U matematici zokrema v opuklomu analizi ponyattya subdiferencialu ta subgradiyentu ye uzagalnennyami vidpovidnih ponyat diferencialu ta gradiyentu klasichnogo analizu Opukla funkciya sinya ta liniyi subgradiyentu v x0 chervoni Zmist 1 Viznachennya 2 Priklad 3 Vlastivosti 4 Div takozh 5 DzherelaViznachennya RedaguvatiNehaj f R n R displaystyle f mathbb R n rightarrow mathbb R nbsp funkciya na evklidovomu prostori R n displaystyle mathbb R n nbsp Vektor g R n displaystyle g in mathbb R n nbsp nazivayetsya subgradiyentom funkciyi f x displaystyle f x nbsp v tochci x R n displaystyle bar x in mathbb R n nbsp yaksho spravdzhuyetsya nerivnist g R n f x f x g T x x x R n displaystyle g in mathbb R n f x f bar x geq g T x bar x quad forall x in mathbb R n nbsp Mnozhina vsih subgradiyentiv nazivayetsya subdiferencialom funkciyi f x v tochci x displaystyle bar x nbsp i poznachayetsya f x displaystyle partial f bar x nbsp Vikoristovuyuchi matematichnu simvoliku mozhna zapisati viznachennya subdiferencialu f x g R n f x f x g T x x x R n displaystyle partial f bar x g in mathbb R n f x f bar x geq g T x bar x quad forall x in mathbb R n nbsp Priklad RedaguvatiDlya funkciyi f R R x x displaystyle f mathbb R rightarrow mathbb R x mapsto x nbsp odniyeyi dijsnoyi zminnoyi mayemo f x 1 x lt 0 1 1 x 0 1 x gt 0 displaystyle partial f bar x begin cases 1 amp bar x lt 0 left 1 1 right amp bar x 0 1 amp bar x gt 0 end cases nbsp Vlastivosti RedaguvatiOpukla funkciya f R R displaystyle f mathbb R to mathbb R nbsp ye diferencijovnoyu v tochci x 0 displaystyle x 0 nbsp todi i tilki todi koli subdiferencijal funkciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp skladayetsya z yedinogo chisla Ce chislo i ye pohidnoyu funkciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp Tochka x 0 displaystyle x 0 nbsp ye tochkoyu globalnogo minimumu opukloyi funkciyi f displaystyle f nbsp todi i tilki todi koli nul vhodit do yiyi subdiferencialu tobto koli na risunku vishe mozhna provesti gorizontalnu dotichnu v tochci x 0 displaystyle x 0 nbsp do grafiku funkciyi f displaystyle f nbsp Yaksho f displaystyle f nbsp i g displaystyle g nbsp ye opuklimi funkciyami z subdiferencialami f x displaystyle partial f x nbsp i g x displaystyle partial g x nbsp to subdiferencialom funkciyi f g displaystyle f g nbsp ye f g x f x g x displaystyle partial f g x partial f x oplus partial g x nbsp de displaystyle oplus nbsp poznachaye sumu Minkovskogo Div takozh RedaguvatiOpukla funkciyaDzherela RedaguvatiMoklyachuk M P Osnovi opuklogo analizu K TviMS 2004 240s M P Moklyachuk Negladkij analiz ta optimizaciya J M Borwein A S Lewis 2000 Convex Analysis and Nonlinear Optimization Springer New York Otrimano z https uk wikipedia org w index php title Subdiferencial amp oldid 35488885
