24

Iaoe?iiaeueia aeaaeai?y iaoe Oe?a?ie

?inoeooo e?aa?iaoeee ?iai? A. I. Aeooeiaa

Ia i?aaao
?oeiieno

II?E?I Aieiaeeie? ?aaiiae/

OAeE
519.8

NOIOANOE*I? IAOIAeE ?ICA’ssCAIIss CAAeA*

IAIIOEEIAI NOIOANOE*IIAI I?IA?AIOAAIIss

OA ?O CANOINOAAIIss

01.05.01 — oai?aoe/i? iniiae ?ioi?iaoeee oa e?aa?iaoeee

Aaoi?aoa?ao aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaith

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Ee?a 1998

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a ?inoeooo? e?aa?iaoeee ?iai? A.I.Aeooeiaa IAI Oe?a?ie.

Iaoeiaee eiinoeueoaio: aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?, aeaaeai?e IAI Oe?a?ie,

??IIEUe?A TH??e Ieoaeeiae/,

?inoeooo e?aa?iaoeee ?iai? A.I.Aeooeiaa

IAI Oe?a?ie, caa?aeoth/ee a?aeae?eii.

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?, aeaaeai?e IAI Oe?a?ie,

EI?IETHE Aieiaeeie? Naiaiiae/,

?inoeooo iaoaiaoeee IAI Oe?a?ie,

aieiaiee iaoeiaee ni?a?ia?oiee;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?, aeaaeai?e IAI Oe?a?ie,

OI? Iaoi Conaeaae/, ?inoeooo

e?aa?iaoeee ?iai? A.I.Aeooeiaa IAI

Oe?a?ie, caa?aeoth/ee a?aeae?eii;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

AeAIEE?I TH??e Ieoaeeiae/,

?inoeooo i?eeeaaeiiai nenoaiiiai

aiae?co IAI Oe?a?ie oa I?iina?oe

Oe?a?ie, i?ia?aeiee iaoeiaee
ni?a?ia?oiee.

I?ia?aeia onoaiiaa: Ee?anueeee iaoe?iiaeueiee oi?aa?neoao

?i. O.A.Oaa/aiea, oaeoeueoao e?aa?iaoeee, eaoaae?a nenoaiiiai

aiae?co ? i?eeiyooy ??oaiue

Caoeno a?aeaoaeaoueny «26» ethoiai 1999 ?. ia 11 aiae. ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.194.02 i?e ?inoeooo? e?aa?iaoeee
?iai? A.I.Aeooeiaa IAI Oe?a?ie

ca aae?anith: 252022 Ee?a 22, i?iniaeo Aeooeiaa, 40.

C aeena?oaoe??th iiaeia iciaeiieoenue o iaoeiai-oaoi?/iiio a?o?a?
?inoeoooo.

Aaoi?aoa?ao ?ic?neaiee «15» ethoiai 1999 ?.

O/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee NEIssANUeEEE A.O.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeena?oaoeiy i?enay/aia noioanoe/iei iaoiaeai ?ica’ycaiiy caaea/
iaiioeeiai noioanoe/iiai i?ia?aioaaiiy, iai?eeeaae, caaea/ eieaeueii? oa
aeiaaeueii? iioeiicaoei? iaaeaaeeeo ooieoe?e i/?eoaaii? ei?eniino? oa
eiia??iino?, noioanoe/ii? iioeiicaoei? ?ic?eaieo ooieoeie,
oeiei/eneiaiai noioanoe/iiai i?ia?aioaaiiy. Oi?iaeueii a aeena?oaoe??
?icaeaathoueny iaoiaee ?ica’ycaiiy caaea/ noioanoe/iiai i?ia?aioaaiiy
aeaeyaeo

minxX [F(x)=Eu(f(x,),)], minxX [P(x)=Pf(x,)B()],

ia iniia? niinoa?aaeaiiy cia/aiue (aai a?aae?aio?a) ooieoe?? f(·,?), aea
ooieoe?? f(·,u), u(·,u), P(x) ? F(x) iiaeooue aooe iaiioeeeie,
iaaeaaeeeie ? iaa?oue ?ic?eaieie, a ci?iia xXRn iiaea aooe iaia?a?aiith
aai aeene?aoiith; u — aeiaaeeiaa aaee/eia; E ? P — ciaee iaoaiaoe/iiai
i/?eoaaiiy oa eiia??iino? ii , B()Rm, f(,):RnRm, u(,):RmR1.

Aeooaeueiinoue oaie. Oai?iy iioeeiai noioanoe/iiai i?ia?aioaaiiy
?ic?iaeaia a ?iaioao Aeae.Aeaioeeaa, TH.I.??iieue?aa, I.Eaeea,
A.I?aeiie, ?.Aaona, Ae.A.THaeiia oa ii. Aae?ana http://mally.eco.rug.nl
o ia?aae? Internet i?noeoue a?ae?ia?ao?th (iiiaae 3000 iineeaiue) ?ia?o
ii noioanoe/iiio i?ia?aioaaiith (neeaaeaia M.H. Van der Vlerk).

Iaeiae inio? aaeeea eieueeinoue i?eeeaaeieo caaea/ iaiioeeiai
noioanoe/iiai i?ia?aioaaiiy. Oea, iai?e eeaae, caaea/i iioeiicaoei?
aeeiaii/ieo nenoai c aeene?aoieie aeiaaeeiaeie iiaeiyie (nenoai
ianeoaiaoaaiiy c /a?aaie, ia?aae ca’yceo, aio/eeo aaoiiaoeciaaieo
ae?iaieoeoa, oaoii/ieo nenoai c aiaeiiaaie oa ii.). Iieacieee
ooieoeiiioaaiiy oaeeo nenoai o caaaeueiiio aeiaaeeo ? iaiioeeeie
iaaeaaeeeie ooieoeiyie aiae ia?aiao?ia nenoai i /anoi iathoue aeaeyae
iaoaiaoe/iiai niiaeiaaiiy (iai?eeeaae, na?aaeiie /an i/ieoaaiiy
ianeoaiaoaaiiy, na?aaeiie /an iaaeoiaeaeaiiy iiaiaeiieaiiy, na?aaeiie
/an aacaiaeiiaii? ?iaioe i o.i.). Oaeiae ? oeieaaith iioeiicaoeiy oeeo
iieacieeia ca ia?aiao?aie i?e iaiaaeaiiyo ia iaeanoue aeiionoeii? ciiie
ia?aiao?ia. Eieaeueia iioeiicaoeiy oeeo iieacieeia o caaaeueiiio
aeiaaeeo iiaeiia i?iaiaeeoeny iaoiaeaie iaaeaaeeiai noioanoe/iiai
i?ia?aioaaiiy (iai?eeeaae, eiio?oeaaiai). Niaoeiaeueiee aeiaaeie
aeaaeeeo iieacieeia ? aea/aiei o ?iaioao X.R.Cao, P.W.Glynn, Y.C.Ho,
I.E.E?eaoeiia, G.Ch.Pflug, R.Rubinstein, R.Suri.

Aaaaoi nenoai c aeene?aoieie iiaeiyie iathoue ?ic?eaii iieacieee
ooieoeiiioaaiiy (aeiaaeeie /a?a a nenoaiao ianiaiai ianeoaiaoaaiiy, /an
ianeoaiaoaaiiy a nenoaiao c aiaeiiaaie oa ?aaaia?aoei?th, ?iaii caiania
ca aeiaaeeiaiai iiieoo a aeiiiii/ieo nenoaiao). Eieaeueia iioeiicaoeiy
oeeo nenoai iio?aao? cianii iiaiai aiaeioe/iiai aia?aoo i aiaeiiaiaeieo
/enaeueieo iaoiaeia (J.P.Aubin, A.Ae.Aaooooii, F.H.Clarke,
A.O.Aeai’yiia, A.O.Ii?aeooiae/, A.I.Ioaie/iee, R.T.Rockafellar,
I.I.?oaiiia, I.C.Oi?, R.J-B.Wets oa ii.)

I?e iioeiicaoei? nenoai c aeene?aoieie iiaeiyie iinoa? ieoaiiy i?i
aeiaaeueio iioeiicaoeith nenoaie, ineieueee ooieoei?, ui
iioeiicothoueny, acaaaei iaiioeei. Iiaeia aoei a canoinoaaoe
caaaeueiiaiaeiii iaoiaee aeiaaeueii? iioeiicaoei?, aea ni?aaa
oneeaaeith?oueny oei, ui ooieoei?, yei iiaeeyaathoue iioeiicaoei?,
yaeythoue niaith iaoaiaoe/ii niiaeiaaiiy, oiaoi aaaaoiaeii?i? iioaa?aee,
oa ?o oi/ia ia/eneaiiy aai aeoaea o?oaeiiinoea, aai i?aeoe/ii
iaiiaeeeaa. Oiio iaiaoiaeii iniaeeai iiaeoiaee aei aeiaaeueii?
iioeiicaoei? oaeeo noioanoe/ieo nenoai. Na?aae iiaeoiaeia, yei aiaeiii
ia aeaiee /an, neiae aiaeiioeoe iaoiaee ea?iaaiiai aeiaaeeiaiai iiooeo
(E.Aaarts, A.I.AEeaeyanueeee, S.Kirkpatrick, H.Kushner, E.A.Iioeeon,
E.A.?ano?eaii, Ae.A.THaeii oa ii.)

Aieueoinoue eeane/ieo caaea/ aeineiaeaeaiiy iia?aoeie, yei /anoi
oi?ioeththoueny ye caaea/i aeene?aoiiai (aai iioaiiai) i?ia?aioaaiiy
(caaea/a i?i ?thecae, i?i i?ecia/aiiy, i?i ?iciiuaiiy, i?i ?iciiaeieaiiy
?ano?nia oa ii.), a caaaeueiiio aeiaaeeo iiaeooue iinoeoe aeiaaeeiai
ia?aiao?e. A oeueiio aeiaaeeo aiie iiaeiii aooe ia?aoi?ioeueiaaii ye
caaea/i noioanoe/iiai aeene?aoiiai i?ia?aioaaiiy (iaeii-, aeai- aai
aaaaoi-aoaiiiai, c eiiai?i?nieie iaiaaeaiiyie, c ooieoeiyie ?eceeo).
Oi?iaeueii caaea/a aeene?aoiiai noioanoe/iiai i?ia?aioaaiiy — oea
caaea/a aeai?o, iai?eeeaae, iiiiiaeueiiai iaoaiaoe/iiai niiaeiaaiiy c
neii/aiii? (ano?iiiii/ii?) iiiaeeie aa?iaioia. I?e iaaaeeeie eieueeinoi
aa?iaioia — oea caaea/a iaoaiaoe/ii? noaoenoeee. Aeayei iaoiaee
?ica’ycaiiy caaea/ noioanoe/iiai oeiei/eneaiiiai i?ia?aioaaiiy
?ic?iaeaii a ?iaioao ?.E.Aaa?aaoa, G.Laporte, F.H.Louveaux,
A.H.G.Rinnoy-Kan, R.L.Schultz, L.Stougie, M.H.Van der Vlerk,
Ae.A.THaeiia oa ii. Aae?ana http://mally.eco.rug.nl o ia?aae? Internet
i?noeoue iiiaae 130 iineeaiue ia ?iaioe ii noioanoe/iiio oe?ei/eneiaiio
noioanoe/iiio i?ia?aioaaiith.

Aoaeue-yea caaea/a noioanoe/ii? iioeiicaoei? ? naia ii niai caaea/ath
aaaaoie?eoa?iaeueii? iioeiicaoei?: ii nooi, eiaeiie ?aaeicaoei?
aeiaaeeiaeo ia?aiao?ia (noeaia?ith) aiaeiiaiaea? naiy oeieueiaa
ooieoeiy i naie ?ica’ycie. O noioanoe/iiio i?ia?aioaaiii, ye i?aaeei,
aa?aaothoue oei aeiaaeeiai oeieueiai ooieoei? ca aeiiiiiaith iia?aoeie
iaoaiaoe/iiai niiaeiaaiiy. Iaeiae oea ia ?aeeiee niinia iiaoaeiae
ocaaaeueiaii? oeieueiai? ooieoei?. Iioa aaaeeeaa ooieoeiy oaeiai ?iaeo —
oea ooieoeiy eiiai?iinoi, yea ae?aaea? eiiai?iinoue oiai, ui aeayea
aeiaaeeiaa aaee/eia, ui caeaaeeoue aiae iaia?a?aieo oa aeene?aoieo
ia?aiao?ia, ia ia?aaieueoo? caaeaii? iaaei aai iaeaaeeoue caaeaiie
iaeanoi. Ca aeiiiiiaith ooieoeie eiiai?iinoi iienothoue iaaeieiinoue
oaoii/ieo nenoai, ?ecee a aeiiiiioei oa aiciani. Eieaeueia oa aeiaaeueia
iioeiicaoeiy ooieoeie eiiai?iinoi — oea nai??iaeia niaoeiaeueia caaea/a
noioanoe/iiai i?ia?aioaaiiy, yea iio?aao? iniaeeaeo iaoiaeia
?ica’ycaiiy. Caaea/a eieaeueii? iioeiicaoei? aeaaeeeo ooieoeie
eiiai?iinoi ?icaeyiooa a ?iaioao A.?.Eiacoia, ?.I.Eiaaeaiea, ?.Eaiia,
E.Marti, I.I.Iaeiia/iiai, A.O.Iieyea, A.Prekopa, A.?aeea, A.Oaii,
N.I.O?ynue?aa, O.Szantai oa ii.

Ooieoei? eiiai?iinoi oa iaoaiaoe/iiai niiaeiaaiiy ? /anoeiaeie
aeiaaeeaie ooieoeie niiaeiaaii? ei?eniinoi, ca aeiiiiiaith yeeo
oi?ioeththoueny caaea/i i?eeiyooy ?ioaiue i?e iaaecia/aiinoi. Ooieoei?
niiaeiaaii? ei?eniinoi iathoue aooe iaiioeeeie, iaaeaaeeeie oa iaa?oue
?ic?eaieie i, ioaea, oaeiae iio?aaothoue niaoeiaeueieo iaoiaeia
iioeiicaoei?.

Oea eiei caaea/ yaey? niaith i?aaeiao aeineiaeaeaiiy aeena?oaoei? c
iaoith ?ic?iaee aiaeiiaiaeieo iaoiaeia aeey ?o ?ica’ycaiiy. Iaoiaeeea
aeine?aeaeaiiy aaco?oueny ia canoinoaaii? aei caaea/ iaiioeeiai
noioanoe/iiai i?ia?aioaaiiy iaoiae?a iaaeaaeeiai aiae?co, aeayeeo
oai?aoeei-eiia??i?nieo ?acoeueoao?a (caeii?a aaeeeeo /enae, oai???
ia?o?iaae?a, oai??? neaaei? ca?aeiino?), oai??? aaaaoicia/ieo
a?aeia?aaeaiue, iaoiaeo iaiioeeeo ooieoe?e Eyioiiaa.

Iaoeiaa iiaecia ?acoeueoaoia ?iaioe iieyaa? a ianooiiiio.

1. ?icaeyiooi iia? i?aeoe/i? eeane caaea/ iaaeaaeei? oa ?ic?eaii?
(iaiioeei?) iioei?caoe??, iia’ycai? c iioei?caoe??th oae caaieo
noioanoe/ieo nenoai c aeene?aoieie iiae?yie. Iieacaii, ui aaeaeaaoiith
iiaeaeeth ia?aiao?e/ieo iaia?a?aieo iieacieeia ooieoeiiioaaiiy
noioanoe/ieo nenoai c aeene?aoieie iiaeiyie oa ooieoeie niiaeiaaii?
ei?eniinoi ? oae caaii ocaaaeueiaii aeeoa?aioeieiaii ooieoei?.

2. Aaaaeaii oa aea/aii aeayei iia? eeane ?ic?eaieo ooieoeie
(eoneiai-iaia?a?aii, iaia?a?aii ca iai?yieaie, neeueii iaiiaiaia?a?aii
cieco).

3. Aeey ?ic?eaieo ooieoe?e oaaaeaii oa aea/aii iiaa iiiyooy
caeaae-aeaiiai noaaeeoa?aioeiaea, aea/aii eiai ca’ycie c
noaaeeoa?aioeiaeaie O.Eea?ea, ?.O.?ieaoaeea?a, Aeae.Aa?ae oa
A.O.Ii?aeooiae/a. Aeey iieno neiaoey?ieo (iane?i/aiieo) cia/aiue
a?aae??io?a ?ic?eaieo ooieoe?e oaaaeaii oa aeineiaeaeaii iiaa iiiyooy
?icoe?aiiai caeaaeaeaiiai noaaeeoa?aioeiaea o «einii/iiio» aaeoi?iiio
i?inoi?i ?.O.?ieaoaeea?a oa ?.Aaona.

4. Ciaeaeaii iaiaoiaeii oiiae aeno?aioio a oa?iiiao caeaaeaeaieo
noaaeeoa?aioeiaeia aeey iaia?a?aieo oa ?ic?eaieo ooieoeie ca iaiioeeeo
iaiaaeaiue.

5. Aeey ia/eneaiiy caeaaeaeaieo noaaeeoa?aioe?ae?a ?ic?eaieo ooieoe?e
cai?iiiiiaaii ne?i/aiii-??cieoeaa? noioanoe/i? ioe?iee.

6. Cai?iiiiiaaii oa iaa?oioiaaii iiaee (ai?ieneiaoe?eiee) i?aeo?ae aei
iioei?caoe?? iaaeaaeeeo oa ?ic?eaieo ooieoe?e iaoaiaoe/iiai niiae?aaiiy,
ooieoe?e eiiai?iinoi oa niiae?aaii? ei?eniino? ca iaiaaeaiue.

7. Iaoiae noioanoe/ieo eaacia?aaei?ioia TH.I.??iieue?aa ocaaaeueiaiee i
iioe?th?oueny ia iaiioee? noioanoe/ii caaea/i iioeiicaoei? ocaaaeueiaii
aeeoa?aioeieiaieo, eieaeueii eiio?oeaaeo oa ?ic?eaieo ooieoeie
iaoaiaoe/iiai niiae?aaiiy ca iaiioeeeo iaiaaeaiue.

8. Aeey ?ica’ycaiiy «ouaeeiieo» caaea/ noioanoe/iiai iaiioeeiai
i?ia?aioaaiiy cai?iiiiiaaii oa iaa?oioiaaii iaoiaee ona?aaeiaieo
noioanoe/ieo a?aae??io?a, iai?eeeaae, noioanoe/i? iaoiaee «aaaeei?
eoeueee» oa «ouaeeiiiai e?ieo».

