Kvaziopukla funkciya uzagalnennya ponyattya opukloyi funkciyi sho znajshlo shiroke vikoristannya v nelinijnij optimizaciyi zokrema pri zastosuvanni optimizaciyi do pitan ekonomiki Kvaziopukla funkciya sho ne ye opukloyuFunkciya sho ne ye kvaziopukloyu mnozhina tochok znachennya funkciyi v yakih ne perevishuye chervonoyi punktirnoyi liniyi ne ye opukloyu Zmist 1 Viznachennya 2 Prikladi 3 Vlastivosti 3 1 Diferencijovni kvaziopukli funkciyi 4 Operaciyi sho zberigayut kvaziopuklist 5 Posilannya 6 LiteraturaViznachennya RedaguvatiNehaj X opukla pidmnozhina R n displaystyle mathbb R n nbsp Funkciya f X R displaystyle f X to mathbb R nbsp nazivayetsya kvaziopukloyu abo unimodalnoyu yaksho dlya dovilnih elementiv x y X displaystyle x y in X nbsp i l 0 1 displaystyle lambda in 0 1 nbsp vikonuyetsya nerivnist f l x 1 l y max f x f y displaystyle f lambda x 1 lambda y leq max big f x f y big nbsp Yaksho takozh f l x 1 l y lt max f x f y displaystyle f lambda x 1 lambda y lt max big f x f y big nbsp dlya x y displaystyle x neq y nbsp i l 0 1 displaystyle lambda in 0 1 nbsp to funkciya nazivayetsya strogo kvaziopukloyu Funkciya f X R displaystyle f X to mathbb R nbsp nazivayetsya kvaziuvignutoyu strogo kvaziuvignutoyu yaksho f displaystyle f nbsp ye kvaziopukloyu strogo kvaziopukloyu Ekvivalentno funkciya ye kvaziuvignutoyu yaksho f l x 1 l y min f x f y displaystyle f lambda x 1 lambda y geq min big f x f y big nbsp i strogo kvaziuvignutoyu yaksho f l x 1 l y gt min f x f y displaystyle f lambda x 1 lambda y gt min big f x f y big nbsp Funkciya yaka odnochasno ye kvaziopukloyu ta kvaziuvignutoyu nazivayetsya kvazilinijnoyu Prikladi RedaguvatiDovilna opukla funkciya ye kvaziopukloyu dovilna uvignuta funkciya ye kvaziuvignutoyu Funkciya f x ln x displaystyle f x ln x nbsp ye kvazilinijnoyu na mnozhini dodatnih dijsnih chisel Funkciya f x 1 x 2 x 1 x 2 displaystyle f x 1 x 2 x 1 x 2 nbsp ye kvazuvignutoyu na mnozhini R 2 displaystyle mathbb R 2 nbsp mnozhina par nevid yemnih chisel ale ne ye ni opukloyu ni uvignutoyu Funkciya x x displaystyle x mapsto lfloor x rfloor nbsp ye kvaziopukloyu i ne ye ni opukloyu ni neperervnoyu Vlastivosti RedaguvatiFunkciya f X R displaystyle f X to mathbb R nbsp de X R n displaystyle X subset mathbb R n nbsp opukla mnozhina kvaziopukla todi i tilki todi koli dlya vsih b R displaystyle beta in mathbb R nbsp mnozhinaX b x X f x b displaystyle X beta x in X f x leqslant beta nbsp Dovedennya Nehaj mnozhina X b displaystyle X beta nbsp opukla dlya bud yakogo b Zafiksuyemo dvi dovilni tochki x 1 x 2 X displaystyle x 1 x 2 in X nbsp ta rozglyanemo tochku x l x 1 1 l x 2 l 0 1 displaystyle x lambda x 1 1 lambda x 2 quad lambda in 0 1 nbsp Tochki x 1 x 2 X b displaystyle x 1 x 2 in X beta nbsp pri b max f x 1 f x 2 displaystyle beta max f x 1 f x 2 nbsp Oskilki mnozhina X b displaystyle X beta nbsp opukla tox X b displaystyle x in X beta nbsp a otzhe f x b m a x f x 1 f x 2 displaystyle f x leqslant beta max f x 1 f x 2 nbsp tobto vikonuyetsya nerivnist u viznachenni i funkciya ye kvaziopukloyu Nehaj funkciya f kvaziopukla Dlya deyakogo b R displaystyle beta in mathbb R nbsp zafiksuyemo dovilni tochki x 1 x 2 X b displaystyle x 1 x 2 in X beta nbsp Todi max f x 1 f x 2 b displaystyle max f x 1 f x 2 leqslant beta nbsp Oskilki X opukla mnozhina to dlya bud yakogo l 0 1 displaystyle lambda in 0 1 nbsp tochka x l x 1 1 l x 2 X displaystyle x lambda x 1 1 lambda x 2 in X nbsp Z oznachennya kvaziopuklosti viplivaye sho f x m a x f x 1 f x 2 b displaystyle f x leqslant max f x 1 f x 2 leqslant beta nbsp tobto x X b displaystyle x in X beta nbsp Otzhe X b displaystyle X beta nbsp opukla mnozhina Neperervna funkciya f X R displaystyle f X to mathbb