⫅ Rn Λ C ͷۃਲ਼(polar cone) ͱ͍͏ɽ • ۃਲ਼ C⋆ ͸ɼC ʹଐ͢Δ͢΂ͯͷϕΫτϧͱ 90◦ Ҏ্ͷ֯Λͳ ͢ϕΫτϧશମͷू߹Ͱ͋Δɽ • ۃਲ਼ C⋆ ͸ਲ਼ͷఆٛΛຬͨ͢ͷͰਲ਼Ͱ͋Δ *4)ɽ • C ͕෦෼ۭؒͰ͋Ε͹ɼC⋆ ͸ C ͷ௚ަิۭؒͱ౳ՁͰ͋Δɽ *4)೚ҙͷ y ∈ C⋆ ʹରͯ͠ɼ⟨y, x⟩ ≦ 0 (∀x ∈ C) ͕੒Γཱͭɽ͜ͷͱ͖ɼ೚ҙͷ α ∈ [0, ∞) ʹରͯ͠ɼα ⟨y, x⟩ = ⟨αy, x⟩ ≦ 0 Ͱ͋Δ͔Βɼαy ∈ C⋆ ͕ݴ͑Δɽ͕ͨͬ͠ ͯɼC⋆ ͸ਲ਼Ͱ͋Δɽ 11 / 20
12. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ਲ਼ͱۃਲ਼ͷੑ࣭ ࣍ͷ 2 ͭͷఆཧ͸ਲ਼ͱۃਲ਼ͷؔ܎Λड़΂ͨ΋ͷͰ͋ΓॏཁͰ͋Δɽ ఆཧ 2.12 ೚ҙͷۭͰͳ͍ਲ਼ C ⊆ Rn ʹରͯ͠ɼ • ͦͷۃਲ਼ C⋆ ͸ดತਲ਼Ͱ͋Δ • ؔ܎ࣜ C⋆ = (co C)⋆ ͕੒Γཱͭ ·ͨɼ2 ͭͷਲ਼ C, D ⊆ Rn ʹରͯ͠ɼC ⊆ D ⇒ C⋆ ⫆ D⋆ ͕੒Γ ཱͭɽ ఆཧ 2.13 ೚ҙͷۭͰͳ͍ਲ਼ C ⊆ Rn ʹରͯ͠ɼͦͷۃਲ਼ C⋆ ͷۃਲ਼ (C⋆)⋆ = C⋆⋆ ͸ɼC ͷತแͷดแ cl co C ʹҰக͢Δ *5)ɽಛʹɼC ͕ดತਲ਼ͳΒ͹ɼC = C⋆⋆ ͕੒Γཱͭɽ *5)ਲ਼ C ͷತแͷดแ cl co C ͱดแͷತแ co cl C ͸Ұக͢Δͱ͸ݶΒͳ͍ɽ 12 / 20
13. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) Farkas’ lemma 2 ͭͷఆཧ͔Β Farkas ͷิ୊ͱݺ͹ΕΔॏཁͳ໋୊͕ಋ͔ΕΔɽ ఆཧ 2.15 Farkas ͷิ୊ ϕΫτϧ a1, . . . , am ∈ Rn ʹΑͬͯੜ੒͞ΕΔดತଟ໘ਲ਼ C = { x ∈ Rn | x = m ∑ i=1 αiai, αi ≧ 0 (i = 1, . . . , m) } ͱɼ͢΂ͯͷ ai ͱ 90◦ Ҏ্ͷ֯Λͳ͢ϕΫτϧશମ͔ΒͳΔดತ ଟ໘ਲ਼ K = { y ∈ Rn | ⟨ ai, y ⟩ ≦ 0 (i = 1, . . . , m) } ʹରͯ͠ɼK = C⋆ ͓Αͼ C = K⋆ ͳΔؔ܎͕੒Γཱͭɽ • ຊิ୊͸ඇઢܗܭը໰୊ͷ࠷దੑ৚݅ (KKT ৚݅) Λಋͨ͘Ί ʹॏཁͳ໾ׂΛՌͨ͢ɽ 13 / 20
14. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) Proof of Farkas’ lemma ূ໌. K = C⋆ ʹର͢Δূ໌͸ (1) K ⫅ C⋆ (2) C⋆ ⫅ K ͷೋஈ֊Ͱߦ͏ɽ (1) ೚ҙͷ y ∈ K ʹରͯ͠ɼy ∈ C⋆ Ͱ͋Δ͜ͱΛࣔ͢ɽy ∈ K Ͱ ͋Δ͔Βɼ೚ҙͷ x = ∑ m i=1 αiai ∈ C ʹରͯ͠ɼ ⟨x, y⟩ = ⟨ m ∑ i=1 αiai, y ⟩ = m ∑ i=1 αi ⟨x, y⟩ ≦ 0 ͕੒Γཱͭɽ͜Ε͸ y ∈ C⋆ Λҙຯ͍ͯ͠ΔͷͰɼK ⫅ C⋆ ͕ࣔ͞ Εͨɽ(2) ೚ҙͷ y ∈ C⋆ ʹରͯ͠ɼy ∈ K Ͱ͋Δ͜ͱΛࣔ͢ɽۃ ਲ਼ C⋆ ͷఆٛΑΓɼ೚ҙͷ x ∈ C ʹରͯ͠ɼ ⟨y, x⟩ = ⟨ y, m ∑ i=1 αiai ⟩ = m ∑ i=1 αi ⟨ y, ai ⟩ ≦ 0 (11) ͕੒Γཱͭɽ͜͜Ͱɼαi ≧ 0 (i = 1, . . . , m) ͸೚ҙͰ͋Δɽ 14 / 20
15. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ্ࣜ (11) ͸ ⟨ y, ai ⟩ ≦ 0 ͕֤ i ∈ I = {1, . . . , m} ʹରͯ͠੒Γཱ ͭ͜ͱΛҙຯ͍ͯ͠ΔɽͳͥͳΒɼ͋Δ j ∈ I ʹରͯ͠ɼ ⟨ aj, y ⟩ > 0 ͳΔఴࣈ͕͋ΔͱԾఆ͢Δͱɼαj = 1, α = 0 (i ̸= j) ͱ͓͘͜ͱʹΑΓɼ m ∑ i=1 ⟨ y, αiai ⟩ = αj ⟨ y, aj ⟩ > 0 ͕੒Γཱͪɼy ∈ C⋆ Ͱ͋Δ͜ͱʹ൓͢Δ͔ΒͰ͋Δɽ͕ͨͬͯ͠ɼ y ∈ K Ͱ͋ΓɼC⋆ ⫅ K ͕੒ΓཱͭɽҎ্ΑΓɼK = C⋆ ͕ࣔ͞ Εͨɽ C = K⋆ Λࣔ͢ʹ͸ɼఆཧ 2.13 Λ K = C⋆ ʹରͯ͠ద༻͢Ε͹Α ͍ɽͭ·ΓɼC ͸ดತਲ਼Ͱ͋Δ͔Βɼఆཧ 2.13 ͕ద༻Ͱ͖ɼ K⋆ = C⋆⋆ = C ͕੒Γཱͭɽ(ূ໌ऴ) 15 / 20
16. 1. ࣹӨ (projection) 2. Farkas ͷิ୊ (Farkas’ lemma) 3. ತؔ਺ (convex function)
17. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ತؔ਺ͷఆٛ ఆٛ: άϥϑɾΤϐάϥϑɾತؔ਺ • ؔ਺ f : Rn → [−∞, +∞] ʹରͯ͠ɼRn+1 ͷ෦෼ू߹ graph f = { (x, β)⊤ ∈ Rn+1 | β = f(x) } Λ f ͷάϥϑ(graph) ͱ͍͏ɽ • f ͷάϥϑΑΓ্ʹ͋Δ఺શମͷू߹ epi f = { (x, β)⊤ ∈ Rn+1 | β ≧ f(x) } Λ f ͷΤϐάϥϑ(epigraph) ͱ͍͏ɽ • Τϐάϥϑ epi f ͕ತू߹Ͱ͋ΔΑ͏ͳؔ਺ f Λ ತؔ਺(convex function) ͱ͍͏ɽ 17 / 20
18. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ತؔ਺Ͱ͋ΔͨΊͷඞཁे෼৚݅ ఆཧ 2.22 ؔ਺ f : Rn → (−∞, +∞] ͕ತؔ਺Ͱ͋ΔͨΊͷඞཁे෼৚݅͸ɼ ೚ҙͷ x, y ∈ Rn ͱ೚ҙͷඇෛ࣮਺ α ∈ [0, 1] ʹରͯ͠ɼ f ((1 − α)x + αy) ≦ (1 − α)f(x) + αf(y) (12) ͕੒Γཱͭ͜ͱͰ͋Δ *6)ɽ • ࣜ (12) ͸ɼf ͷάϥϑʹଐ͢Δ೚ҙͷ 2 ఺Λ݁Ϳઢ෼͕ f ͷ Τϐάϥϑʹؚ·ΕΔ͜ͱΛҙຯ͍ͯ͠Δɽ • ຊఆཧ͸ತؔ਺ͷఆ͔ٛΒূ໌Ͱ͖Δ (෇࿥ B)ɽ • ༩͑ΒΕͨؔ਺͕ತؔ਺Ͱ͋Δ͔Ͳ͏͔൑ఆ͢Δ͜ͱ͸Ұൠత ʹ༰қͰ͸ͳ͍ͨΊɼఆཧ 2.22 ͸༗༻Ͱ͋Δɽ *6)ࣜ (12) Λತؔ਺ͷఆٛͱͯ͠༻͍͍ͯΔจݙ΋ଟ͍ɽ 18 / 20
19. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ತؔ਺: ޯ഑ʹΑΔಛ௃෇͚ ఆཧ 2.29 f : Rn → (−∞, +∞] Λ࣮ޮఆٛҬ *7)dom f ͕։ತू߹Ͱ͋ΔΑ͏ ͳؔ਺ͱ͢Δɽf ͕ dom f ʹ͓͍ͯඍ෼ՄೳͰ͋Δͱ͖ɼf ͕ತؔ ਺Ͱ͋ΔͨΊͷඞཁे෼৚݅͸ɼx ̸= y Ͱ͋ΔΑ͏ͳ೚ҙͷ x, y ∈ dom f ʹରͯ͠ɼҎԼͷෆ౳͕ࣜ੒ཱ͢Δ͜ͱͰ͋Δɿ f(y) − f(x) ≧ ⟨∇f(x), y − x⟩ . (13) • ࣜ (13) ͸௒ฏ໘ H = {(y, α) ∈ Rn+1 | α = f(x) + ⟨∇f(x), y − x⟩} ͕఺ (x, f(x))⊤ ∈ Rn+1 ʹ͓͚Δ epi f ͷࢧ࣋௒ฏ໘Ͱ͋Δ͜ ͱΛҙຯ͍ͯ͠Δɽ *7)ؔ਺ f ʹରͯ͠ɼ{x ∈ Rn | f(x) < +∞} Ͱఆٛ͞ΕΔू߹ 19 / 20