9. ?icaeyiooi iia? (noioanoe/i? iae?i?ei?) caaea/? aeene?aoii?
iioei?caoe?? (noioanoe/io aeaiaoaiio caaea/o i?i ?thecae, noioanoe/io
caaea/o oiiainoaaiiy i?iaeoia, noioanoe/io caaea/o ?iciiuaiiy
ae?iaieoeoaa, eiiieio noioanoe/io aeaiaoaiio caaea/o, noioanoe/io
caaea/o i?i ?aeiino?oeoe?th ia?aae?, noioanoe/io caaea/o i?i
iaeneiaeueiee iio?e o ia?aae? oa ?i.).

10. Cai?iiiiiaaii iiaee i?eeii ia?anoaiiai/ii? ?aeaenaoe??
(ia?a-noaiiaee iia?aoi??a i?i?i?caoe?? oa iaoaiaoe/iiai niiae?aaiiy aai
eii-a??iino? (aai i?aenoiiaoaaiiy)) aeey ioe?iee cieco iioeiaeueieo
cia/aiue caaea/ noioanoe/ii? aeene?aoii? oa noioanoe/ii? aeiaaeueii?
iioei?caoe??.

11. Ca aeiiiiiaith ia?anoaiiai/ii? ?aeaenaoei? oa eaa?aiaeaai?
?aeaenaoei? iaea?aeaii iiai noioanoe/ii ioeiiee iioeiaeueieo cia/aiue
aeene?aoieo oa iaia?a?aieo caaea/ noioanoe/iiai i?ia?aioaaiiy,
aeeth/ath/e noioanoe/io aeaiaoaiio caaea/o i?i ?thecae, noioanoe/io
caaea/o oiiainoaaiiy i?iaeoia, noioanoe/io caaea/o ?iciiuaiiy
ae?iaieoeoaa, eiiieio noioanoe/io aeaiaoaiio caaea/o, caaea/o
iioeiicaoei? ooieoei? eiiai?iinoae.

12. Ia aaci noioanoe/ieo ioeiiie iioeiaeueieo cia/aiue ?ic?iaeaii iiaee
noioanoe/iee aa??aio iaoiaeo aieie oa iaae aeey ?ica’ycaiiy caaea/
noioanoe/iiai aeene?aoiiai i?ia?aioaaiiy oa noioanoe/ii? aeiaaeueii?
iioeiicaoei? ca aeaoa?iiiiaaieo oa noioanoe/ieo iaiaaeaiue.
Aeineiaeaeaii oiiae caiaeiinoi iaoiaeo iaeaea iaiaaia.

13. ?ic?iaeaii aa?iaioe noioanoe/iiai iaoiaeo aieie oa iaae aeey
eiie?aoieo caaea/ noioanoe/iiai i?ia?aioaaiiy, oaeeo, ye noioanoe/ia
aeaiaoaiia caaea/a oiiainoaaiiy i?iaeoia, noioanoe/ia caaea/a ?iciiuaiiy
ae?iaieoeoaa, caaea/a iioeiicaoei? eiiai?iinoae, caaea/a iioeiicaoei? ca
eiiai?i?nieo iaiaaeaiue. I?iaaaeaii aenia?eiaioaeueii ?ic?aooiee, ui
iiaeoaaaeaeothoue aei?caeaoiinoue oa aoaeoeaiioue ?ic?iaeaiiiai
noioanoe/iiai iaoiaeo aieie oa iaae.

14. Aeineiaeaeaii oiiae oa oaeaeeinoue caiaeiinoi iaoiaeo aiii?e/ieo
na?aaeiio o aeiaaeeo neeaaeieo ooieoeie ?eceeo, a oaeiae iiaa
aeiiiiiaeia iiiyooy ii?iaeiciaaii? caiaeiinoi aeiaaeeiaeo aaee/ei.

I?aeoe/ia cia/aiiy ?iaioe iieyaa? a oiio, ui ?ic?iaeaii
ia/e-nethaaeueii iaoiaee ?ica’ycaiiy neeaaeieo caaea/ iioeiicaoei?
oaoii/ieo nenoai c aeene?aoieie aeiaaeeiaeie iiaeiyie oa iaaeaaeeeie
iieacieeaie yeinoi ooieoeiiioaaiiy, ia/enethaaeueii iaoiaee iioeiicaoei?
?eceeo, ui ? ae?aaeaiei ca aeiiiiiaith ooieoeie ?eceeo, ia/enethaaeueii
iaoiaee ?ica’ycaiiy noioanoe/ieo caaea/ aeene?aoiiai i?ia?aioaaiiy. Oei
iaoiaee ia iio?aaothoue oi/iiai ia/eneaiiy ooieoei?, ui iioeiico?oueny,
o eiaeiie oi/oei, a aeei?enoiaothoue aeiaaeeiai ?aaeicaoei? ooieoei? aai
?o a?aaei?ioia, ui io?eiothoueny c iiioaoeieii? iiaeaei nenoaie, yea
iioeiico?oueny.

Ai?iaaoeiy ?iaioe. ?acoeueoaoe ?iaioe aeiiiaiaeaeeny ia ?V-VII
Iiaeia?iaeieo eiioa?aioeiyo ii noioanoe/iiio i?ia?aioaaiith (Aii-A?ai?,
NOA, 1989; Oaeiia, Ioaeiy, 1992; Iaoa?aey, Ic?a?eue, 1995), ia
I?aeia?iaeiiio iaoaiaoe/iiio eiia?an? (Aa?e?i, I?ia//eia, 1998), ia XV
Iiaeia?iaeiiio neiiicioii ii iaoaiaoe/iiio i?ia?aioaaiith (Aii-A?ai?,
NOA, 1994), ia iiaeia?iaeieo neiiicioiao «Aeiaaeueia iioeiicaoeiy»
(Oii?ii, Oai?ueia, 1990), «Ai?ieneiaoeiy caaea/ noioanoe/iiai
i?ia?aioaaiiy» (Eaenaiao?a, Aano?iy, 1993), «Noioanoe/ia i?ia?aioaaiiy»
(Aa?eii, Iiia//eia, 1994), «Iaaeaaeea oa ?ic?eaia iioeiicaoeiy i
canoinoaaiiy» (Eaenaiao?a, Aano?iy, 1995), «Noioanoe/ia iioeiicaoeiy:
ia/enethaaeueii iaoiaee oa oaoii/ia canoinoaaiiy» (Iuethaiaa?a/Ithioai,
Iiia//eia, 1996), ia iiaeia?iaei?e eiioa?aioei? INFORMS (Nai Ae??ai,
NOA, 1997). Iaoa??aee aeena?oaoe?? iaaiai?thaaeeny ia iaoeiaeo nai?ia?ao
a ?inoeooo? e?aa?iaoeee ?i. A.I.Aeooeiaa IAI Oe?a?ie, ?inoeooo?
iaoaiaoeee IAI Oe?a?ie, Iaoe?iiaeueiiio oaoi?/iiio oi?aa?neoao? Oe?a?ie,
?inoeooo? nenoaiiiai aiae?co IAI Oe?a?ie oa I?iina?oe Oe?a?ie, a oaeiae
ia oaeoeueoao? e?aa?iaoeee Ee?anueeiai iaoe?iiaeueiiai oi?aa?neoaoo ?i.
O.A.Oaa/aiea.

Ioaeieaoei?. ?acoeueoaoe ?iaioe iioaeieiaaii o 25 ae?oeiaaieo ?iaioao
(ic ieo 13 c niiaaaoi?aie), o oiio /enei a iaeiie iiiia?aoi? (c
niiaaaoi?aie), 13 aeo?iaeueieo ioaeieaoeiyo, 7 noaooyo a cia?aiiyo
i?aoeue, 4 i?ai?eioao.

No?oeoo?a oa ianya ?iaioe. Aeena?oaoeiy neeaaea?oueny c anooio, 6
?icaeieia, aeniiaeia oa nieneo eioa?aoo?e. Ianya ?iaioe — 250 noi?iiie,
3 ?enoiee, 4 oaaeeoei. Nienie eioa?aoo?e iinoeoue 337 iaeiaioaaiue.

CIINO ?IAIOE

O anooii iiyniaii aeooaeueiinoue oaie aeena?oaoei?, ei?ioei aeeeaaeaii
ciino aeena?oaoei?, aeaii iaeyae eioa?aoo?e ca oaiith aeena?oaoei? oa
aeacaii ca’ycie ?iaioe c iioeie aeineiaeaeaiiyie.

1. Caaea/? iaiioeei? noioanoe/ii? iioei?caoe??.

O ?icaeiei 1 iaaiai?th?oueny caaaeueia iaoiaeieiaiy i?eeiyooy ?ioaiue
ca oiia noioanoe/ii? iaaecia/aiinoi ca aeiiiiiaith oae caaieo ooieoeie
?eceeo, ooieoeie niiaeiaaii? ei?eniinoi oa ia/ioeeo oeieueiaeo ooieoeie,
a oaeiae iaaaaeai? i?eeeaaee iaiioeeeo caaea/ iioei?caoe?? nenoai c
aeene?aoieie iiae?yie, caaea/ noioanoe/ii? aeene?aoii? oa noioanoe/ii?
aeiaaeueii? iioei?caoe??.

Iiae ooieoei?th ?eceeo F(x), ui caeaaeeoue aiae ia?aiao?ia x, ia?oueny
ia oaaci aoaeue-yeee ooieoeiiiae aiae ?iciiaeieo aeiaaeeiaeo
«caeoeia-aea?aoo» f(x,), ui aiaeiiaiaea? oiio aai iioiio aeai?o
ia?aiao?ia x oa ?aaeicaoei? aeiaaeeiaeo ia?aiao?ia . Ca aeiiiiiaith
ooieoeie ?eceeo caeienith?oueny aeai? ?aoeiiiaeueieo ?ioaiue ca oiia
noioanoe/ii? iaaecia/aiinoi.

O i?ae?icae?e? 1.1 oa o [18] iaaiai?ththoueny iaoaiaoe/ii aeanoeainoi
ooieoeie ?eceeo oa ?icaeyaeathoueny iaoiaee i?eeiyooy ?ioaiue ia aaci
ooieoeie ?eceeo. Oeiiaeie i?aaenoaaieeaie ooieoeie ?eceeo ? ooieoei?
iaoaiaoe/iiai niiaeiaaiiy F(x)=Ef(x,), aeenia?ni? F(x)=E(f(x,)Ef(x,))2,
eiiai?iinoi F(x)=Pf(x,)c oa ii., ui caeaaeaoue aiae ia?aiao?ia xRn,
i?e/iio aeiaaeeia? ooieoe?? iiaeooue aooe iaiioeeeie oa iaaeaaeeeie. O
iiae?icaeiei 1.5 oa a [10, 11, 25] iaaaaeaii i?eeeaaee caaea/
iioeiicaoei? aeayeeo nenoai c aeene?aoieie aeiaaeeiaeie iiaeiyie — DES,
o yeeo aeiaaeeiai ooieoei? f(x,) aoaeothoueny c aeaaeeeo ooieoeie ca
aeiiiiiaith iia?aoeie iaeneioio oa iiiiioio. Iai?eeeaae,

caaea/a iaenei?caoe?? na?aaeiueiai /ano aeeooy ia?aae? ia? aeaeyae

maxx X [F(x)= E maxpP minep fep(x,)],
(1)

aea p — aeayeee oeyo c iiiaeeie P oeyo?a, ui aaaeooue a?ae aoiaeo aei
aeoiaeo ia?aae?; fep(x,) — aeiaaeeiaee /an aeeooy aeaiaioa e oeyoo p. O
aeaiiio aeiaaeeo ooieoe?y ei?eniino? u(t)=maxpP minep tep ? iaiioeeith
oa iaaeaaeeith. I?iaeaia iieyaa? a ia/eneaii? noioanoe/ieo ocaaaeueiaieo
a?aae??io?a iaiioeei? iaaeaaeei? ooie?? F(x).

I?ae ooieoei?th niiaeiaaii? ei?eniinoi iathoueny ia oaaci ooieoei?
aeaeyaeo F(x)=Eu(f(x,)), aea ooieoeiy u(·) ae?aaea? ei?eniinoue (aeey
iniae, ui i?eeia? ?ioaiiy) oeo aai iioeo cia/aiue f(·,·). ssnii, ui
ei?eniinoue u(f(x,)) iiaea aooe iaiioeeith ii x, iaaioue yeui ooieoeiy
f(x,) iioeea ii x. sseui u(·)= c(·), aea c(t)=0 aeey t<0 oa 1 aeey t ? 0, aiaeiiaiaeia ooieoeiy niiaeiaaii? ei?eniinoi niiaiaaea? c ooieoei?th eiiai?iinoi. Eiee u(t)=max(0,t) aiaeiiaiaeia ooieoeiy ei?eniinoi ? ooieoeiy ?eceeo aeaeyaeo F(x)=Ef(x)c(f(x,)). I/aaeaeii, ooieoeiy eiiai?iinoi iiaea aooe iaaeaaeeith i iaaioue ?ic?eaiith. Aeanoeainoi ooieoei? eiiai?iinoi oa iioeo ooieoeie niiaeiaaii? ei?eniinoi (iaia?a?aiinoue, ocaaaeueiaia aeeoa?aioeieiaiinoue, eaacioaiaiooinoue) aeaoaeueii ?icaeyaeathoueny o iiae?icaeiei 4.4. Ooieoeiy u(·) iiaea aooe ooieoei?th iaeaaeiinoi (0u()1) ?icieoi? oeiei aaaeaii? iaeanoi cia/aiue e?eoa?ith f , oiaei u(f(x,)) ae?aaea? nooiiiue aeinyaiaiiy ?icieoi? oeiei i?e ?ica'yceo x, a F(x)= Eu(f(x,)) ae?aaea? niiaeiaaio nooiiiue aeinyaiaiiy oeiei. O iiae?icaeiei 1.4 oa a [4] iaaiai?ththoueny iaoaiaoe/ii aeanoeainoi caaea/i ia/ioei? iioeiicaoei?. Ia i?aeoeoei caaea/a i?eeiyooy ?ioaiue aai iioeiicaoei? ooieoeiiioaaiiy aeayei? noioanoe/ii? nenoaie caiaeeoueny aei iioeiicaoei? oeo aai iioeo ooieoeie ?eceeo aai ?o eiiaiiaoeie ca iaiaaeaiue. Ooieoei? ?eceeo iiaeooue aooe iaiioeeeie, iaaeaaeeeie i iaaioue ?ic?eaieie oa iiaeooue caeaaeaoe aiae iaia?a?aieo oa aeene?aoieo ia?aiao?ia. O i?ae?icaeieao 1.5 - 1.8 iaaaaeaii i?eeeaaee caaea/ noioanoe/ii? iioeiicaoei? c oaeeie ooieoeiyie ?eceeo (iioeiicaoeiy coieiee iaaacia/iiai ae?iaieoeoaa, noioanoe/ieo ia?aae c aiaeiiaaie, ooieoeiiioaaiiy ia?aae ca'yceo, i?inoi? o?ainii?oii? eiii?, aiini? caa?oaeithaa/ia, ?iciiuaiiy ae?iaieoeoaa, ?aeiino?oeoei? noioanoe/ii? ia?aaei, ?eceeo o no?aoiaiio aiciani, oiiainoaaiiy i?iaeoia, ii?ooaey oeiiieo iaia?ia oa ii.). Aeooaeueiith ? caaea/a ?ic?iaee iaoiaeia eieaeueii? iioeiicaoei? oaeeo neeaaeieo ooieoeie ?eceeo. Oeueiio i?enay/aii ?icaeiee 2 - 4. Ineieueee ooieoei? ?eceeo iiaeooue aooe iaiioeeeie, ie ia iiaeaii oieeiooe i?iaeaie aeiaaeueii? iioeiicaoei? oaeeo ooieoeie, ui aea/a?oueny a ?icaeiei 5. sseui ia?aiao? x i?eeia? aeene?aoii cia/aiiy, oi caaea/a aeai?o, neaaeiii, iiiiiaeueiiai iaoaiaoe/iiai niiaeiaaiiy F(x)= Ef(x,) ia iniiai ?aaeicaoeie f(x,) ? eeane/iith caaea/ath iaoaiaoe/ii? noaoenoeee. Aiaeiiiiinoue caaea/i noioanoe/ii? aeene?aoii? iioeiicaoei? iieyaa? a oiio, ui /enei iiaeeeaeo ?ica'yceia x iiaea aooe ano?iiiii/ii aaeeeei. Iai?eeeaae, noioanoe/ia aeaiaoaiia caaea/a ea?oaaiiy i?iaeoaie ia? aeaeyae iaenei?caoe?? i/?eoaaiiai noeoiiiai aeioiaeo (xj=1, yeui i?iaeo j o?iaino?oueny ia ia?oiio aoai?, xj=0 - o i?ioeaiiio aeiaaeeo): maxx [F(x)=1ncjxj+E f2(x,)], i?e ?ano?nieo iaiaaeaiiyo ia?oiai aoaio 1naijxj b1i, i=1,...,m, xj0,1, j=1,...,n, aea aeiaaeeiaee aeioiae f2(x,) a?ae i?iaeiaaeoaaieo ia ae?oaiio aoai? i?iaeo?a ? ?ica'ycie i?e o?eniaai?e ?aae?caoe?? aeiaaeeiaeo (aenia?oieo) ia?aiao??a dj(), eij(), b2i() ianooiii? aaaaoiaei??ii? caaea/? i?i ?thecae (yj=1, yeui i?iaeo j i?iaeiaaeo?oueny ia ae?oaiio aoai?, yj=0 - o i?ioeaiiio aeiaaeeo): f2(x,)=maxy 1ndj()yj, 1neij()yj b1i+b2i()1naijxj, i=1,...,m, yj0,xj, j=1,...,n. Aeey ?ica'ycaiiy iiae?aieo caaea/ iaiaoiaeiee iaoaiici nei?i/aiiy ia?aai?o iiaeeeaeo cia/aiue x. Oaeee iaoaiici iaaea?oueny noioanoe/iei iaoiaeii aieie oa iaae, ?icaeyiooei o ?icaeiei 5. Uia canoinoaaoe oeae iaoiae aei caaea/ noioanoe/ii? aeene?aoii? iioeiicaoei?, a iiae?icaeiei 5.5 iiaoaeiaaii niaoeeoi/ii noioanoe/ii ioeiiee iioeiaeueieo cia/aiue aeey ?iciiiaiioieo eeania caaea/ noioanoe/ii? iioeiicaoei?. O ?icaeiei 6 aeineiaeaeo?oueny iaoiae aiii?e/ieo na?aaeiio aeey iiiiiicaoei? neeaaeieo ooieoeie ?eceeo, yeee caiaeeoue caaea/o noioanoe/iiai i?ia?aioaaiiy aei aeaoa?iiiiaaii? caaea/i iioeiicaoei?. 2. Iaoiaee noioanoe/ieo ocaaaeueiaieo a?aaei?ioia. O caaaeueiie oi?ii caaea/i iioeiicaoei? nenoai c aeiaaeeiaeie aeene?ao-ieie iiaeiyie, ?icaeyiooi o iiae?icaeieao 1.2 - 1.8 aeena?oaoei?, iiaeia noi?ioethaaoe o aeaeyae? minx X [F(x)= E f(x,)], aea x - aaeoi? ea?iaaieo ia?aiao??a (??oaiiy); - aeiaaeeiaee ia?aiao?, aecia/aiee ia eiiai?i?niiio i?inoi?i (?,O,P); f(x,) - aeiaaeeiaa ooieoeiy yeinoi ?ioaiiy x i?e cia/aiii aeiaaeeiaiai ia?aiao?a ; F(x) - iaoaiaoe/ia niiaeiaaiiy ooieoei? yeinoi; X - aeiionoeia iiiaeeia c Rn. Nooo?aith iniaeeainoth caaea/i ? aiaenooiinoue o ooieoei? f(·,) oi?ioeo aiaeioe/ieo aeanoeainoae, o caaaeueiiio aeiaaeeo aiia iiaea aooe iaiioeeith, iaaeaaeeith i iaaioue ?ic?eaiith. Aea iiiaei ooieoeiy iaoaiaoe/iiai niiaeiaaiiy F(x) iiaea aooe aeaaeeith. Oeae aeiaaeie ?icaeyiooi o ?iaioao Y.C.Ho, X.R.Cao, R. Suri, P. Glasserman, P.W. Glynn, R.Y. Rubinstein, G.Ch. Pflug, I.E. E?eaoeiia oa ?i. O aeiaaeeo iaaeaaeei? ooieoei? F(x) aaaeeeaei ? aeiaaeie eiio?oeaaeo ooieoeie iaoaiaoe/iiai niiaeiaaiiy, ?icaeyiooee A.I. Aoiaeii Aoiae A.I. Noioanoe/aneea iaoiaeu ?aoaiey iaaeaaeeeo yeno?aiaeueiuo caaea/. - Eeaa: Iaoe. aeoiea, 1979. - 150 n. . Aieueoa oiai, ye iieacaii a iiae?icaeiei 1.6 (aeea. (1), (2)) oa a [10, 11, 25], ie /anoi ia?ii ni?aao ia c caaaeueiei eeanii eiio?oeaaeo ooieoeie, a c ?o iiaeeeanii, ooai?aiei c aeayeeo aaciaeo (iaia?a?aii aeeoa?aioeieiaieo) ooieoeie ca aeiiiiiaith iia?aoeie iaeneioio, iiiiioio oa aeaaeeeo o?ainoi?iaoeie. Oaei ooieoei? iaeaaeaoue aei eeano oae caaieo ocaaaeueiaii aeeoa?aioeieiaieo ooieoeie, aea/aieo a [1]. sse i?eeeaae aeiaaaeaia ocaaaeueiaia aeeoa?aioe?eiai?noue oa ia/eneai? noioanoe/i? ocaaaeueiai? a?aae??ioe ooieoe?iiaeo F(x)=E min1tT ut(R(t,x,)) , (2) a?ae i?ioeano (no?aoiaiai) ?eceeo R(t,x,) = x1+c2x2t1iN(t,) minx2Li(),x3c3x3t, 1tT, ci >0,