R nbsp de X opukla mnozhina v R displaystyle mathbb R nbsp kvaziopukla todi i tilki todi koli vikonuyetsya odna z takih umov f nespadna f nezrostayucha isnuye taka tochka c X displaystyle c in X nbsp sho dlya vsih t X t c displaystyle t in X t leqslant c nbsp funkciya f nezrostayucha i dlya vsih t X t c displaystyle t in X t geqslant c nbsp funkciya f nespadna Diferencijovni kvaziopukli funkciyi Redaguvati Nehaj f X R displaystyle f X to mathbb R nbsp diferencijovana funkciya na X de X R n displaystyle X subset mathbb R n nbsp vidkrita opukla mnozhina Todi f kvaziopukla na X todi i tilki todi koli spravdzhuyetsya spivvidnoshennya f y f x f x y x 0 displaystyle f y leqslant f x Rightarrow left langle f x y x right rangle leqslant 0 nbsp dlya vsih x y X displaystyle x y in X nbsp Nehaj f dvichi diferencijovna funkciya Yaksho f kvaziopukla na X to vikonuyetsya umova f x y 0 f x y y 0 displaystyle left langle f x y right rangle 0 Rightarrow left langle f x y y right rangle geqslant 0 nbsp dlya vsih x X y R n displaystyle x in X y in mathbb R n nbsp Neobhidni i dostatni umovi kvaziopuklosti i kvaziuvignutosti mozhna takozh dati cherez tak zvanu obramlenu matricyu Gesse Dlya funkciyi f x 1 x m displaystyle f x 1 ldots x m nbsp viznachimo dlya 1 n m displaystyle 1 leqslant n leqslant m nbsp viznachniki D n 0 f x 1 f x 2 f x n f x 1 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n f x 2 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n f x n 2 f x n x 1 2 f x n x 2 2 f x n 2 displaystyle D n begin vmatrix 0 amp frac partial f partial x 1 amp frac partial f partial x 2 amp cdots amp frac partial f partial x n frac partial f partial x 1 amp frac partial 2 f partial x 1 2 amp frac partial 2 f partial x 1 partial x 2 amp cdots amp frac partial 2 f partial x 1 partial x n frac partial f partial x 2 amp frac partial 2 f partial x 2 partial x 1 amp frac partial 2 f partial x 2 2 amp cdots amp frac partial 2 f partial x 2 partial x n vdots amp vdots amp vdots amp ddots amp vdots frac partial f partial x n amp frac partial 2 f partial x n partial x 1 amp frac partial 2 f partial x n partial x 2 amp cdots amp frac partial 2 f partial x n 2 end vmatrix nbsp Todi spravedlivi tverdzhennya Yaksho funkciya f kvaziopukla na mnozhini X todi Dn x 0 dlya vsih n i vsih x z X Yaksho funkciya f kvaziuvignuta na mnozhini X todi D1 x 0 D2 x 0 1 mDm x 0 dlya vsih x z X Yaksho Dn x 0 dlya vsih n i vsih x z X to funkciya f kvaziopukla na mnozhini X Yaksho D1 x 0 D2 x 0 1 mDm x 0 dlya vsih x z X funkciya f kvaziuvignuta na mnozhini X Operaciyi sho zberigayut kvaziopuklist RedaguvatiMaksimum zvazhenih kvaziopuklih funkcij z nevid yemnimi vagami tobtof max w 1 f 1 w n f n displaystyle f max left lbrace w 1 f 1 ldots w n f n right rbrace nbsp de w i 0 displaystyle w i geqslant 0 nbsp kompoziciya z nespadnoyu funkciyeyu yaksho g R n R displaystyle g mathbb R n rightarrow mathbb R nbsp kvaziopukla h R R displaystyle h mathbb R rightarrow mathbb R nbsp nespadna todi f h g displaystyle f h circ g nbsp ye kvaziopukloyu minimizaciya yaksho f x y ye kvaziopukloyu C opukla mnozhina todi h x inf y C f x y displaystyle h x inf y in C f x y nbsp ye kvaziopukloyu Posilannya RedaguvatiM P Moklyachuk Osnovi opuklogo analizu Arhivovano 5 Lipnya 2016 u Wayback Machine K TviMS 2004 240s M P Moklyachuk NEGLADKIJ ANALIZ TA OPTIMIZACIYa Arhivovano 4 Bereznya 2016 u Wayback Machine Stephen Boyd Lieven Vandenberghe Convex Optimization Arhivovano 13 Lipnya 2017 u Wayback Machine Literatura RedaguvatiAlpha C Chiang Fundamental Methods of Mathematical Economics Third Edition McGraw Hill Book Company 1984 Otrimano z https uk wikipedia org w index php title Kvaziopukla funkciya amp oldid 35019354