20. ࣹӨ (projection) Farkas ͷิ୊ (Farkas’ lemma) ತؔ਺ (convex function) ತؔ਺: Hesse ߦྻʹΑΔಛ௃෇͚ ఆཧ 2.30 f : Rn → (−∞, +∞] Λ࣮ޮఆٛҬ dom f ͕։ತू߹Ͱ͋ΔΑ͏ͳ ؔ਺ͱ͢Δɽf ͕ dom f ʹ͓͍ͯ 2 ճ࿈ଓతඍ෼ՄೳͰ͋Δͱ͖ɼ f ͕ತؔ਺Ͱ͋ΔͨΊͷඞཁे෼৚݅͸ɼx ̸= y Ͱ͋ΔΑ͏ͳ೚ҙ ͷ x, y ∈ dom f ʹରͯ͠ɼHesse ߦྻ ∇2f(x) ͕൒ਖ਼ఆ஋ߦ ྻ *8)ɼ͢ͳΘͪ ∇2f(x) ≽ O ͱͳΔ͜ͱͰ͋Δɽ ྫ ؔ਺ f(x) = 4x2 1 − 2x1x2 + x2 2 − 10x1 − 8x2 − 100 ʹର͢Δ Hesse ߦྻ͸ ∇2f(x) = ( 8 −2 −2 2 ) ≽ O Ͱ͋Δ *9) ͷͰɼఆཧ 2.30 ΑΓؔ਺ f ͸ತؔ਺Ͱ͋Δɽ *8)n ࣍ਖ਼ํߦྻ A ͸ɼ೚ҙͷ x∈Rn ʹରͯ͠ɼ⟨x, Ax⟩ ≧ 0 ͕੒Γཱͭͱ͖൒ਖ਼ఆ஋Ͱ ͋Δͱ͍͍ɼA ≽ O ͱॻ͘ɽ *9)ݻ༗஋ΛٻΊΔͱ λ = 5 ± √ 13 > 0 ΑΓ ∇2f(x) ≽ O Ͱ͋Δɽ 20 / 20
21. 4. ิ଍ (Appendix)
22. ෇࿥ A ೦ͷҝɼࣜ (7)(8) ͔Βࣜ (9) Λಋ͘աఔΛࡌ͓ͤͯ͘ɽࣜ (7) ͸࣍ ࣜͱ౳ՁͰ͋Δ͜ͱʹ஫ҙ͢Δɽ ⟨P s(x) − x, P s(x) − P s(y)⟩ ≦ 0 (7’) ࣜ (7’)(8) ͷลʑΛՃ͑Δ͜ͱʹΑΓɼ ⟨P s(x) − P s(y), P s(x) − x − P s(y) + y⟩ ≦ 0 ͕੒Γཱͭ (P s(x) − P s(y) Λڞ௨Ҽࢠͱͯ͘͘͠Γग़͢)ɽޙ͸ɼ ಺ੵͷੑ࣭Λར༻ͯ͠ܭࢉ͢Ε͹ɼ ∥P s(x) − P s(y)∥2 ≦ ⟨P s(x) − P s(y), x − y⟩ ͕ಘΒΕΔɽ
23. ෇࿥ B ূ໌ ఆ͔ٛΒ؆୯ʹಋग़Ͱ͖Δɽತؔ਺ͷఆٛΑΓɼؔ਺ f ͕ತ ؔ਺Ͱ͋Δ͜ͱ͸ɼ೚ҙͷ (x, β)⊤ ∈ epi f, (y, γ)⊤ ∈ epi f ͱ࣮਺ α ∈ [0, 1] ʹରͯ͠ɼ (1 − α) ( x β ) + α ( y γ ) = ( (1 − α)x + αy (1 − α)β + αγ ) ∈ epi f ͱͳΔ͜ͱͰ͋ΔɽΤϐάϥϑͷఆٛΑΓ্ࣜ͸ɼ f ((1 − α)x + αy) ≦ (1 − α)β + αγ (14) ͱ౳ՁͰ͋Δɽ͍·ɼf(x) ≦ β, f(y) ≦ γ Λຬͨ͢೚ҙͷ x, y ∈ Rn ͱ࣮਺ β, γ ∈ R ͓Αͼ α ∈ [0, 1] ʹରͯ͠ɼෆ౳ࣜ (14) ͕ຬͨ͞Εͳ͚Ε͹ͳΒͳ͍ɽ͕ͨͬͯ͠ɼॴ๬ͷෆ౳ࣜ f ((1 − α)x + αy) ≦ (1 − α)f(x) + αf(y) ͕ಘΒΕΔɽ(ূ໌ऴ)