aea ooieoe?? ei?eniino?

ut(R)=R aeey R?0 oa ut(R)=(1+et)R aeey R<0, t0 , Li() - aeiaaeeia? ao?aoe; N(t,) - aeiaaeeiaa /enei oaeeo ao?ao ca /an t; x1 - ii/aoeiaee eai?oae no?aoiai? eiiiai??; x2 - cano?aoiaaia aeiey ao?ao; x3 - ??aaiue ia?ano?aooaaiiy; x=(x1,x2,x3). O ?iaio? [1] aoee ?ic?iaeaii oa aeineiaeaeaii aeayei noioanoe/i? ocaaaeueiaii a?aaei?ioii iaoiaee iioeiicaoei? ocaaaeueiaii aeeoa?aioeieiaieo ooieoeie, iai?eeeaae, noioanoe/i? iaoiaee "aaaeei? eoeueee" oa "ouaeeiiiai e?ieo" aeey iioei?caoe?? "ouaeeiieo" ooieoe?e. Oe? iaoiaee aeei?enoiaothoue ocaaaeueiai? a?aae??ioe i?ae?ioaa?aeueii? ocaaaeueiaii aeeoa?aioe?eiaii? ooieoe?? f(x,). O ?icaeiei 2 oa a [11,25] aeiaiaeeoueny caiaeiinoue ua iaeiiai iaoiaeo iiiiiicaoei? ocaaaeueiaii aeeoa?aioeieiaieo ooieoeie - iaiioeeiai aiaeiao a?aeiiiai iaoiaeo noioanoe/ieo eaacia?aaei?ioia TH.I.??iieue?aa A?iieueaa TH.I. Iaoiaeu noioanoe/aneiai i?ia?aiie?iaaiey. - I.: Iaoea, 1976. - 240 n. : xk+1X(xkkg(xk,k)), g(xk,k)xf(xk,k), xkxkk, k=0,1,..., aea X - (aaaaoicia/iee) iia?aoi? i?iaeooaaiiy ia iaiioeeo aeiionoeio iiiaeeio X; g(x,) - aei??iee ca noeoii?noth ci?iieo ia?a??c noaaeeoa?aioe?aea xf(x,); k - iacaeaaei? niinoa?aaeaiiy aeiaaeeiaiai ia?aiao?a ; iaa?ae'?ii? /enea k, k caaeiaieueiythoue oiiaaii limk k = limk k =0, 0k =+, 0k2 <+. (3) Iaoiae iaeaea iaiaaia ca?aa?oueny aei iiiaeeie oi/ie X*=x| 0F(x)+NX(x), ui caaeiaieueiythoue iaiao?aeiei oiiaai iioeiaeueiino?, oa ?nio? a?aieoey limkF(xk)F(X*) i.i. Aeey ia/eneaiiy iioeiaeueiiai cia/aiiy limkF(xk) aeei?enoiaothoueny ioe?iee, ui aeieeaathoue ?c ianoaoe?iia?iiai caeiio aaeeeeo /enae [12]: Fk+1=(1k)Fk+ +kf(xk+1,k), F0=0, k=0,1,..., aea /enea sk caaeiaieueiythoue oiiaai 0k1, limkk=0, 0k=+, 0Ek1+|f(xk,k) F(xk)|1+<+, 0<1. Oe? ioe?iee iiaeooue ?icaeyaeaoeny ye ocaaaeueiaiiy iaoiaeo Iiioa Ea?ei aeey ia/eneaiiy F(X*) ca niinoa?aaeaiiyie f(xk,k), oaeeie, ui E f(xk,k)| x0,..., xk=F(xk) F(X*) i.i. Oiiae ca?aeiino? iaoiaeo ? a?eueo neaaeeie i?ae o aeiaaeeo caaaeueieo e?io?oeaaeo ooieoe?e. Oey i?ioeaaeo?a oaeiae ocaaaeueith? ?acoeueoaoe ?.I.Io?iiinueeiai Io?ieineee A.A. *eneaiiua iaoiaeu ?aoaiey aeaoa?ieie?iaaiiuo e noioanoe/an-eeo ieieiaeniuo caaea/. - Eeaa: Iaoe. aeoiea, 1979. - 161 n. oa I.A.Aei?ioa?aa Aei?ioaaa I.A. I iaeioi?uo naienoaao iaiauaiiiai a?aaeeaioiiai iaoiaea // AEo?i. au/ene. iaoaiaoeee e iao. oeceee. - 1985. - 25, N 2. - C. 181-189., iaea?aeaii aeey aeiaaeeo eaaci-aeeoa?aioeieiaieo ooieoeie F(x), yei ia ioiieththoue i?eeeaaeia oeio (1), (2) oa ?ioeo, ?icaeyiooeo o iiae?icaeiei 1.5. Iiaeaeueoa ocaaaeueiaiiy oeueiai iaoiaeo ia aeiaaeie eaac?iioeeeo oa ?ic?eaieo ooieoe?e ?eceeo iiaoaeiaaii a ?icae?eao 3, 4. I?iaeaia aeiaaeueii? iioei?caoe?? iaiioeeeo ooieoe?e ?eceeo ?icaeyaea?oueny a ?icae?e? 5. 3. Noioanoe/ii iaoiaee caeaaeaeoaaiiy. O ?icaeiei 3 ?icaeaa?oueny iiaeoiae [2, 9] aei iioeiicaoei?, aeania eaaeo/e, ?ic?eaieo ooieoeie ?eceeo oeyoii ai?ieneiaoei? ?o aeaaeeeie ona?aaeiaieie ooieoeiyie. ?aeay i?aeoiaeo iieyaa? a ai?ieneiaoe?? aeo?aeii? caaea/? noeoii?noth iaiioeeeo caaea/ noioanoe/iiai i?ia?aioaaiiy ? canoinoaaii? aei inoaii?o anueiai a?naiaeo i?eeii?a, iai?aoeueiaaieo o noioanoe/iiio i?ia?aioaaii? (iaoiaeeea ia/eneaiiy noioanoe/ieo a?aae??io?a aac ia/eneaiiy aaaaoiaei??ieo ?ioaa?ae?a, oaoi?ea iaiioeeiai noioanoe/iiai ae?oaiai iaoiaeo Eyioiiaa aeey aeiaaaeaiiy ca?aeiinoe iaeaea iaiaaia noioanoe/ieo aeai?eoi?a iioei?caoe??, iaoiaeeea i?enei?aiiy ca?aeiino? noioanoe/ieo iaoiae?a, iaoiaeeea ianoaoe?iia?ii? noioanoe/ii? iioei?caoe??). E??i oiai, o i?ae?icaeieao 3.6, 3.7 oa a [9, 10, 21] ona?aaeiaii ooieoei? aeei?enoiaothoueny aeey iiaoaeiae coaaeeoa?aioeiaeia ?ic?eaieo ooieoeie. Onoaiiaeaii iaiaoiaeii oiiae iioeiaeueiinoi a oa?iiiao oae caaieo caeaaeaeaieo noaaeeoa?aioeiaeia ?ic?eaieo ooieoeie. Aeiaaaeaia caiaeiinoue ai?ieneiaoeieii? noaie, oiaoi caiae-iinoue iiiiioiia iaaeeaeaieo caaea/ aei iiiiioiia ii/aoeiai?. Aeiaaaeaia caiaeiinoue iaeaea iaiaaia noioanoe/ieo eiioeaai-?icieoeaaeo i?ioeaaeo? iioeiicaoei? ?ic?eaieo ooieoeie. Ae?oaee i?aeo?ae aei iioei?caoe?? ?ic?eaieo ooieoe?e, iiaoaeiaaiee ia aac? eieaeueii? ai?ieneiaoe?? ?ic?eaii? ooieoe?? e?i?eieie ooieoe?yie, ?icaeaa?oueny A.Ae.Aaoooo?iei oa ?i. Aaooooei A.Ae., Iaeai?iaea E.A. ?ac?uaiua yeno?aiaeueiua caaea/e. - Naieo-Iaoa?ao?a: Aeiiie?ao, 1995. - 208 n. O iaaeaaeeiio aiaeici aaiaeyoueny oa aea/athoueny ?icii eeane iaaee-oa?aioeieiaieo ooieoeie. Oa ae naia o?aaa i?i?iaeoe i aeey ?ic?eaieo ooieoeie. Ie iaiaaeo?ii eiiai?io ?ic?eaiinoue ooieoeie aeiaaeeii oae caaieo neeueii iaiiaiaia?a?aieo cieco ooieoeie. Icia/aiiy 1 [9]. Ooieoeiy F : Rn R1 iaceaa?oueny neeueii iaiia-iaia?a?aiith cieco a oi/oei x, yeui aiia iaiiaiaia?a?aia cieco a x oa inio? iineiaeiaiinoue xk x, oaea, ui F - iaia?a?aia a xk (aeey onio k) oa F(xk) F(x). Ooieoeiy F iaceaa?oueny neeueii iaiiaiaia?a?aiith cieco (neeueii iic) ia X Rn, yeui oea ia? iinoea aeey anio x X. O iiae?icaeiei 3.2 oa o [10, 20] oaeiae aaiaeyoueny eeane iaia?a?aieo ca iai?yieaie oa eoneiai-iaia?a?aieo ooieoeie. Aeanoeainoi neeueii? iaiiaiaia?a?aiinoi cieco, iaia?a?aiinoi ca iai?yieaie, eoneiai? iaia?a?aiinoi caa?iaathoueny i?e iaia?a?aieo ia?aoai?aiiyo. O iiae?icaeiei 3.2 aeaii aeinoaoii oiiae neeueii? iaiiaiaia?a?aiinoi aeayeeo neeaaeieo ooieoeie (o /anoeiaiio aeiaaeeo, ooieoeie iaoaiaoe/iiai niiaeiaaiiy oa eiia??iino?) i ?ic?iaeaii /eneaiiy aeey oeeo ooieoeie. Ona?aaeiai? ooieoe?? aeaaii aeei?enoiaothoueny a iaoaiaoeoe?. O aeai?e ?iaio? aiie aaeeaathoueny aeey iioei?caoe?? ?ic?eaieo ooieoe?e. Ona?aaeiaii ooieoei? f , R1 aecia/athoueny oeyoii eiiaiethoei? aeaii? ooieoei? f c aeayeei niiaenoaii yaea? (mollifiers) , >0, oaeeo, ui

lim +0 ||z|| (z)dz = 1 > 0.

Iai?eeeaae, iaoae () aoaea aaonianueeith uieueiinoth eiiai?iinoi,
(x)=(x/), > 0. Aeey iaiaaeaii? eieaeueii ?ioaa?iaii? ooieoei? f(x)
?icaeyiaii ianooiia niiaenoai ona?aaeiaieo ooieoeie:

f (x)=(1/ n) f(y)((x-y)/)dy, > 0.
(4)

Oiaei eiaeia ooieoeiy f ? aiaeioe/iith c a?aaei?ioii

f(x)=E(1/)[f(x+)f(x)]= E(1/2)[f(x+)f(x)], (5)

aea aeiaaeeiaa aaee/eia ia? noaiaea?oiee ii?iaeueiee ?iciiaeie, E —
iaoaiaoe/ia niiaeiaaiiy ii .

sseui ooieoeiy f ia ? iaia?a?aiith, oi ie ia iiaeaii i/ieoaaoe, ui
ona?aaeiaii ooieoei? f(x) caiaathoueny aei f ?iaiiii?ii. Aea aeey oeieae
iioeiicaoei? oea i ia iio?iaii. Iio?iaia oaea caiaeiinoue ona?aaeiaieo
ooieoeie aei f, yea oyaia ca niaith caiaeiinoue iiiiioiia f(x) aei
iiiiioiia f. Oea aa?aioo?oueny oae caaiith aiicaiaeiinoth ooieoeie.

Icia/aiiy 2 Rockafellar R.T., Wets R.J-B. Variational Analysis. —
Berlin: Springer, 1998.

733 p.

. Iineiaeiaiinoue ooieoeie f k: Rn R=R+,
picscalex1000100090000035000000000000e0000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02000220010e00000026060f001200ffffffff00000800
0000c0ffc0ffe000c0010b00000026060f000c004d61746854797065000050000a000000
26060f000a00ffffffff010000000000030000000000aii-caiaathoueny aei
ooieoei? f: Rn R a oi/oei x, yeui

1) liminfk f k(xk) f(x) aeey on?o xk x;

2) limk f k(xk) = f(x) aeey aeayei? iine?aeiaiino? xk x.

Iineiaeiaiinoue f kk N aii-caiaa?oueny aei f, yeui oea ni?aaaaeeeai o
eiaeiie oi/oei xRn.

Ianooiia aaaeeeaa aeanoeainoue aiicaiaeieo ooieoeie iieaco?, ui
ii-oeiicaoeiy ?ic?eaii? ooieoei? F(x) ca iaiaaeaiiyi xK, acaaaei, iiaea
aooe aeeiiaia ca aeiiiiiaith ai?ieneiaoei? F(x) aiicaiaeieie ooieoeiyie
Fk(x) (iai?eeeaae, ona?aaeiaieie ooieoeiyie) oa ?aeaenaoe?? iaiaaeaiue
caaea/?.

Oai?aia 1 [10, 21]. sseui iineiaeiaiinoue ooieoeie Fk: Rn R
aii-caiaa?oueny aei F: Rn R, oi aeey aoaeue-yeiai eiiiaeoo X Rn

lim0 (liminfk (infX() Fk))= lim0 (liminfk (supX() Fk))= infX F ,

aea X()=X+B, B=xRn| ||x|| 1. sseui Fk(xk ) infX() Fk +k , aea xk
X(), k 0, oi

limsup0 (limsupk xk ) argminX F ,

aea (limsupk xk ) aecia/a? iiiaeeio K a?aie/ieo oi/ie iineiaeiaiinoae
xk oa ( limsup0 K ) aecia/a? iiiaeeio a?aie/ieo oi/ie niiaenoaa K ,
R+, eiee 0.

Ianooiia oai?aia iieaco?, ui ona?aaeiaii ooieoei? iiaeooue aooe
aeei?enoaii aeey iiiiiicaoei? neeueii iaiiaiaia?a?aieo cieco ooieoeie a
iaaeao ai?ieneiaoeieii? noaie oai?aie 1, oae ye aiicaiaeiinoue oyaia ca
niaith caiaeiinoue iioeiaeueieo cia/aiue oa caiaeiinoue iiiiioiia. O
iiae?icaeiei 3.4 aea/athoueny ai?ieneiaoeieii noaie aeey ?ic?eaieo
aeno?aiaeueieo caaea/ c ooieoeiiiaeueieie iaiaaeaiiyie, a o iiae?icaeiei
3.5 — iaoiae ?ic?eaieo oo?aoieo ooieoeie.

Oai?aia 2 [9]. Aeey iaiaaeaii? neeueii iaiiaiaia?a?aii? eieaeueii
iioaa?iaii? ooieoei? f: RnR aoaeue-yea anioeieiaaia iineiaeiaiinoue
ona?aae-iaieo ooieoeie f k := f(k) , (k) R+ aii-caiaa?oueny aei f,
eiee q(k) 0.

O iiae?icaeieao 3.6, 3.7 ona?aaeiaii ooieoei? aeei?enoiaothoueny
aeey iiaoaeiae ocaaaeueiaieo iioiaeieo oa noaaeeoa?aioeiaeia ?ic?eaieo
ooieoeie, yei a naith /a?ao a iiae?icaeiei 3.8 aeei?enoiaothoueny aeey
oi?ioethaaiiy iaiaoiaeieo oiia aeno?aioio.

