Fritz Jonning shartlari - Fritz John conditions
The Fritz Jonning shartlari (qisqacha FJ shartlari ), in matematika , a zarur shart ichida hal qilish uchun chiziqli bo'lmagan dasturlash bolmoq maqbul .[1] Karush-Kann-Taker sharoitlari , lekin ular o'zlariga tegishli.
Biz quyidagilarni ko'rib chiqamiz optimallashtirish muammosi :
minimallashtirish f ( x ) uchun mavzu: g men ( x ) ≤ 0 , men ∈ { 1 , … , m } h j ( x ) = 0 , j ∈ { m + 1 , … , n } { displaystyle { begin {aligned} { text {minimize}} & f (x) , { text {: mavzuga bo'ysunadi:}} va g_ {i} (x) leq 0, i in left {1, dots, m right } & h_ {j} (x) = 0, j in left {m + 1, dots, n right } end {hizalangan}} } qayerda ƒ bo'ladi funktsiya minimallashtirish, g men { displaystyle g_ {i}} cheklovlar va h j { displaystyle h_ {j}} Men { displaystyle { mathcal {I}}} Men ′ { displaystyle { mathcal {I '}}} E { displaystyle { mathcal {E}}} indekslar to'plamlar faol bo'lmagan, faol va tenglik cheklovlari va x ∗ { displaystyle x ^ {*}} f { displaystyle f} λ = [ λ 0 , λ 1 , λ 2 , … , λ n ] { displaystyle lambda = [ lambda _ {0}, lambda _ {1}, lambda _ {2}, nuqtalar, lambda _ {n}]}
{ λ 0 ∇ f ( x ∗ ) + ∑ men ∈ Men ′ λ men ∇ g men ( x ∗ ) + ∑ men ∈ E λ men ∇ h men ( x ∗ ) = 0 λ men ≥ 0 , men ∈ Men ′ ∪ { 0 } ∃ men ∈ ( { 0 , 1 , … , n } ∖ Men ) ( λ men ≠ 0 ) { displaystyle { begin {case} lambda _ {0} nabla f (x ^ {*}) + sum limit _ {i in { mathcal {I}} '} lambda _ {i} nabla g_ {i} (x ^ {*}) + sum limitlar _ {i in { mathcal {E}}} lambda _ {i} nabla h_ {i} (x ^ {*}) = 0 [10pt] lambda _ {i} geq 0, i in { mathcal {I}} ' cup {0 } [10pt] i in left ( {0,1, ldots, n } teskari burilish { mathcal {I}} o'ng) chap ( lambda _ {i} neq 0 o'ng) end {holatlar}}} λ 0 > 0 { displaystyle lambda _ {0}> 0} agar The ∇ g men ( men ∈ Men ′ ) { displaystyle nabla g_ {i} (i in { mathcal {I}} ')} ∇ h men ( men ∈ E ) { displaystyle nabla h_ {i} (i in { mathcal {E}})} chiziqli mustaqil yoki umuman olganda, a cheklash malakasi ushlab turadi.
Nomlangan Fritz Jon , bu shartlar ga teng Karush-Kann-Taker sharoitlari holda λ 0 > 0 { displaystyle lambda _ {0}> 0} λ 0 = 0 { displaystyle lambda _ {0} = 0} Mangasarian-Fromovits cheklovlari malakasi (MFCQ). Boshqacha qilib aytganda, Fritz Jon sharti KKT yoki MFCQ bo'lmagan maqbullik shartiga tengdir.[iqtibos kerak
Adabiyotlar
Qo'shimcha o'qish
Rau, Nikolay (1981). "Lagrange ko'paytmalari". Matritsalar va matematik dasturlash . London: Makmillan. 156–174 betlar. ISBN 0-333-27768-6
![lambda = [ lambda_0, lambda _1, lambda _2, nuqtalar, lambda _n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e25076573b9439185d47b40d8f29939905230dbe)
![{ displaystyle { begin {case} lambda _ {0} nabla f (x ^ {*}) + sum limit _ {i in { mathcal {I}} '} lambda _ {i} nabla g_ {i} (x ^ {*}) + sum limitlar _ {i in { mathcal {E}}} lambda _ {i} nabla h_ {i} (x ^ {*}) = 0 [10pt] lambda _ {i} geq 0, i in { mathcal {I}} ' cup {0 } [10pt] i in left ( {0,1, ldots, n } teskari burilish { mathcal {I}} o'ng) chap ( lambda _ {i} neq 0 o'ng) end {holatlar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d83fc06dc1e0fe79f3a7894e8a105379b58edca6)