Icia/aiiy 3 [9]. Iaoae f k : = f(k) , (k) 0 ? aeaaeeeie.
-caeaae-aeaiei (?aaoey?iei) noaaeeoa?aioeiaeii ooieoei? f o oi/oei x
iaceaa?oueny iiiaeeia

f(x):= Limsupk f k(xk)| xk x,

oiaoi f(x) ? iiiaeeia a?aie/ieo oi/ie onio iineiaeiaiinoae fk(xk),
oaeeo, ui xk x.

Iiiaeeia f(x) -caeaaeaeaieo noaa?aaei?ioia caieiaia oa o
caaaeueiiio aeiaaeeo caeaaeeoue aiae aeai?o iineiaeiaiinoi k,
aeei?enoaii? i?e eiai iiaoaeiai (a iioeeiio aeiaaeeo ia caeaaeeoue oa
ia? iinoea f(x) = f(x)).. Iiiaeeia y f(x) caaaeaee ia ii?iaeiy, yeui
ooieoeiy f iaeaea ne?icue aeaaeea oa ?? a?aaei?ioe eieaeueii iaiaaeaii
ia iiiaeeii, aea aiie iniothoue.

Icia/aiiy 4 [9]. Iaoae ona?aaeiaii ooieoei? f k:= f(k) , (k) 0 ?
aeaaeeeie; -caeaaeaeaiith iioiaeiith ooieoei? f o oi/oei x oa o
iai?yieo u iaceaa?oueny aaee/eia

f'(x;u):= supx(k)x limsupk (f k)'(x(k);u),

aea (f k)'(x;u) ? iioiaeia f k a oi/oei x ii iai?yieo u, noi?aioi
aa?aoueny ii aiaeiioaiith aei anio iineiaeiaiinoae x(k)x.

Oai?aia 3 [9]. Ni?aaaaeeeaa aeeth/aiiy

conv f(x) G(x):= gRn| g,u f'(x;u) uR n ,

aea conv icia/a? iioeeo iaieiieo. sseui iiiaeeia G(x) iaiaaeaia, oi
conv f(x) = G(x).

Aeey iaia?a?aii? ooieoei? f aaaaeaiee noaaeeoa?aioeiae niiaiaaea? c
iioiaeiith iiiaeeiith Aeae.Aa?ae Aa?aa Aeae. Iioeiaeueiia oi?aaeaiea
aeeooa?aioeeaeueiuie e ooieoeeiiaeueiuie o?aaiaieyie. — I.: Iaoea, 1977.
— 624 n. , a aeey eieaeueii eiio?oeaai? ooieoei? conv f(x) ni?aiaaea? c
noaaeeoa?aioe?aeii O.Eea?ea Eea?e O. Iioeiecaoeey e iaaeaaeeee aiaeec.
— I.: Iaoea, 1988. -280 n. Cf(x)..

Iaiaoiaeii oiiae aeno?aioio (o oiio /ene? aeey ?ic?eaieo ooieoe?e)
aeaoaeueii ?icaeyioo? A.I.Ioaie/iei Ioaie/iue A.I. Iaiaoiaeeiua oneiaey
yeno?aioia: 2 ecae. — I.: Iaoea, 1982. — 144 n., ?.O.?ieaoaeea?ii
Rockafellar R.T. The theory of subgradients and its application to
problems of optimization. — Berlin: Holderman, 1981. — 107 p. oa
A.O.Ii?aeooiae/ai Ii?aeooiae/ A.O. Iaoiaeu aii?ieneiaoeee a caaea/ao
iioeiecaoeee e oi?aaeaiey. — I.: Iaoea, 1988. — 360 n. , aea io?eiai?
?acoeueoaoe ia ae/a?iothoue i?iaeaio. Oae iaiaoiaeii oiiae aeno?aioio
aeey caaea/i iiiiiicaoei? aac iaiaaeaiue ?ic?eaii? ooieoei? f(x) o
oa?i?iao caeaaeaeaiiai noaaeeoa?aioe?aea iathoue noaiaea?oiee aeaeyae: 0
f(x). Aea i?e iioeiicaoei? ca iaiaaeaiue caaaeueieo iaia?a?aieo oa
?ic?eaieo ooieoeie ie noeea?iiny c i?iaeaiith neiaoey?ieo a?aae??io?a.
?icaeyiaii caaea/o: minx1/3|x0. Ca aoaeue-yeiai ?icoiiiai icia/aiiy
ocaaaeueiaieo a?aaei?ioia a?aaei?io ooieoei? x1/3 a oi/oei x=0
aei?iaith? +, oiaoi ?aaoey?ia neeaaeiaa a?aae??ioa ii?iaeiy, a
neiaoey?ia aei?iaith? +. Ioaea, aeey oiai, uia noi?ioethaaoe iaiaoiaeii
oiiae iioeiaeueiinoi aeey oaeeo caaea/ oa caaea/, yei aeeth/athoue
?ic?eaiinoi, iio?iaia iiaa, ui iinoeoue ianeii/aiii aaee/eie. Oaeith
i?eaeaoiith eiino?oeoei?th ? iiiyooy «einii/iiai» aaeoi?iiai i?inoi?o
Rn Rockafellar R.T., Wets R.J-B. Cosmic convergence /Optimization and
Nonlinear Analysis; eds. A.Ioffe, M.Marcus and S.Reich; Pitman Research
Notes in Mathem. Ser. 244. Essex: Longman Scientific & Technical, 1991.
— P. 249-272.

.

Icia/aiiy 5. Iicia/eii R+ =x R| x 0 oa R+ =R+ +. Iicia/eii
«einii/iee» i?inoi? Rn ye iiiaeeio ia? x=(x,a), aea xRn, ||x||=1 i aR+ .
Oni ia?e aeaeyaeo (x,0) aaaaeathoueny iaeaioe/ieie oa iicia/athoueny ye
0. Iineiaeiaiinoue (xk,,ak ) Rn iaceaa?oueny «einii/ii» caiaeiith aei
aeaiaioo (x,a) Rn, yeui x=limkxk , a=limk ak R+ aai limk ak =0 (a
inoaiiueiio aeiaaeeo limk (xk ,,ak )=0). Iicia/eii

Limsupk (xk,ak)=(x,a)Rn| k(m): (x,a)=limm (xk(m),ak(m)).

Icia/aiiy 6 [10, 21]. ?icoe?aiei -caeaaeaeaiei noaaeeoa?aioeiaeii F o
oi/oei x iaceaa?oueny iiiaeeia

F(x):= Limsupk (Fk(xk)/||Fk(xk)||,||Fk(xk)||) | xk x ,

aea ae?ac Fk(xk)/||Fk(xk)|| caiiith?oueny aoaeue yeei iaeeie/iei
aaeoi?ii, yeui Fk(xk) =0, oiaoiF(x) neeaaea?oueny ic a?aie/ieo oi/ie
(o «einii/iiio» i?inoi?i Rn ) onio iiaeeeaeo iineiaeiaiinoae

(Fk(xk)/||Fk(xk)|| ,||Fk(xk)||), oaeeo, ui xk x.

?icoe?aiee caeaaeaeaiee noaaeeoa?aioeiae F(x) caaaeaee ? iaii?iaeiy
caieiaia iiiaeeia o Rn oa aiaeia?aaeaiiy x F(x) caieiaia..

Oai?aia 4 [10, 21]. Iaoae X — caieiaia iiiaeeia o Rn. I?eionoeii, ui
iaiaaeaia eieaeueii iioaa?iaia ooieoeiy F ia? eieaeueiee iiiiioi
aiaeiinii X o aeaye?e oi/oei x X oa inio? iineiaeiaiinoue xk X, xk
x, oaea, ui F iaia?a?aia a xk oa F(xk) F(x). Oiaei aeey aoaeue-yei?
iineiaeiaiinoi k aeaaeeeo yaea?, ia? iinoea

F(x) NX(x) ,
(6)

aea -F(x)= (-g,a) Rn | (g,a) F(x) , NX(x)=(y,a)Rn | yNX (x),
||y||=1, aR+ , NX (x) — ii?iaeueiee eiion aei iiiaeeie X o oi/oei x.

Ianeiaeie 1. Aeey iaia?a?aii? ooieoei? F oiiaa (6) ? iaiaoiaeiith,
oiaoi (6) caaeiaieueiy?oueny aeey aoaeue yeeo eieaeueieo iiiiioiia F ia
K.

Oai?aia 5 [10, 21]. sseui F — neeueii iaiiaiaia?a?aia cieco oa
iiiaeeia K eiiiaeoia, oi iiiaeeia X* oi/ie, ui caaeiaieueiy? oiiaai
iioeiaeueiinoi (6), ia ii?iaeiy oa iinoeoue i?eiaeii? iaeei aeiaaeueiee
iiiiioi ooieoei? F ia K.

O ?icaeiei 3.8 coi?ioeueiaaii a oa?iiiao ?icoe?aiiai noaaeeoa?aioeiaea
oiiae iioeiaeueiinoi (oeio Eoia — Oaea?a) aeey ?ic?eaieo caaea/ c
iaiaaeaiiyie-ia?iaiinoyie.

Oaeei /eiii, aeey caaea/ aac iaiaaeaiue oiiae iioeiaeueiinoi
caiaeyoueny aei aiaeiiiai aeaeyaeo 0F(x) oa ? iaiaoiaeieie, oiaoi
aeeiiothoueny aeey onio eieaeueieo iiiiioiia caaea/i. Oiiae
iioeiaeueiinoi (6) ? iaiaoiaeieie oaeiae o aeiaaeeo iaia?a?aiinoi
oeieueiai? ooieoei? F(x). O ?ic?eaiiio aeiaaeeo neooaoeiy a?eueo
neeaaeia: ia iiaeia aa?aiooaaoe, ui oni eieaeueii iiiiioie aai iaaioue
oni aeiaaeueii iiiiioie caaeiaieueiythoue (6). I?e/eia oeueiai iieyaa? o
oiio, ui ia oni a?aie/ii eieaeueii oa aeiaaeueii iiiiioie F ia K iiaeia
aiaeiaeoe ca aeiiiiiaith iiiiiicaoei? ona?aaeiaieo ai?ieneiaoeie F, ui
eaaeaoue a iniiai iiiyooy caeaaeaeaiiai noaaeeoa?aioeiaea. Aea iiiaeeia
X* oi/ie, ui caaeiaieueiythoue (6), ia ii?iaeiy, iinoeoue i?eiaeii?
iaeei aeiaaeueiee iiiiioi. Oiiae iioeiaeueiinoi (6) eiino?oeoeaii, aiie
aeathoue iaeath iaaeeaeaiiai ia/eneaiiy aeaiaioia X*.

?icaeyiaii caaea/o aacoiiaii? iiiiiicaoei? neeueii iaiiaiaia?a?aii?
cieco ooieoei? f(x), xRn. Caaea/a oiiaii? iioeiicaoei? iiaea aooe
caaaeaia aei caaea/i aacoiiaii? iioeiicaoei? ca aeiiiiiaith iaoiaeo
?ic?eaieo oo?aoieo ooieoeie, ?icaeyiooiai a ?icaeiei 3.5. I?eionoeii, ui
iineiaeiaiinoue aeaaeeeo ona?aaeiaieo ooieoeie f , yea iiaoaeiaaia ic f
ca aeiiiiiaith yaea? , aii-caiaa?oueny aei f.

Iaoae iineiaeiaiinoue iaaeeaeaiue x aoaeo?oueny ca ianooiiei
i?aaeeii. Eiaeia ooieoeiy f iiiiiico?oueny, ii/eiath/e c iiia?aaeiueiai
iaaeeaeaiiy x-1, aei oi/ee x, oaei?, ui ||f (x)|| , aea 0. Ii/aoeiaa
oi/ea x0 aa?aoueny aeiaieueii. O oeueiio iaoiaei, yeui x’=limkx(k),
oi, ii icia/aiith caeaaeaeaiiai noaaeeoa?aioe?aea,

limk f (k)(x(k)) = 0 f(x’).

O iiiia?aoi? [1] iaaaaeaii aaaaoi iaoiaeia oiiaii? oa aacoiiaii?
iioeiicaoei? eiio?oeaaeo ooieoeie ia iniiai caeaaeaeoaaiiy. O
iiae?icaeiei 3.10 ia i?eeeaaei caaea/i ?ic?eaii? iioeiicaoei? c i?inoeie
iaiaaeaiiyie (oaeeie, ui ia ieo eaaei caeienithaaoue i?iaeooaaiiy)
iieacaii, ye aoaeothoueny noioanoe/ii iaoiaee oiiaii? ?ic?eaii?
iioeiicaoei?, ui aeei?enoiaothoue iaeath caeaaeaeoaaiiy ?ic?eaii?
ooieoei?.

?icaeyiaii iioeiicaoeieio caaea/o

f(x) min xX
(7)

eieaeueii? iiiiiicaoei? ooieoei? f:Rn R ia iioeeie eiiiaeoiie iiiaeeii
X Rn.

I?eionoeii, ui f(x) — iaiaaeaia neeueii iaiiaiaia?a?aia cieco ooieoeiy
ia Rn. Iaoae (x) — ooieoeiy uieueiinoi eiiai?iinoi. Aecia/eii niiaenoai
ona?aaeiaieo ooieoeie (4). Iaoae aaciaa uieueiinoue (x) aeae?a?oueny
oaeith, ui ona?aaeiaii ooieoei? f (x) iaia?a?aii aeeoa?aioeieiaaii oa
aeey onio x X oa aeayeiai C<+ | f(x) f(x)| C|-|/max(,), ||f(x) f(y) || (C/) ||x-y||, ||f(x) f(x) || C||/max(,)2. O aiaeiiaiaeiinoi c oai?aiith 4 iicia/ei iiiaeeio (?ica'yce?a) X*=x X| F(x) NX(x) . ?icaeyiaii ianooiia ocaaaeueiaiiy iaoiaeo noioanoe/ieo eaacia?aaei?ioia TH.I.??iieue?aa aeey ciaoiaeaeaiiy eieaeueieo iiiiioiia ?ic?eaii? caaea/i (7) (x0 X): xk+1 X(xkk gk), gk = f(k) (xk)+k, k=0,1,..., aea aaeoi?e k oa iaaiae'?iii /enea k , (k) caaeiaieueiythoue oiiaai k k = +, k (k/(k))2 = +, limk(k)=0, limk |(k)-(k+1)| /((k)k)=0, limkk = 0. Oai?aia 6. Inio? iiaeiineiaeiaiinoue xk(s), oaea, ui lims ||xk(s+1) -xk(s) || /k(s) = 0, oa eiaeia oaea iiaeiineiaeiaiinoue caiaa?oueny aei iiiaeeie ?ica'yce?a X*. Aeaiee iaoiae iio?aao? ioeiiee gk a?aaei?ioia f(k)(x). O caaaeueiiio aeiaaeeo oea aeineoue neeaaeia oa o?oaeiiinoea i?ioeaaeo?a, ui aeiaaa? ia/eneaiiy aaaaoiaeii?ieo iioaa?aeia. Iaeiae aneiioioe/ii caiaeii ioeiiee iiaeia iiaoaeoaaoe ia?aeaeueii c iiaoaeiaith iniiaii? iineiaeiaiinoi x ca aeiiiiiaith oae caaii? i?ioeaaeo?e ona?aaeiaiiy noioanoe/ieo a?aaei?ioia: gk+1 = gk k(gk k(xk)), g0=0, k=0,1,..., aea Ek(x) = f(k) (x), 0 k 1, aeiaaeeia? aaeoi?e k(x) iiaeooue aooe iiaoaeiaai? ia aac? (5). O i?ae?icaeiei 3.11 anoaiiaeaii oiiae caiaeiinoi oei?? i?ioeaaeo?e i?eeiath/e aei oaaae oo ianoaaeio, ui a?aaei?ioe f(k)(xk) iiaeooue c?inoaoe aac iaiaaeaiue. 4. Eieaeueia iioeiicaoeiy ooieoeie niiaeiaaii? ei?eniinoi oa eiiai?iinoi. Ooieoei? niiaeiaaii? ei?eniinoi aeaaii aeei?enoiaothoueny a oai?i? i?eeiyooy ?ioaiue. Iaeiae aei oeueiai /ano yeinue iaei caa?oaee oaaao ia oa, ui aiie iiaeooue aooe iaaeaaeeeie, iaiioeeeie oa iaaioue ?ic?eaieie. Ooieoeiy eiiai?iinoi ? /anoeiaei aeiaaeeii ooieoei? niiaeiaaii? ei-?eniinoi oa cia?aaeo? eiiai?iinoue oiai, ui aeayea aeiaaeeiaa aaee/eia, ui caeaaeeoue aiae ea?iaaieo ia?aiao?ia, ia?aaieueoo? aeayeee caaeaiee ?iaaiue aai iaeaaeeoue caaeaiie iiiaeeii. Ooieoei? eiiai?iinoi neoaeaoue aeey oi?iaeueiiai cia?aaeaiiy iiiyooy ?eceeo a oai?i? i?eeiyooy ?ioaiue. ?icaeyiaii ooieoeith niiaeiaaii? ei?eniinoi aeaeyaeo F(x)= =Eu(f(x,)), aea ooieoeiy (ei?eniinoi) u:RmR1 cia?aaeo? aiaeiioaiiy iniae, ui i?eeia? ?ioaiiy, aei ?icieo cia/aiue "i?eaooeia (ao?ao)" f(x,) i?e ?ica'yceo x. A ie?aieo aeiaaeeao c u(t)=c(t)=0 i?e t<0, 1 o i?ioeeaaeiiio aeiaaeeo, iaea?aeo?ii ooieoeith eiiai?iinoi P(x)= =Pf(x,) 0, c u(t)=t - ooieoeith iaoaiaoe/iiai niiaeiaaiiy F(x)= Ef(x,), a c u(t)=max0,t - ooieoeith ?eceeo aeaeyaeo F(x)=Ef(x,)(f(x,)). Ooieoe?y ei?eniinoi u(·) o caaaeueiiio aeiaaeeo iiaea aooe ?ic?eaiith ia aeaye?e iiiaeeii DRm. ?? iiaeia ??cieie niiniaaie ai?ieneioaaoe iaia?a?aieie ooieoe?yie ue , aea - aeayeee ia?aiao?, oae, ui u(y) u(y) aeey an?o yD i?e 0. Oiae? ooieoe?y F(x) ai?ieneio?oueny ooieoe?yie F(x)=E u(f(x,)) ? cai?noue F(x) iiaeia iioei?coaaoe F(x). Iicia/eii D=Rm|dist(,D), aea dist(,D)=infzD ||-z||. Eaia 1. sseui ??aiii??ii ii x aeeiiaii lim0 P f(x,)D=0 ? ??aiii??ii ii z D ni?aaaaeeeai lim0 u(z)=u(z), oi F(x) iaia?a?aia ? lim0 F(x)=F(x) ??aiii??ii ii x. sseui ooieoe?? u(z) ? f(x,) ocaaaeueiaii aeeoa?aioe?eiai?, oi noia?iiceoe?y u(f(x,)) ocaaaeueiaii aeeoa?aioe?eiaia c noaaeeoa?aioe?aeii xu(f(x,)), eio?ee iiaeia ia/eneeoe ca aeiiiiiaith eaioethaiaiai i?aaeea aeeoa?aioe?thaaiiy neeaaeii? ooieoe??. sseui noaaeeoa?aioe?ae xu(f(x,)) iaaei?o?oueny eiinoaioith (E?ioeoey), yea ?ioaa?o?oueny, oi (aeea. [1]) ooieoe?y F(x) oaeiae ocaaaeueiaii aeeoa?aioe?eiaia c noaaeeoa?aioe?aeii F(x)=Exu(f(x,)). Aeey iioei?ecaoe?? ocaaaeueiaii aeeoa?aioe?eiaii? ooieoe?? Fe(x) canoiniai? ocaaaeueiaii a?aae??ioi? iaoiaee c ?icae?eo 2.o 2. O iiae?icaeiei 4.6 oa o [22, 24] ai?ieneiaoe?eiee i?aeo?ae ?icae?eo 3 aaeaioo?oueny aeey iioei?caoe?? ooieoe?e eiia??iino?, yea o caaaeueiiio aeiaaeeo oaeiae iiaea aooe iaaeaaeeith, iaiioeeith oa iaaioue ?ic?eaiith. Aea ca aeayeeo oiia aiia ? iaia?a?aiith oa eaacioaiaiooith (-oaiaiooith). O oaeiio aeiaaeeo ie iiaee a canoinoaaoe noaa?aaei?ioiee iaoiae aeey ?? iaeneiicaoei?. Aea ia/eneaiiy noaa?aaei?ioia ooieoei? eiiai?iinoi aei oeueiai /ano ? i?iaeaiith. Oiio ie ?iaiiii?ii ai?ieneio?ii aeoiaeio ooieoeith eiiai?iinoi iineiaeiai?noth ocaaaeueiaii aeeoa?aioeieiaieo (eaacioaiaiooeo, -oaiaiooeo) ooieoeie niiaeiaaii? ei?eniinoi, aeey yeeo ia/eneaiiy noaa?aaei?ioia aeeiio?oueny ii?iaiyii eaaei. Aeey ?ica'ycaiiy iaaeeaeaieo caaea/ ie canoiniao?ii aoaeoeaiee noioanoe/iee noaa?aaei?ioiee iaoiae c ona?aaeiaiiyi o?a?eoi?i?. Iaea?aeaii ?acoeueoaoe i?i caiaeiinoue oa oaeaeeinoue caiaeiinoi oeueiai iaoiaeo. Oaiaiooi oa eaacioaiaiooi ooieoei? aiaeia?athoue cia/io ?ieue o oai?i? aeno?aiaeueieo caaea/. Aeey caaea/ c ooieoeiyie eiiai?iinoi oa ei?eniinoi iiaeiaio ?ieue aiaeia?athoue eiaa?eoi?/ii oaiaiooi Prekopa A. Stochastic Programming. Dordrecht: Kluver Academic Publishers, 1995. 600 p. oa aieueo caaaeueii -oaiaiooi ooieoei? (oa ii?e), aea - aeieniee ia?aiao?. O iiae?icaeiei 4.2 oa o [4] iiaea?oueny iaeyae oai?i? -oaiaiooeo ooieoeie oa ii?. Iaaiae'?iia ooieoeiy F(x) ? -oaiaiooith ia XRn (-<<+) oiaei i oieueee oiaei, eiee F(x) oaiaiooa aeey >0, ln F(x) oaiaiooa aeey =0 oa F(x)
iioeea ia X aeey <0. Iaaiae'?iia eaacioaiaiooa ooieoeiy, ca icia/aiiyi, ? (-)-oaiaiooith. Ia oieueee ooieoei?, aea e ii?e iiaeooue iaoe aeanoeainoi oaiaiooinoi. O ?iaio? [4] -oaiaiooi ooieoe?? aeei?enoiaothoueny aeey aiae?co caaea/ ia/?oei? iioei?caoe??. Aeyaey?oueny, ui aaaaoi eeane/ieo ?iciiaeieia eiiai?iinoae (?iaiiii?ia, ii?iaeueia, -?iciiaeie, Nothaeaioa, Ia?aoi, F-?iciiaeie) iathoue -oaiaiooi uieueiinoi i, oaeei /eiii, aaia?othoueny '-oaiaiooeie ii?aie ia aiaeiiaiaeiiio iini?. O ?icaeiei 4.4 iaaiai?ththoueny aeanoeainoi ooieoeie eiiai?iinoi (iaia?a?aiinoue, eaaciaeeoa?aioeieiaiinoue, eaacioaiaiooinoue, -oaiaiooinoue). Aiaeiii, ui aeey eaacioaiaiooi? ca noeoii?noth ciiiieo ooieoei? f(x,) nooiiiue oaiaiooinoi aiaeiiaiaeii? ooieoe?? eiiai?iinoi P(x) aecia/a?oueny nooiaiai oaiaiooinoi eiiai?i?nii? ii?e P. O caaaeueiiio aeiaaeeo ooieoeiy eiiai?iinoi P(x) iiaea aooe ?ic?eaiith, oae ye ?? cia?aaeaiiy o aeaeyaei iaoaiaoe/iiai niiaeiaaiiy iinoeoue ?ic?eaio iiaeeeaoi?io ooieoeith c(·). C oei?? ae i?e/eie ia iiaeia aeeoa?aioeithaaoe P(x) oeyoii ia?anoaiiaee iia?aoeie aeeoa?aioeithaaiiy oa iioaa?oaaiiy o ae?ac?, ui aecia/a? P(x). Oiio ie caiiith?ii ?ic?eaio ooieoeith c(·) iaia?a?aieie ai?ieneiaoeiyie u(·), i, oaeei /eiii, iaea?aeo?ii iaia?a?aio ai?ieneiaoeith aeey P(x). Iaoae aeayea uieueiinoue eiiai?iinoi (), R1, aaia?o? ii?o ia R1 ca oi?ioeith (A):= A() d, AR1. Aecia/eii ?? ooieoeith ?iciiaeieo ye u(t):=(-,t]. Oaia? ?icaeyiaii ianooiio ai?ieneiaoeith ooieoeie eiiai?iinoi ooieoeiyie (niiaeiaaii? ei?eniinoi) F(x):= u(f(x,)/)P(d), (8) aea >0 — /enaeueiee ia?aiao?. Ianooiia eaia anoaiiaeth? oiiae
?iaiiii?ii? caiaeiinoi F(x) aei P(x).

Eaia 2. Iaoae ooieoeiy f(x,) iaia?a?aia ii x iaeaea aeey onio .
I?eionoeii, ui aeey anio xX oa onio c’, ui aeecuee? 0, aeeiio?oueny
Pf(x,)=c’=0. Oiaei aeey aoaeue-yeiai eiiiaeoo KX ooieoei? F(x)
?iaii-ii?ii caiaathoueny o K aei iaia?a?aii? ooieoei? eiiai?iinoi P(x)
i?e 0.

Ca aeayeeo oiia ooieoei? ei?eniinoi F(x) ? -oaiaiooeie c aeayeei . A
naia, yeui f(x,) oaiaiooa ii noeoiiinoi ciiiieo, a aeiaooie ii? P —
-oaiaiooa ii?a, oi ooieoei? niiaeiaaii? ei?eniinoi F(x) -oaiaiooi. O
?icaeiei 4.2 oa o [4] iaaaaeaii aeayei aeinoaoii oiiae -oaiaiooinoi
aeiaooeia ii?.

Ianooiia oai?aia aea? aeinoaoii oiiae noaaeeoa?aioeieiaiinoi ooieoeie
niiaeiaaii? ei?eniinoi (8).

Oai?aia 7 [24]. Iaoae ooieoeiy uieueiinoi eiiai?iinoi () iaaiae’?iia,
iaiaaeaia oa iaia?a?aia, a ooieoeiy f(·,) — oaiaiooa ia xX aeey
aoaeue-yeiai ?. Iaoae

sup g gx f(y,), y-x L(x,), >0,

aea ooieoeiy L(x,) iioaa?iaia ii aeey eiaeiiai x X. Oiaei F(x) ?
eiio?oeaaith (-)?aaoey?iith (oa ocaaaeueiaii aeeoa?aioeieiaiith)
ooieoei?th, oa ?? noaaeeoa?aioeiae caaea?oueny oi?ioeith

F(x):= (1/) (f(x,)/) x f(y,)P(d).

?icaeyiaii caaea/o iioeiicaoei? ooieoei? niiaeiaaii? ei?eniinoi:

maxxX [F(x)= Eu(f(x,))].

Iaoae F* — iioeiaeueia cia/aiiy, a X* — iioeiaeueia iiiaeeia oei??
caaea/i.

sseui ooieoei? u(·) ? f(·,) ocaaaeueiaii aeeoa?aioeieiaii, oa f(·,)
ia? eiinoaioo Eiio?oey, yea ?ioaa?o?oueny, oi u(f(·,)) ? F(x)= Eu(f(x,))
ocaaaeueiaii aeeoa?aioeieiaii oa aeey iioeiicaoei? F(x) iiaeia
canoinoaaoe noioanoe/ii ocaaaeueiaii a?aaei?ioii iaoiaee ic [1] oa
?icaeieo 2.

O iiae?icaeiei 4.6 iaoiae noioanoe/ieo eaacia?aaei?ioia TH.I.??iieue?aa
(c ona?aaeiaiiyi A?oea — Iaie?ianueeiai — THaeiia Iaie?ianeee A.N.,
THaeei Ae.A. Neiaeiinoue caaea/ e yooaeoeaiinoue iaoiaeia iioeiecaoeee.
— I.: Iaoea, 1979. — 383 n.

) iioe?th?oueny aeey iioeiicaoei? ooieoeie niiaeiaaii? ei?eniinoi oa
iaaeeaeaii? iioeiicaoei? ooieoeie eiiai?iinoi.

I?eionoeii, ui ooieoeiy ei?eniinoi (?eceeo) F(x) ? -oaiaiooith, a X
— caieiaia iioeea iiiaeeia. Iiaoaeo?ii ianooiio iineiaeiaiinoue
ai?ieneiaoeie (x0=x0 X):

xk+1 = X(xk+k k),

x k+1 = (1k+1)xk +k+1 xk+1, k=0,1,…,

aea noioanoe/iee eaacia?aaei?io k ooieoei? F(x) o oi/oei xk, oaeee, ui

Ek | x0,…, xkF(xk), Ek2| x0,…, xkC< , X - iia?aoi? i?iaeooaaiiy ia iiiaeeio X, k=k /0ki , iaaiae'?iii /enea k caaeiaieueiythoue oiiaai (3). Iai?eeeaae, noioanoe/iee eaacia?aaei?io k = k(xk,k) ooieoei? (8) iiaeia acyoe o aeaeyaei (x,) = (1/)(f(x,)/) g(x,), g(x,) x f(x,), aea g(x,) - aeii?iee ii (x,) ia?a?ic aaaaoicia/iiai aiaeia?aaeaiiy xf(x,) i k, k=0,1,..., - iacaeaaeii iaeiaeiai ?iciiaeieaii niinoa?aaeaiiy aeiaaeeiaiai ia?aiao?a .. Oai?aia 8 [24]. Ca c?iaeaieo i?eiouaiue aeey iaeaea onio o?a?eoi?ie xk, ui ii?iaeaeaii aeuacacia/aiei noioanoe/iei eaacia?aaei?ioiei iaoiaeii, aeeiiaii 1) oni a?aie/ii oi/ee iineiaeiaiinoae xk, xk iaeaaeaoue iiiaeeii iaeneioiia argmax xX F(x); 2) limk F(xk)=limk F(xk)=maxxX F(x). Oai?aia 9 [24] (oaeaeeinoue caiaeiinoi). I?eionoeii, ui ooieoeiy F(x) -oaiaiooa oa F(x) >0, x X, oiaei aeey aoaeue-yeiai iaeneioia x* oa
aoaeue-yei? iine?aeiaiino? e?ie?a k 0 neooii ioeiiee

F(x*)EF(xk) (F(x*)/)max(1- ,0)(Ex0 x*2 +C0ki2)/(20ki).

5. Noioanoe/ii iaoiaee aieie oa iaae.

Iaoith oeueiai ?icaeieo ? ?ic?iaea noioanoe/ii? aa?ni? iaoiaeo aieie oa
iaae aeey ?ica’ycaiiy iaiioeeeo caaea/ iioeiicaoei?, ui iinoyoue
aeene?aoii oa iaia?a?aii ciiiii, a oaeiae iaaecia/aii ia?aiao?e.
I?iiiiiaaiee iaoiae iiaea aooe canoiniaaiee, eiee cae/aeii
aeaoa?iiiiaaii iiaeoiaee noeeathoueny c o?oaeiiuaie i?e ia/eneaiii
oi/ieo iaae cia/aiue oeieueiai? ooieoei?. Oei neooaoei? ? oeiiaeie aeey
iioei?caoei? aeene?aoieo oa iaia?a?aieo noioanoe/ieo nenoai.

*anoeiaeie aeiaaeeaie ?icaeyiooi? caaea/i ? caaea/a ia?aai?ee aiiioac,
caaea/a iaa/aiiy aaoiiaoia, caaea/a i?i aaaaoi?oeiai aaiaeeoa, aeey yeeo
iniothoue niaoeiaeiciaaii iaoiaee ?ica’ycaiiy. O iiae?icaeiei 1.6
?icaeyiooi iio? canoinoaaiiy oa iiaeaei noioanoe/ii? aeene?aoii?
iioeiicaoei?.

Aaaeeeaa aeanoeainoue oaeeo caaea/ noioanoe/ii? aeene?aoii?
iioeiicaoei? iieyaa? a oiio, ui /enei N iiaeeeaeo ?ioaiue (aeie) iiaea
aooe iaaecae/aeii aaeeeei, a a iaia?a?aiiio aeiaaeeo — i ianeii/aiiei.
Ioaea, aeei?enoaiiy noaiaea?oii? oaoiiee ia?aai?ee aiiioac aai oaoiiee,
ui ?ic?iaeaia aeey caaea/ iaa/aiiy aaoiiaoia, noa? i?aeoe/ii iaiiaeeeaei
oiio, ui aiie a?oioothoueny ia iineiaeiaiiio niinoa?aaeaiii ianeiaeeia
onio aeiionoeieo aeie.

Oi?iaeueii caaea/a, ui ?icaeyaea?oueny, ia? aeaeyae iaoiaeaeaiiy
aeiaaeueiiai iii?ioio

minx [F0(x)= E f0 (x,)],
(9)

ca iaiaaeaiue

Fk(x)= E fk (x,) 0, k=1,…,K;
(10)

x XD Rn,
(11)

aea fk(·,), k=0,1,…,K, — aeayei iaiioeei (iai?eeeaae, eaaciiioeei)
ooieoei?; X — aeene?aoia aai iaia?a?aia iiiaeeia i?inoi? no?oeoo?e
(iai?eeeaae, ia?aoei aeayei? aeene?aoii? ?aoioee oa ia?aeaeaiiiaaeo a
Rn); iiiaeeia D=xRn|G(x)0 caaea?oueny aeayeith aeaoa?iiiiaaiith
ooieoei?th G:RnR1, E — iaoaiaoe/ia niiaeiaaiiy noiniaii aeiaaeeia?e
aaee/eie , ui aecia/aia ia aeayeiio eiiai?i?niiio i?inoi?i (,,P).

Iniaeeai oeieaaee ie?aiee aeiaaeie caaea/i, eiee aeayei ooieoei? Fk (x)
iathoue aeaeyae eiiai?iinoi:

Fk(x)= P fk(x,)=(fk1 (x,),…, fkm (x,)) B() Rm .

Oaeei /eiii, ooieoeiy Fk (x) iiaea aooe iaaeaaeeith, iaiioeeith oa
iaaioue ?ic?eaiith.

Aeey ?ica’ycaiiy caaea/i (9) — (11) ?icaeaa?oueny noioanoe/ia aa?niy
iaoiaeo aieie oa iaae. Aeey oeueiai ie aeaiaeeii niaoeiaeueii
noioanoe/ii ieaeii oa aa?oii iaaei iioeiaeueiiai cia/aiiy caaea/
aeaeyaeo (9) — (11), iiaoaeiaaii ca aeiiiiiaith ia?anoaiiaee iia?aoeie
iaeneiicaoei? oa iaoaiaoe/iiai niiaeiaaiiy (aai iia?aoi?a eiiai?iinoi,
aai iia?aoi?a iiaenoiiaoaaiiy) — oae caaia ia?anoaiiai/ia ?aeaenaoeiy. O
aeaiiio aeiaaeeo iaaei iathoue aeaeyae iaoaiaoe/ieo niiaeiaaiue aiae
aeayeeo aeiaae eiaeo aaee/ei, iai?eeeaae, iioeiaeueieo cia/aiue aeayeeo
aeiaaeeiaeo iioeiicaoeieieo caaea/. Oaei iaaei iiaeia ia/eneeoe oi/ii
oieueee a ie?aieo aeiaaeeao, iai?eeeaae, aeey aeene?aoieo ?iciiaeieia
aeiaaeeiaeo aaee/ei, a o caaaeueiiio aeiaaeeo iiaeia aeei?enoaoe oieueee
?o noaoenoe/ii ioeiiee. Oaei iaaei aoee aaaaeaii oa aea/aii o ?iaioao
[13, 14, 19, 20] aeey ?ica’ycaiiy aeayeeo noioanoe/ieo aeene?aoieo oa
iaia?a?aieo caaea/ aeiaaeueii? iioeiicaoei?. O ca’yceo c aeiaaeeiaei
oa?aeoa?ii iaae, iaoiae aieie oa iaae iaaoaa? noioanoe/ieo ?en.

?icaeyiaii, iai?eeeaae, caaea/o noioanoe/iiai aeene?aoiiai
i?ia?aioaaiiy (9) — (11) aac iaiaaeaiue (10):

max xXD [F(x)= E f (x,)],
(12)

Iaoae X* — iiiaeeia ?ioaiue, a F* — iioeiaeueia cia/aiiy caaea/i.
C?iaeii ianooiii i?eiouaiiy.

I1. Iniothoue ooieoei? L:2X R oa U:2X R, oaei, ui aeey eiaeiiai ZX,

L(Z)F*(Z)=maxxZF(x)U(Z), U(Z)=F(x) aeey aeayeiai xZ,

i yeui Z ae?iaeaeo?oueny o oi/eo, oi L(Z)=F*(Z)=U(Z). Aeiionea?ii
oaeiae, yeui ZD = , oi oey neooaoeiy iiaea aooe iaeaioeoieiaaia i, ii
icia/aiith, L(Z)=U(Z)=+ .

Ooieoei? L oa U acaaaei aecia/athoueny ca aeiiiiiaith ?ica’ycaiiy
aeayeeo aeiiiiiaeieo noioanoe/ieo iioeiicaoeieieo caaea/, ui icia/aii ia
iiaeiiiaeeiao Z X. O iiae?icaeiei 5.5 ?icaeyaeathoueny niiniae
iiaoaeiae oaeeo aeiiiiiaeieo caaea/. O caaaeueiiio aeiaaeeo iiaeia
oieueee i?eioneaoe inioaaiiy noaoenoe/ieo ioeiiie aaee/ei L(Z) oa U(Z).

I2. O aeayeiio eiiai?i?niiio i?inoi?i (,,P) aeey eiaeii? iiaeiiiaeeie Z
X inio? iineiaeiaiinoue aeiaaeeiaeo aaee/ei l(Z), l=1,2,…, oa m(Z),
m=1,2,…, oaeeo, ui iaeaea iaiaaia

liml l(Z) = L(Z) , limm m(Z) = U(Z) .

Eiioeaiooaeueiee aeai?eoi aieie oa iaae.

O iaoiaei aieie oa iaae aeoiaeia «i?inoa» iiiaeeia X ioa?aoeaii
?icaeaa?oueny ia iiaeiiiaeeie Z X, ui ooai?ththoue ?icaeooy Pk
iiiaeeie X aai eiai /anoeie. Eeth/iao ?ieue a aeai?eoii a?aeia?athoue
?aei?aeii iiiaeeie Yk, oiaoi iiiaeeie, ui iathoue iaeiaioo ieaeith
iaaeo. Iaaeeaeaii ?ica’ycee xk caaea/i aeae?athoueny ic iiiaeei Xk, ui
iathoue iaeiaioo aa?oith iaaeo. A ii?o oiai ye ?aei?aeia iiiaeeia
?icaeaa?oueny ia iaioi iiaeiiiaeeie, aaia?othoueny iiai ioeiiee
oeieueiai? ooieoei? aeey onio iiaeiiiaeei, ciaoiaeyoueny iiai iaaeeaeaii
?ica’ycee, aeae? ioa?aoei? iiaoi?ththoueny. Oae ye iaaei aeiaaeeiai, oi
i ?aei?aeii iiiaeeie aeiaaeeiai, ioaea oni ia’?eoe, ui aaia?othoueny
aeai?eoiii, ? aeiaaeeiaeie.

Iiioeiaeicaoeiy. Noi?ioaaoe ii/aoeiaa ?icaeooy P0 =X. Ia/eneeoe iaaei 0
= l(0)(X) oa 0 = m(0)(X). Iieeanoe k=0.

?icaeooy. Aea?aoe ?aei?aeio iiaeiiiaeeio Ykargmink (Z): ZPk oa
aiaeiia?aeiee iaaeeaeaiee ?ica’ycie xkXkargmin k(Z): ZPk. sseui
?aei?aeia iiaeiiiaeeia ? oi/eith, oi iieeanoe Pk = Pk i eoe ia e?ie
Ioeiiea iaae. O i?ioeaiiio aeiaaeeo iiaoaeoaaoe ?icaeooy Pk(Yk)=Yik,
i=1,2,… ?aei?aeii? iiiaeeie Yk, oaea, ui Yk =i Yik i YikYjk= aeey
Yik, YjkPk(Yk), i j. Aecia/eoe iiaa iiaia ?icaeooy Pk = (PkYk )
Pk”(Yk).

Ioeiiea iaae. Aeey onio iiaeiiiaeei ZPk aeaa?aii aeayei ioeiiee

k (Z) = l(k,Z)(Z) i k(Z) = m(k,Z)(Z) aeey L(Z) i U(Z) aiaeiiaiaeii.

Aeaeaeaiiy iiaeiiiaeei. I/enoeoe ?icaeooy Pk aiae iaaeiionoeieo
iiaeiiiaeei, oiaoi aecia/eoe ?icaeooy Pk = Pk ZPk : ZD=. Iieeanoe
k:=k+1 i eoe ia e?ie ?icaeooy.

Caoaaaeaiiy. sseui ioeiiee ? oi/ieie, oiaoi k (Z) L(Z) i k(Z)U(Z),
oiaei ia e?ioei Aeaeaeaiiy iiaeiiiaeei iiaeia oaeiae aeaeaeyoe oni
iiaeiiiaeeie Z, aeey yeeo

k (Z) > k(Z).
(13)

Caiaeiinoue iaoiaeo anoaiiaeth?oueny ianooiiith oai?aiith.

Oai?aia 10 [13]. I?eionoeii, ui iiaeaene l(k,Z) i m(k,Z) aeae?athoueny
oaeei /eiii, ui yeui ZPk aeey ianeii/aiii? e?eueeino? k, oi iaeaea
iaiaaia limk l(k,Z) = limk m(k,Z) = +. Oiaei c eiiai?i?noth
iaeeieoey inio? iiia? ioa?aoei? k*, oaei?, ui aeey onio k k*:

1) ?aei?aeii iiiaeeie Yk ? oi/eaie i Yk X*;

2) iaaeeaeaii ?ica’ycee xk X*.

Eiiai?iinii ioeiiee oi/iinoi iioi/ieo iaaeeaeaiue xk iiaeooue aooe
iaea?aeaii ca ianooiieo aeiaeaoeiaeo i?eiouaiue.

I3. Aeey eiaeiiai Z X aeiaaeeiai aaee/eie k (Z), k(Z), k=0,1,…,
iacaeaaeii i ii?iaeueii ?iciiaeieaii c na?aaeiiie cia/aiiyie L(Z), U(Z)
oa aeenia?niyie k(Z), k (Z) aiaeiiaiaeii.

I4. Aeey aeenia?nie k(Z), k (Z), Z X, aiaeiii aa?oii iaaei:

k(Z) k(Z), k (Z) k (Z).

Aicueiaii aeiai?/i iaaei aeey L(Z), U(Z) o aeaeyaei

k (Z) =k (Z)c(k)k(Z), k(Z) = k(Z) c(k)k (Z).

aea eiinoaioe c(k), k=0,1,…, oaei, ui

(ck)=(1/2)-c(k) e-/2 d = 1 k , 0 < k < 1. Eaia 3 [13]. A i?eiouaiiyo I3, I4 aeey eiaeiiai k ni?aaaaeeeaa ianooiia ioeiiea oi/iinoi: PF(xk) F* k(Xk) mink (Z)|ZPk (1 k). Aaaeeeaith ?enith iaoiaeo aieie oa iaae ? iiaeeeainoue aeaeaeyoe ca oiiaith (13) iaia?niaeoeaii iiaeiiiaeeie ic iioi/iiai ?icaeooy ca aeiiiiiaith ieaeiio oa aa?oiio iaae iioeiaeueieo cia/aiue oeieueiai? ooieoei? ia iiaeiiiaeeiao ?icaeooy. A noioanoe/iiio aeiaaeeo, iaeia/a, a neeo aeiaaeeiainoi iaae aeaeaeaiiy iiaeiiiaeei iiaea i?ecaanoe aei ao?aoe iioeiaeueiiai ?ica'yceo. O ianooiiiio i?aaeei aeaeaeaiiy iiaeiiiaeei ie ia aeaeaey?ii ?o ia eiaeiie ioa?aoei?, a ?iaeii oea oieueee iiney aeeiiaiiy aeineoue aaeeei? eieueeinoi ioa?aoeie N, a oaeiae iiney ia/eneaiiy iacaeaaeii? ioeiiee cia/aiiy oeieueiai? ooieoei? o oi/oei iioi/iiai iaaeeaeaiiai ?ica'yceo. C?iaeii ianooiia aeiaeaoeiaa i?eiouaiiy. I5. Inio? ?iaiiii?ia iaaea 2 aeey aeenia?nie onio aeiaaeeiaeo aaee/ei k (Z) i k(Z), Z X, k=1,2,... I?aaeei aeaeaeaiiy iiaeiiiaeei. Iiney N e?ieia coieiy?iiny, aicueiaii ?aei?aeio iiaeiiiaeeio XN (aai YN ) ic oiiaeueiiai ?icaeooy PN oa c?iaeii N aeiaeaoeiaeo niinoa?aaeaiue Ni(XN ), i=1,...,N, i ia/eneeii iiao ioeiieo aeey U(XN ): U(XN)= (1/N) 1N Ni (XN ). Oiaei aeey aeayei? oi/iinoi (0,1) aeaeaeeii oni iiaeiiiaeeie ZPN oaei, ui LN(Z) > U(XN) + 2cN , aea cN = 2/(N).

Eaia 4 [14]. Iaoae x* — aeayeee ?ica’ycie caaea/i (12). Oiaei

P x* ao?a/a?oueny iiae /an aeaeaeaiiy 2.

O iiae?icaeiei 5.3 oa o [20] noioanoe/iee iaoiae aieie oa iaae
iioe?th?oueny ia caaea/i c iaeiiieieie noioanoe/ieie iaiaaeaiiyie (10),
a o ?icaeiei 5.4 oa o [14] — ia iaia?a?aii aaaaoiaeno?aiaeueii caaea/i
noioanoe/iiai i?ia?aioaaiiy.

Iniiaia i?iaeaia o noioanoe/iiio iaoiaei aieie oa iaae — oea ioeiiea
iaae iioeiaeueieo cia/aiue aeey caaea/ noioanoe/iiai i?ia?aioaaiiy.

sse aa?oith iaaeo U(X) iioeiaeueiiai cia/aiiy F*(X) iiaeia acyoe
cia/aiiy oeieueiai? ooieoei? o aeayeie aeiionoeiie oi/oei x’ X:

U(X) = F(x’) = E f(x’,).

Aaaeeeai aea?aoe oi/eo x’ oaeei /eiii, uia cia/aiiy F(x’) aoei yeiiiaa
iaioei. Oaei oi/ee iiaeooue aooe iaeaeaii aoaeue-yeei (iiaeeeai,
aa?inoe/iei) iaoiaeii eieaeueii? noioanoe/ii? iioeiicaoei?.

Aeey iiaoaeiae ieaeiio iaae a iiae?icaeiei 5.5 ?icaeyaeathoueny aeaa
caaaeueieo iiaeoiaee: ia?anoaiiaea iia?aoi?ia iiiiiicaoei? oa
iaoaiaoe/iiai niiaeiaaiiy aai iia?aoi?a eiiai?iinoi (ia?anoaiiai/ia
?aeaenaoeiy) i aeai?noi ioeiiee. E?ii oiai, aeayei aiaeiii
aeaoa?iiiiaaii iaoiaee iiaoaeiae iaae (oaei, ye ?aeaenaoeiy oiia
oeiei/enaeueiinoi) iiaeooue aooe aeei?enoaii o eiiaiiaoei? c cacia/aieie
iiaeoiaeaie.

Iaeay ia?anoaiiai/ii? ?aeaenaoei? aeey caaea/i (12) ?ethno?iaaia
ianooiiith ia??ai?noth [13, 14, 19]:

F*(X)=minxX Ef(x,) E minxXf(x,) = Ef(x*(),), (14)

aea x*( ) argmin xX f(x, ). Oaeei /eiii, aaee/eia L(X)=Ef(x*(),) aea?
ieaeith ioeiieo iioeiaeueiiai cia/aiiy F*(X). O aaaaoueio aeiaaeeao aeey
oieniaaiiai ?ica’ycie x*( ) iiaea aooe ciaeaeaii ii?iaiyii eaaei.
Neeaaei?noue caaea/ noioanoe/i? aeene?aoii? iioei?caoe?? aeyaey?oueny
ooo o oiio, ui aeey io?eiaiiy ioe?iee (14) iaiao?aeii ?ica’ycaoe aaaaoi
(oiaoi aeey eiaeiiai ) aeaoa?i?iiaaieo ioe?ii/ieo caaea/ aeaeyaeo minxX
f(x,).

I?inoee oeyo iieiioeoe ieaeith iaaeo (14) oa ?? ioeiieo Iiioa -Ea?ei
iieyaa? o aeei?enoaiii M iacaeaaeieo eiiie l aeiaaeeiai? aaee/eie :

F*(X)=(1/M)minxX 1mEf(x, l)

EM minxX(1/M)1Mf(x,l) =LM(X),
(15)

aea EM — iia?aoi? iaoaiaoe/iiai niiaeiaaiiy c on?o l . Aieueoa oiai,
caieueooth/e /enei niinoa?aaeaiue M ana?aaeeii iia?aoei? iiiiiicaoei?,
iiaeia c?iaeoe oi/iinoue aiii?e/ieo ioeiiie LM (X) ye caaaiaeii
aenieith.

Aeei?enoaiiy aiii?e/ieo ioeiiie (15) — ia ?aeeiee niinia aeei?enoaoe
e?aoii niinoa?aaeaiiy. Iaoae ciiao l, l=1,…,M (iaia?ia) — iacaeaaeii
eiii? . Oiaei caaea/a iiiiiicaoei? F(x) aeaiaaeaioia iiiiiicaoei? c x X
ooieoei?

(F(x))M=(E f(x,))M = 1M E f(x,l ) = E 1M f(x,l ),

aea a inoaiiie ?iaiinoi ie aeei?enoaee iacaeaaeiinoue l, l=1,…,M.
Ia?anoaiiaea iia?aoi?ia iiiiiicaoei? oa iaoaiaoe/iiai niiaeiaaiiy
i?ecaiaeeoue aei ianooiii? noioanoe/ii? iaaei:

minxX F(x) (E minxX 1M f(x,l ))1/M.
(16)

sseui ln f(·,) — oaiaiooa ooieoeiy, oi aioo?ioiy iioeiicaoeieia caaea/a
a (16) aeaiaaeaioia caaea/i iioeeiai i?ia?aioaaiiy: minxX1Mlnf(x,l).
Inio? oe?ieee eean oae caaieo -oaiaiooeo ooieoeie f(·,) (ye?
?icaeyaeaeeny o iiae?icaeiei 4.2), aeey yeeo aioo?ioiy caaea/a a (16)
oaeiae caiaeeoueny aei caaea/i iioeeiai i?ia?aioaaiiy.

O iiae?icaeiei 5.5 iaoiae ia?anoaiiai/ii? ?aeaenaoei? i?iiethno?iaaii
ia caaea/ao oiiainoaaiiy i?iaeoia, ?iciiuaiiy ae?iaieoeoaa, iioeiicaoei?
ii?ooaey oeiiieo iaia?ia, eiio?ieth iaae aeeeaeii caa?oaeiaiue.

Ieaeii iaaei aeey F*(X) iiiaei iiaeia io?eiaoe aeoiaey/e ic aeanoeainoi
iiiioiiiinoi aeiaaeeiai? oeieueiai? ooieoei? f(x,) (i, ioaea,
iiiioiiiinoi F(x)). Iiiioiiiinoue aeiaaeeiaeo iieacieeia ooieoeiiioaaiiy
nenoai f(x,) iiaea aeiioneaoeny uiaei oaeeo ciiiieo x, ye iiaanoeoei?,
?ano?ne, i?iaeoeoeaiinoue i o. ii. Ieaeii iaaei aeey F*(X) oaeiae
iiaeooue aooe io?eiaii ca aeiiiiiaith noioanoe/ieo aeioe/ieo iiii?aio
ooieoeie f(x,) oa iaoiaeo ia?anoaiiai/ii? ?aeaenaoei?. Aeaoa?iiiiaaii
aeioe/ii iiii?aioe o eiioaenoi aeiaaeueii? iioeiicaoei? aoee aaaaeaii
N.A.Iiyanueeei Ieyaneee N.A. Ia iaeiii aeai?eoia iienea aaniethoiiai
yeno?aioia ooieoeee // AEo?i. au/ene. iaoaiaoeee. e iao. oeceee. — 1972.
— 12, N 4. — C.888-896. oa aeei?enoaii o [8]. A i?ae?icaeiei 5.5 oa o
[19] ie aeei?enoiao?ii noioanoe/ii aeioe/ii iiii?aioe oeieueiai?
ooieoei? F(x) aeey iio?aa aeiaaeueii? noioanoe/ii? iioeiicaoei?.

Aeai?noi ioeiiee a eiiaiiaoei? c iaoiaeaie iaaeaaeei? iioeiicaoei?
oe?iei aeei?enoothoueny a aeaoa?iiiiaaiiio aeene?aoiiio i?ia?aioaaiii
Aeea., iai?eeeaae, Shor N.Z. Nondifferentiable optimization and
polynomial problems. — Dordrecht: Kluver Academic Publishers, 1998. —
600 p.

. O iiae?icaeiei 5.5 oaeiae ?icaeyaeathoueny iniaeeainoi aeai?noeo
ioeiiie o eiiaiiaoei? c iaoiaeii ia?anoaiiai/ii? ?aeaenaoei? noiniaii
aei caaea/ noioanoe/ii? iioeiicaoei?.

O iiae?icaeiei 5.6 aoaeothoueny iaaei aeey eiiai?iinoae. ?icaeyiaii
caaea/o

maxxX[P(x)=P f(x, )B]= P*(X),

aea X Rn, f(x,)=(f1(x,),…,fm(x,)) — aeiaaeeiaa aaeoi?-ooieoeiy; B —
caieiaia iiaeiiiaeeia a Rm.

Ioeiieii caa?oo P*(X) ca aeiiiiiaith ia?anoaiiaee iia?aoi?ia
iaeneiicaoei? oa eiiai?iinoi (ia?anoaiiai/ia ?aeaenaoeiy). I/aaeaeii,

P*(X) P x()X: f(x(),) B= U(X)

P x()conv X: f(x(),) B= U(X), (17)

aea conv X — iioeea iaieiiea iiiaeeie X.

Iai?eeeaae, uia ia/eneeoe noioanoe/ii ioeiiee (X’,)=A(X’)() aaee/eie
U(X’), o?aaa ia?aai?eoe aeey aeaiiai aeiionoeiinoue oiia f(x’,)B, x’X.
sseui ooieoei? fi(x,), i=1,…,m, eiiieii ii x, X i B — aaaaoia?aiii
iiiaeeie, oi caaea/a ia?aai?ee niieueiinoi oiia f(x’,)B, x’X ? caaea/ath
eiiieiiai oeiei/eneiaiai i?ia?aioaaiiy (i eiiieiith caaea/ath aeey oiia
f(x’,)B, x’ conv X).

Iiaeia c?iaeoe iaaei (17) oi/iioeie, iaeii/anii aeei?enoiaoth/e
aeaeieueea iacaeaaeieo niinoa?aaeaiue (1,…,l )= l aeiaaeeiai?
aaee/eie . Oiaei

Ul(X)= P1/l x()X: f(x(l),1)B,…, f(x(l),l)B

? aa?oiy iaaea aeey eiiai?iinoae P(x), xX.

O iiae?icaeiei 5.7, a oaeiae o [13, 14, 20] iaaiaeyoueny i?eeeaaee
/enaeueiiai ?ica’ycaiiy caaea/ noioanoe/iiai aeene?aoiiai i?ia?aioaaiiy,
noioanoe/ii? aeiaaeueii? iioeiicaoei? oa aeiaaeueii? iioeiicaoei?
eiiai?iinoae noioanoe/iei iaoiaeii aieie oa iaae.

6. Iaoiae aiii?e/ieo na?aaeiio.

O ?icaeiei 6 oa o [6] aiaeiiee iaoiae aiii?e/ieo na?aaeiio (aai
noaoen-oe/iee iaoiae) iioe?th?oueny ia caaea/i noioanoe/iiai
i?ia?aioaaiiy, ui iinoyoue neeaaeii (neeaaeaii i iaaeeoa?aioeieiaii)
ooieoei? ?eceeo. Caiaeiinoue iaoiaeo aiii?e/ieo na?aaeiio o?aeoo?oueny
ye ianeiaeie noieeinoi aeayei? caaea/i ooieoeiiiaeueiiai ia?aiao?e/iiai
i?ia?aioaaiiy. Oaeaeeinoue caiaeiinoi iaoiaeo aea/a?oueny ia iniiai
iiiyooy ii?iaeiciaaii? caiaeiinoi aeiaaeeiaeo aaee/ei.

?icaeyiaii caaea/o noioanoe/iiai i?ia?aioaaiiy aeaeyaeo

F(x) = E0(x,E0(x,),…,Em(x,),) minxX,
(18)

aea

fj(x) = Ej(x,) = 0(x,)P(d), jI=1,…,m;

g0(x,y) = E0(x,y,)= 0(x,y,)P(d), yY Rm;

X — eiiiaeo a Rn; Y — aeayea iiiaeeia a Rm; ?, (,,P) — aeayeee
eiiai?i?niee i?inoi?; j :X R1 i 0 :XY R1 — iioaa?iaii i?e
eiaeiiio x X i y Y ooieoei?. A /anoeiaeo aeiaaeeao iiaea aooe F(x) =
E0(x,), F(x) =maxjJ Ej(x,), F(x)=D0(x,)=E02(x,) (E0(x,))2 i o.i.

Iaoiae aiii?e/ieo na?aaeiio (aai noaoenoe/iee iaoiae) aeey
?ica’ycaiiy caaea/i (18) iieyaa? a ianooiiiio: aiia caiiith?oueny
iineiaeiaiinoth caaea/ aeaeyaeo

FS(x)=(1/s)1s0(x,(1/s)1s0(x,k),…,(1/s)1sm(x,k),k) minxX,

(19)

aea k — iacaeaaeii iaeiaeiai ?iciiaeieaii aeiaaeeiai aaee/eie
(niinoa?aaeaiiy), ?iciiaeie yeeo aecia/a?oueny ii?ith P.

Iicia/eii F*, Fs* i X*, Xs* iioeiaeueii cia/aiiy oa ?ica’ycee caaea/
(18), (19), X*=xX| F(x) F*+, Xs*= xX| Fs(x) Fs*+ — iaaeeaeaii
?ica’ycee caaea/ (18), (19), 0.

Iaoa iaoa iieyaa? a oiio, uia aeineiaeeoe caiaeiinoue, o aeayeiio

eiiai?i?niiio naini, aeiaaeeiaeo aaee/ei Fs*, Xs*, Xs* aei
aiaeiiaiaeieo cia/aiue F*, X*, X*.

Ianooiia oai?aia aecia/a? oiiae caiaeiinoi iaeaea iaiaaia iaoiaeo
aiii?e/ieo na?aaeiio aeey ?ica’ycaiiy caaea/i iiiiiicaoei? neeaaeii?
ooieoei? ?eceeo i?e aeaoa?iiiiaaieo iaiaaeaiiyo.

Oai?aia 11 (caiaeiinoue). I?eionoeii, ui

1) ooieoei? j(x,), j=0,…,m, iaia?a?aii ii xX, a 0(x,y,)
iaia?a?aia ii (x,y)X Rm ca anio , oa aeii?ii ii ca oieniaaieo x,y;

2) iniothoue iioaa?iaii ooieoei? Cj(), j=0,…,m, oaei, ui aeey
anio xX, |j(x,)| Cj();

3) aeey aoaeue-yeiai eiiiaeoo YRm inio? iaaiae’?iia iioaa?iaia
ooieoeiy D(), oaea, ui aeey anio (x,y)XY, |0 (x,y,)|D().

Oiaei ooieoei? 0 , j, F iaia?a?aii, Fs* F* i aiaeoeeaiiy (Xs*,X*)0
iaeaea iaiaaia.

Ianooiia oai?aia aecia/a? oiiae caiaeiinoi oa oaeaeeinoue caiaeiinoi
(ii?yaeeo 1/s ) ca ooieoeiiiaeii aeey iaoiaeo aiii?e/ieo na?aaeiio.

Oai?aia 12 (oaeaeeinoue caiaeiinoi). I?eionoeii, ui

1) ooieoei? j(x,), j=0,…,m, iaia?a?aii ii xX, a 0(x,y,) iaia?a?aia
ii (x,y)X Rm ca anio , oa aeii?ii ii ca oieniaaieo x,y;

2) ooieoei? j(x,) i 0(x,y,) iioaa?iaii (ii ) a eaaae?aoi aeey anio xX
i yRm;

3) iniothoue iioaa?iaii a eaaae?aoi ooieoei? Lj(), j=1,…,l, oaei,
ui aeey aoaeue-yeeo x1,x2X aeeiiaii

|j(x1,) j(x2,)| Lj() x1 x2;

4) aeey aoaeue-yeiai eiiiaeoo Y Rm iniothoue iioaa?iaii a
eaaae?aoi ooieoei? M0() oaei, ui aeey aoaeue-yeeo x1, x2 X ? y1, y2
Y aoaea

|0(x1,y1,) 0(x2,y2,)| M0() (x1 x2 + y1 y2 ).

Oiaei Fs* F* i (Xs*,X*) 0 iaeaea iaiaaia, i e?ii oiai, Fs*()
F* ii?iaeiciaaii (aeea. icia/aiiy 7) ci oaeaeeinoth 1/s . sseui
aeiaeaoeiai aei oiia 1)-4) aeeiiaii

5) ooieoeiy F(x) a caaea/i (17) iioeea ia iioeeie eiiiaeoiie
iiiaeeii X Rn,

oi a?aeoeeaiiy (Xs*,X*) 0 ii?iaeiciaaii ci oaeaeeinoth 1/s.

O iiae?icaeiei 6.6 aea/a?oueny oae caaia ii?iaeiciaaia caiaeiinoue
aeiaaeeiaeo aaee/ei, ui aeei?enoiao?oueny a oai?aii 12.

O oai?i? eiiai?iinoae aiaeiii ianooiii iniiaii aeaee caiaeiinoi
aeiaaeeiaeo aaee/ei: iaeaea iaiaaia, a na?aaeiueiio, ca eiiai?i?noth oa
?iciiaeieii. I?e aea/aiii oaeaeeinoi caiaeiinoi (neaaeiii, aei ioey)
aeiaaeeiai? iineiaeiaiinoi n , n=1,2,…, oaeiae ?icaeyaeathoue
caiaeiinoue (iaeaea iaiaaia i o.i.) iineiaeiaiinoae aeaeyaeo nn, aea
/eneiaa iineiaeiaiinoue n 0 oaea, ui limn 1/n=0. O oiio aeiaaeeo, eiee
limnPnn =0 >0, aiai?youe i?i caiaeiinoue n aei ioey ca eiiai?i?noth ci
oaeaeeinoth 1/n. *anoi i?e aeineiaeaeaiii noioanoe/ieo ioa?aoeieieo
aeai?eoiia iioeiicaoei? i a i?eeeaaeieo canoinoaaiiyo iaoiaeo aiii?e/ieo
na?aaeiio io?eiothoueny ioeiiee oaeaeeinoi caiaeiinoi, yei (iiney
aeaiaioa?ieo ia?aoai?aiue) iathoue aeaeyae

Pnn 0. Ci caiaeiinoi o na?aaeiueiio aeieeaa? ia
oieueee caiaeiinoue ca eiiai?i?noth (ui caaaeueiiaiaeiii), aea e
ii?iaeiciaaia caiaeiinoue aiaeiiaiaeieo aeiaaeeiaeo aaee/ei.
Ii?iaeiciaaia caiaeiinoue caa?iaa?oueny i?e aaeueaea?iaeo ia?aoai?aiiyo
aeiaaeeiaeo aaee/ei. Aiia caa?iaa?oueny oaeiae i?e aeiaeaaaiii,
iiiaeaiii, aeieaiii oa aeaea?oiaiio aeiaooeo ii?iaeiciaaii caiaeieo
iineiaeiaiinoae aeiaaeeiaeo aaee/ei. Eiaeia a?aie/ia oai?aia oai?i?
eiiai?iinoae oyaia ca niaith ii?iaeiciaaio caiaeiinoue aiaeiiaiaeieo
aeiaaeeiaeo aaee/ei. Aeey ii?iaeiciaaii? caiaeiinoi ia? iinoea
e?eoa?ie caiaeiinoi oeio e?eoa??th Eioi.

Ioaea, na?aae iineiaeiaiinoae, ui caiaathoueny ca eiiai?i?noth c
aeayeith oaeaeeinoth, ii?iaeiciaaia caiaeiinoue aeeo/a? ii-?iciiio
caiaeii iineiaeiaiinoi. Aeey ii?iaeiciaaii caiaeieo aeiaaeeiaeo aaee/ei
iiaeia aoaeoaaoe aeiai?/i iaeanoi (/iai iaiiaeeeai c?iaeoe aeey
caaaeueieo, caiaeieo ca eiiai?i?noth c aeayeith oaeaeeinoth, aeiaaeeiaeo
aaee/ei), i, e?ii oiai, o canoinoaaiiyo /anoi aeiaiaeeoueny aieueoa,
iiae caiaeiinoue ca eiiai?iinoth c aeayeith oaeaeeinoth, a naia
ii?iaeiciaaia caiaeiinoue.

AENIIAEE

1. O aeena?oaoe?? ?icaeyiooi aaeeeo e?euee?noue i?eeeaaeieo caaea/
iaiioeeiai noioanoe/iiai i?ia?aioaaiiy c iaaeaaeeeie, ?ic?eaieie aai
aeene?aoieie ooieoe?yie, aeey yeeo iaia? aaeaeaaoieo iaoiae?a
?ica’ycaiiy.

2. Aeine?aeaeaii aeanoeaino? caaea/ iaiioeeiai noioanoe/iiai
i?ia?a-ioaaiiy: iiaeae? iaiioeeeo iaaeaaeeeo caeaaeiinoae,
noaaeeoa?aioe?aeuei? aeanoeaino? ooieoe?e i/?eoaaii? ei?eniino?,
iaiao?aei? oiiae iioeiaeueiino?, ioe?iee iioeiaeueieo cia/aiue.

3. ?ic?iaeai? oa aeine?aeaeai? noioanoe/i? iaoiaee ?ica’ycaiiy caaea/
iaiioeeiai noioanoe/iiai i?ia?aioaaiiy, iiaoaeiaai? ia aeei?enoaii?
?aae?caoe?e aeiaaeeiaeo ooieoe?e caaea/? aai ?o a?aae??io?a.

4. Cie?aia, a?aeiiee iaoiae noioanoe/ieo eaac?a?aae??io?a
TH.I.??iieue?aa iioe?aiee ia caaea/? eieaeueii? iioei?caoe??
noioanoe/ieo nenoai c aeene?aoieie iiae?yie, caaea/? iioei?caoe??
ooieoe?e i/?eoaaii? ei?eniino?, caaea/? iioei?caoe?? eiia??iinoae c
iaiioeeeie iaaeaaeeeie oa ?ic?eaieie ooieoe?yie.

5. Cie?aia, ?ic?iaeaii oa iaa?oioiaaii noioanoe/iee iaoiae a?eie oa
iaae aeey ?ica’ycaiiy caaea/ noioanoe/ii? aeiaaeueii? oa noioanoe/ii?
aeene?aoii? iioei?caoe??, aeiaaeueii? iioei?caoe?? eiia??iinoae.

I, ia caeii/aiiy, aaoi? aaey/iee ia?aae/anii a?ae?eoeiio ni?aaaoi?o
iiiia?ao?? [1] aeaaeai?eo Aieiaeeie?o Na?a?eiae/o Ieoaeaae/o.

Aaoi? ue?i aeyeo? nai?i eieaaai oa niiaaaoi?ai TH.I.??iieue?ao,
A.I.Aoiaeo, I.A.?i?ieo, ?.Aaono, A.?oueinueeiio, A.Ioethao ca iaioeiieio
aeiiiiiao i niia?iaioieoeoai i?ae /an aeeiiaiiy oei?? ?iaioe.

Iniiai? ?acoeueoaoe aeena?oaoe?? iioae?eiaai? a oaeeo i?aoeyo:

1. Ieoaeaae/ A.N., Aoiae A.I., Ii?eei A.E. Iaoiaeu iaauioeeie
iioeiecaoeee. — I.: Iaoea, 1987. — 286 n.

2. Aoiae A.I., Ii?eei A.E. Aeai?eoi ieieiecaoeee ?ac?uaiuo ooieoeee //
Eeaa?iaoeea. — 1977. — N 2. — C.73-75.

3. A?iieueaa TH.I., Ii?eei A.E. Ii?iaeeciaaiiay noiaeeiinoue
neo-/aeiuo aaee/ei e aa i?eiaiaiey //Eeaa?iaoeea. -1990.-N 6.-C.89-93.

4. Ii?eei A.E., ?iaiei I.A. a-aiaiooua ooieoeee e ia?u e eo
i?eiaiaiey // Eeaa?iaoeea e nenoaiiue aiaeec. — 1991. — N 6. —

N.77-88.

5. Ermoliev Yu.M., Norkin V.I. Normalized convergence in stochastic
optimization // Ann. of Oper. Res. 1991. 30. P.187-198.

6. Ii?eei A.E. Ia oneiaeyo e nei?inoe noiaeeiinoe iaoiaea
yiie-?e/aneeo n?aaeieo a iaoaiaoe/aneie noaoenoeea e noioanoe/aneii
i?i-a?aiie?iaaiee // Eeaa?iaoeea e nenoaiiue aiaeec. — 1992. — N 2. —
C.107-120.

7. Ii?eei A.E. Ii?iaeeciaaiiay noiaeeiinoue neo/aeiuo aaee/ei //
Eeaa?iaoeea e nenoaiiue aiaeec. — 1992. — N 3. — C.84-92.

8. Ii?eei A.E. I iaoiaea Ieyaneiai aeey ?aoaiey iauae caaea/e
aeiaaeueiie iioeiecaoeee //AEo?i. au/ene. iaoaiaoeee e iao. oeceee. —
1992. — 32, N 7. — C. 992-1006.

9. Ermoliev Yu.M, Norkin V.I. and Wets R.J-B. The minimization of
semi-continuous functions: Mollifier subgradients // SIAM J. Contr. and
Opt. 1995. N 1. P.149-167.

10. Ermoliev Yu.M., Norkin V.I. Om nonsmooth and discontinuous problems
of stochastic systems optimization // European J. of Oper. Research.
1997. 101. P. 230-244.

11. A?iieueaa TH.I., Ii?eei A.E. Noioanoe/aneee iaiauaiiue
a?aaeeaioiue iaoiae aeey ?aoaiey iaauioeeuo iaaeaaeeeo caaea/
noioan-oe/aneie iioeiecaoeee//Eeaa?iaoeea e nenoaiiue aiaeec.- 1998. —

N 2.- N.50-71.

12. A?iieueaa TH.I., Ii?eei A.E. I ianoaoeeiia?iii caeiia aieue-oeo
/enae aeey caaeneiuo neo/aeiuo aaee/ei e aai i?eiaiaiee a
noi-oanoe/aneie iioeiecaoeee //Eeaa?iaoeea e nenoaiiue aiaeec. — 1998. —
N 4. — N.94-106.

13. Norkin V.I., Ermoliev Yu.M. and Ruszczynski A. On Optimal
Allocation of Indivisibles under Uncertainty //Operations Research.
1998. 46, N 4. P.381-395.

14. Norkin V.I., Pflug G.Ch. and Ruszczynski A. A Branch and Bound
Method for Stochastic Global Optimization // Mathematical Programming.
1998. 83. P.425-450.

15. Ii?eei A.E. Iino?iaiea ?aeaenaoeeiiiuo iaoiaeia iaauioeeie
iaaeaaeeie iioeiecaoeee // Iao. iaoiaeu aiaeeca e iioeiecaoeee nenoai,
ooieoeeiie?othueo a oneiaeyo iaii?aaeaeaiiinoe / Iiae ?aae.
TH.I.A?iieueaaa, E.I.Eiaaeaiei. — Eeaa: Ei-o eeaa?iaoeee A.I.Aeooeiaa AI
ONN?, 1986. — N.35-41.

16. Ii?eei A.E. Iaoiae i?eaaaeaiiiai a?aaeeaioa aeey ?aoaiey caaea/
iaaeaaeeiai e noioanoe/aneiai i?ia?aiie?iaaiey // Iao. iaoiaeu aiaeeca e
iioeiecaoeee neiaeiuo nenoai, ooieoeeiie?othueo a oneiaeyo
iaii?aaeaeaiiinoe. — Eeaa: Ei-o eeaa?iaoeee ei. A.I.Aeooeiaa AI ONN?,
1989. — N. 4-9.

17. Ii?eei A.E. I oeaei/eneaiiii noioanoe/aneii i?ia?aiie?iaaiee //
Iaoaiaoe/aneea iaoiaeu iiaeaee?iaaiey e nenoaiiiai aiaeeca a oneiaeyo
iaiieiie eioi?iaoeee. — Eeaa: Ei-o eeaa?iaoeee ei. A.I.Aeooeiaa AI ONN?,
1991. — N. 28-34.

18. Ii?eei A.E. Iioeiecaoeey ooieoeee ?enea // Iiaeaee e iaoiaeu
enneaaeiaaiey iia?aoeee, oai?ee ?enea e iaaeaaeiinoe. — Eeaa: Ei-o
eeaa?iaoeee ei. A.I.Aeooeiaa AI Oe?aeiu, 1992. — N.13-23.

19. Ii?eei A.E. Aeiaaeueiay noioanoe/aneay iioeiecaoeey: iaoiae aaoaae
e aa?iyoiinoiuo a?aieoe // Iaoiaeu oi?aaeaiey e i?eiyoey ?aoaiee a
oneiaeyo ?enea e iaii?aaeaeaiiinoe. — Eeaa: Ei-o eeaa?iaoeee ei.
A.I.Aeooeiaa AI Oe?aeiu, 1993. — N. 3-12.

20. Norkin V.I. Global Optimization of Probabilities by the Stochastic
Branch and Bound Method / Lecture Notes in Econ. and Mat. Systems 458.
Stochastic Progr. and Tech. Appl. (K.Marti and P.Kall, eds.).Berlin:
Springer, 1998. P.186-201.

21. Ermoliev Yu.M., Norkin V.I. Constrained optimization of
discontinuous systems / Lecture Notes in Econ. and Mat. Syst. 458.
Stochastic Progr. and Tech. Appl. (K.Marti and P.Kall, eds.). Berlin:
Springer, 1998.P.128-142.

22. Ii?eei A.E. Iioeiecaoeey aa?iyoiinoae. — Eeaa, 1989. — 19 c. —
(I?ai?./ AI ONN?. Ei-o eeaa?iaoeee ei. A.I.Aeooeiaa; 89-6).

23. Ii?eei A.E. Onoie/eainoue noioanoe/aneeo iioeiecaoeeiiiuo iiaeaeae
e noaoenoe/aneea iaoiaeu noioanoe/aneiai i?ia?aiie?iaaiey. — Eeaa,
1989. — 24n. — (I?ai?./ AI ONN?. Ei-o eeaa?iaoeee ei. A.I. Aeooeiaa;
89-53).

24. Norkin V.I. The anaysis and optimization of probability functions
//Working Paper WP-93-6, Int. Inst. for Appl. Syst. Anal. Laxenburg,
Austria, 1993. 23 p.

25. Ermoliev Yu.M. and Norkin V.I. Stochastic generalized gradient
method with application to insurance risk management // Interim Report
IR-97-021, Int. Inst. for Appl. Syst. Anal. Laxenburg, Austria, 1997.
19p.

?iaioe [5, 9, 10, 13, 14, 21, 24, 25] iioa?eiaai? oaeiae o aeaeyae?

i?ai?eio?a o I?aeia?iaeiiio ?inoeooo? i?eeeaaeiiai nenoaiiiai aiae?co
(Eaenaiao?a, Aano??y) ? aeinooii? /a?ac aae?ano o ia?aae? Internet
http://www.iiasa.ac.at/Publications/ .

O ?iaioao, iioaeieiaaieo o niiaaaoi?noai, iniaenoi caeiaoaa/ai io?eiaii
ianooiii ?acoeueoaoe:

[1] — ia?aa?aoe 1, 14, 24, iiaiinoth aeaae 3, 6; [2, 11, 25] — oai?aie
caiaeiinoi iaoiae?a; [3, 5] — iiiyooy ii?iaeiciaaii? caiaeiinoi
(aaaaeaii a [23]), aeineiaeaeaiiy ?? aeanoeainoae; [4] — icia/aiiy
a-oaiaiooeo ooieoeie /a?ac oaiaiooi oa iioeei ooieoei?, aeineiaeaeaiiy
?o noaaeeoa?aioeiaeueieo aeanoeainoae, canoinoaaiiy aei ia/ioei?
iioeiicaoei?; [9, 10, 21] — iiiyooy neeueii iaiiaiaia?a?aieo,
iaia?a?aieo ca iai?yieaie, eoni/ii iaia?a?aieo ooieoeie, caeaaeaeaiiai i
?icoe?aiiai caeaaeaeaiiai noaaeeoa?aioeiaeia, caeaaeaeaii? ocaaaeueiaii?
iioiaeii?, aeineiaeaeaiiy ?o ca’yceia oa aeanoeainoae, oaeo
aiicaiaeiinoi ona?aaeiaieo ooieoeie, iaiaoiaeii oiiae aeno?aioio aeey
?ic?eaieo ooieoeie; [12] — aeiaaaeaiiy ianoaoeiiia?ieo caeiiia aaeeeeo
/enae iaoiaeii ooieoe?e Eyioiiaa; [13, 14] — iaeay ia?anoaiiai/ii?
?aeaenaoei? (aeneiaeaia a [19]) a ?aieao iaoiaeo aieie oa iaae, ??
?aaeicaoeiy aeey ?iciiiaiioieo eeania caaea/ noioanoe/ii? iioeiicaoei?.

Ii?e?i A.?. Noioanoe/i? iaoiaee ?ica’ycaiiy caaea/ iaiioeeiai
noioanoe/iiai i?ia?aioaaiiy oa ?o canoinoaaiiy. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.05.01 — oai?aoe/i?
iniiae ?ioi?iaoeee ? e?aa?iaoeee. — ?inoeooo e?aa?iaoeee ?i.
A.I.Aeooeiaa IAI Oe?a?ie, Ee?a, 1998.

Aeena?oaoeith i?enay/aii iaoiaeai ?ica’ycaiiy caaea/ iaiioeeiai
noioanoe/iiai i?ia?aioaaiiy, aeeth/ath/e eieaeueio oa aeiaaeueio
noioanoe/io iioeiicaoeith, oeiei/eneaiia noioanoe/ia i?ia?aioaaiiy,
eieaeueio oa aeiaaeueio iioeiicaoeith eiiai?iinoae oa ooieoe?e
niiae?aaii? ei?eniino?, noioanoe/io iioeiicaoeith ?ic?eaieo ooieoeie.
Iaoiae noioan-oe/ieo eaacia?aaei?ioia TH.I.??iieue?aa ocaaaeueiaii oa
iioe?aii ia iaiioee? noioanoe/ii caaea/i eieaeueii? iioeiicaoei?
ocaaaeueiaii aeeoa?ai-oeieiaieo, eieaeueii eiio?oeaaeo oa ?ic?eaieo
ooieoeie iaoaiaoe/iiai niiae?aaiiy ca iaiioeeeo iaiaaeaiue. Ia aaci
noioanoe/ieo ioeiiie iioe-iaeueieo cia/aiue, iaea?aeaieo ca aeiiiiiaith
ia?anoaiiaee iia?aoi??a i?i?i?caoe?? ? iaoaiaoe/iiai i/?eoaaiiy,
?ic?iaeaii iiaee noioanoe/iee aa??aio iaoiaeo aieie oa iaae aeey
?ica’ycaiiy caaea/ noioanoe/iiai aeene?aoiiai i?ia?aioaaiiy oa
noioanoe/ii? aeiaaeueii? iioeiicaoei? ca aeaoa?iiiiaaieo oa noioanoe/ieo
iaiaaeaiue. Cai?iiiiiaaii oa iaa?oioi-aaii ai?ieneiaoe?eiee i?aeo?ae aei
iioei?caoe?? iaaeaaeeeo oa ?ic?eaieo ooieoe?e iaoaiaoe/iiai niiae?aaiiy,
ooieoe?e eiiai?iinoi oa niiae?aaii? ei?eniino? ca iaiaaeaiue.

Eeth/ia? neiaa: iaiioeea noioanoe/ia i?ia?aioaaiiy, noioanoe/ia
aeiaaeueia iioei?caoe?y, noioanoe/ia aeene?aoia iioei?caoe?y, ooieoe??
niiae?aaii? ei?eniino?, ooieoe?? eiia??iino?, noioanoe/i? ocaaaeueiaii
a?aae??ioi? iaoiaee, noioanoe/ee iaoiae a?eie oa iaae, iaoiae aii??e/ieo
na?aaei?o.

Ii?eei A.E. Noioanoe/aneea iaoiaeu ?aoaiey caaea/ iaauioeeiai
noioanoe/aneiai i?ia?aiie?iaaiey e eo i?eiaiaiey. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie aeieoi?a
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.05.01 — oai?aoe/aneea
iniiau eioi?iaoeee e eeaa?iaoeee. — Einoeooo eeaa?iaoeee ei.
A.I.Aeooeiaa IAI Oe?aeiu, Eeaa, 1998.

A aeenna?oaoeee ?anniio?aii aieueoia /enei i?eeeaaeiuo caaea/
iaauioeeiai noioanoe/aneiai i?ia?aiie?iaaiey n iaaeaaeeeie, ?ac?ua-iuie
e aeene?aoiuie ooieoeeyie oeaee e ia?aie/aiee, iai?eia?, caaea/e
iioeiecaoeee aeeiaie/aneeo nenoai n aeene?aoiuie niauoeyie
(noioanoe/aneea naoe n ioeacaie, eiiioieeaoeeiiiua naoe, o?ainii?o-iay
eeiey, auyaeaiea iaia?aoeiuo eciaiaiee, iioeiaeueiay inoaiiaea iianiiai
i?iecaiaenoaa), caaea/e iioeiecaoeee iaeeanne/aneeo ooieoeee iaeeaeaaiie
iieaciinoe (noioanoe/aneay ia/aoeay iioeiecaoeey, noioanoe/aneay
eaeneeia?aoe/aneay iioeiecaoeey), caaea/e iioeiecaoeee aa?iyoiinoae,
caaea/e noioanoe/aneie aeene?aoiie iioeiecaoeee (noioanoe/aneay caaea/a
oeiaine?iaaiey i?iaeoia, noioanoe/aneay caaea/a ?aciauaiey, caaea/a
?aeiino?oeoeee noioanoe/aneie naoe, aeaooyoaiiay noioanoe/aneay caaea/a
i ieieiaeueiii iioiea a naoe), noioanoe/aneea caaea/e aeiaaeueiie
iioeiecaoeee (iai?a?uaiay noioanoe/aneay caaea/a ?aciauaiey, caaea/a
auai?a ii?ooaey oeaiiuo aoiaa). Iiieii iaauioeeinoe o?oaeiinoue ?aoaiey
yoeo caaea/ ninoieo a oii, /oi ooieoeee caaea/ yaeythony iaoaiaoe/aneeie
iaeeaeaieyie, o.a. iiiaiia?iuie eioaa?aeaie, oi/iia au/eneaiea eioi?uo
eeai i/aiue o?oaeiaiei, eeai i?aeoe/anee iaaiciiaeii. Aeey caaea/
noioanoe/aneie iioeiecaoeee n aeaaeeeie ooieoeeyie a eeoa?aoo?a iienaiu
iaeioi?ua noioanoe/aneea a?aaeeaioiua iaoiaeu. Aeey caaea/ n iaauioeeuie
iaaeaaeeeie eee ?ac?uaiuie ooieoeeyie auai? iaoiaeia e?aeia ia?aie/ai.
Caaea/e noioanoe/aneie aeiaaeueiie iioeiecaoeee i?aeoe/anee ia
eco/aeenue. A iaeanoe noioanoe/aneie aeene?aoiie iioeiecaoeee
nouanoaotho eeoue ioaeaeueiua ioaeeeaoeee.

A aeenna?oaoeee ?ac?aaioaiu e eco/aiu noioanoe/aneea iaoiaeu ?aoaiey
caaea/ iaauioeeiai noioanoe/aneiai i?ia?aiie?iaaiey n iaaeaaeeeie,
?ac?uaiuie e aeene?aoiuie ooieoeeyie oeaee e ia?aie/aiee. Yoe iaoiaeu ia
o?aaotho oi/iiai au/eneaiey iaoaiaoe/aneeo iaeeaeaiee, a eniieuecotho
neo/aeiua ?aaeecaoeee cia/aiee iiaeuioaa?aeueiuo ooieoeee eee eo
a?aaeeaioia, eioi?ua iiaeii iieo/eoue ec eiieoaoeeiiiie iiaeaee
iioeiece?oaiie nenoaiu.

A /anoiinoe, ecaanoiue iaoiae noioanoe/aneeo eaacea?aaeeaioia
TH.I.A?iieueaaa iaiauaaony aeey eieaeueiie iioeiecaoeee nenoai n
aeene?aoiuie niauoeyie, caaea/ iioeiecaoeee iaeeaeaaiie iieaciinoe,
caaea/ iioeiecaoeee aa?iyoiinoae n iaauioeeuie iaiauaiii
aeeooa?ai-oee?oaiuie, eaaceaiaioouie, eeioeoeaauie e ?ac?uaiuie
ooieoeeyie. Noiaeeiinoue (ii/oe iaaa?iia) iaoiaea onoaiiaeaia n
iiiiuueth iaauioeeiai noioanoe/aneiai aa?eaioa iaoiaea ooieoeee
Eyioiiaa. Enneaaeiaaia nei?inoue noiaeeiinoe iaoiaea a neo/aa
eaaceaiaioouo ooieoeee e aeey eiia/ii-oaaiaiai iaoiaea. Oaeia iaiauaiea
iio?aaiaaei aaaaeaiey e eco/aiey iaeioi?uo eeannia iaauioeeuo iaaeaaeeeo
(a oii /enea ?ac?uaiuo) ooieoeee, enneaaeiaaiey eo
noaaeeooa?aioeeaeueiuo naienoa, iaiaoiaeeiuo oneiaee yeno?aioia, a
oaeaea ?acaeoey niiniaia au/eneaiey noioanoe/aneeo a?aaeeaioia (a oii
/enea eiia/ii-?aciinoiuo) aeey iaauioeeuo iaaeaaeeeo e ?ac?uaiuo
ooieoeee. ?ac?aaioaiu noioan-oe/aneea iaoiaeu iioeiecaoeee ia?aaeiuo
ooieoeee iaoaiaoe/aneiai iaeeaeaiey — yoi noioanoe/aneea aiaeiae
iaoiaeia «oyaeaeiai oa?eea» e «ia?aaeiiai oaaa».

I?aaeeiaeai e iainiiaai aii?ieneiaoeeiiiue iiaeoiae (n iiiiuueth
on?aaeiaiiuo ooieoeee) e iioeiecaoeee iaaeaaeeeo e ?ac?uaiuo ooieoeee
iaoaiaoe/aneiai iaeeaeaiey, ooieoeee aa?iyoiinoe e iaeeaeaaiie
iieaciinoe i?e ia?aie/aieyo. A ?acoeueoaoa caaea/a naiaeeony e
iaauioeeie caaea/a noioanoe/aneiai i?ia?aiie?iaaiey, e eioi?ie
i?eiaiythony ?ac?aaioaiiua a aeenna?oaoeee iaoiaeu ?aoaiey. Onoaiiaeaia
noiaeeiinoue aii?ieneiaoeeiiiie noaiu, o.a. onoaiiaeaiu oneiaey, i?e
eioi?uo ieieioiu i?eaeeaeaiiuo caaea/ noiaeyony e ieieioiai enoiaeiie.

Aeey ?aoaiey caaea/ noioanoe/aneie aeiaaeueiie iioeiecaoeee, a oaeaea
caaea/ noioanoe/aneiai aeene?aoiiai i?ia?aiie?iaaiey a aeenna?oaoeee
?ac?aaioai noioanoe/aneee aa?eaio iaoiaea aaoaae e a?aieoe. Ieacaeinue,
/oi aeey iiiaeo caaea/ noioanoe/aneiai i?ia?aiie?iaaiey iiaeii
iino?ieoue niaoeeoe/aneea ioeaiee iioeiaeueiuo cia/aiee, eioi?ua eiatho
aeae iaoaiaoe/aneeo iaeeaeaiee io iaeioi?uo aniiiiaaoaeueiuo neo/aeiuo
aaee/ei. Iieo/aiea oaeeo ioeaiie iniiaaii ia oae iacuaaaiie
ia?anoaiiai/iie ?aeaenaoeee caaea/ noioanoe/aneiai i?ia?aiie?iaaiey,
eiaaea iia?aoeee ieieiecaoeee e iaoaiaoe/aneiai iaeeaeaiey (eee
aa?iyoiinoe, eee noiie?iaaiey) iaiythony ianoaie. Ia?anoaiiai/iay
?aeaenaoeey iiaeao eiiaeie?iaaoueny n eaa?aiaeaaie ?aeaenaoeeae e
?aeaenaoeeae oneiaee oeaei/eneaiiinoe. Iineieueeo ioeaiee aaoaae
yaeythony neo/aeiuie aaee/eiaie, oi e iaoiae aaoaae e a?aieoe
i?eia?aoaao noioanoe/aneee oa?aeoa? e noiaeeony a iaeioi?ii
aa?iyoiinoiii niunea (ii/oe iaaa?iia, n aa?iyoiinoiuie ioeaieaie
oi/iinoe ?aoaiey). ?ac?aaioai aa?eaio noioanoe/aneiai iaoiaea aaoaae e
a?aieoe aeey ?aoaiey caaea/ n iaeeiaeiuie ia?aie/aieyie.
?aaioiniiniaiinoue noioanoe/aneiai iaoiaea aaoaae e a?aieoe
i?ieeethno?e?iaaia /eneaiiuie ?aoaieyie noioanoe/aneie aeaooyoaiiie
aoeaaie caaea/e oeiaine?iaaiey i?iaeoia, noioanoe/aneie iai?a?uaiie
caaea/e ?aciauaiey, noioanoe/aneie oeaei/eneaiiie caaea/e nieaeaiey
caa?yciaiee.

Eeth/aaua neiaa: iaauioeeia noioanoe/aneia i?ia?aiie?iaaiea,
noioanoe/aneay aeiaaeueiay iioeiecaoeey, noioanoe/aneay aeene?aoiay
iioeiecaoeey, ooieoeee iaeeaeaaiie iieaciinoe, ooieoeee aa?iyoiinoe,
noioanoe/aneea iaiauaiii a?aaeeaioiua iaoiaeu, noioanoe/aneee iaoiae
aaoaae e a?aieoe, iaoiae yiie?e/aneeo n?aaeieo.

Norkin V.I. Stochastic methods for solution of nonsmooth stochastic
programming problems and their applications. — Manuscript.

Thesis for a doctor degree by speciality 01.05.01 — theoretical bases
of informatics and cybernetics. — Institute of Cybernetics of the
Ukrainian Academy of Sciences, Kiev, 1998.

The dissertation is devoted to solution of nonconvex stochastic
programming problems, including local and global stochastic
optimization, stochastic integer programming, local and global
optimization of probability and expected utility functions, stochastic
optimization of discontinuous functions. Stochastic quasigradient method
by Yu.Ermoliev is generalized to nonconvex stochastic local optimization
problems with generalized differentiable, locally Lipschitzian and
discontinuous functions subject to constraints. A new stochastic branch
and bound method based on interchange of minimization and expectation
operations is developed for solution of stochastic discrete programming
problems and stochastic global optimization problems. Approximation
approach is proposed to optimization of nonsmooth and discontinuous
mathematical expectation and probability functions.

Key words: nonconvex stochastic programming, stochastic global
optimization, stochastic discrete optimization, expected utility
functions, probability functions, stochastic generalized gradient
method, stochastic branch and bound method, empirical mean method.

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