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Асимптотически касательный конус второго порядка к множествам и условия оптимальности в задачах оптимизации с ограничениями.

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Y,lI,K 517.272
B. B.
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C
Q'(xO)
T O'feK
B Ka'feCTBe ,ll,OnOJlHHTeJlbHOH JIOKaJIbHOH
Q
annpOKCHMa~H
MHO)KeCTBa ,ll,OnYCTHMbIX
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Q c X. )1,.I/,Jl, moao "tmo6'bt mo"t'ICa xO E Q aOCma8.1/,Jl,.I/,a .l/,o'ICa.l/,'b'/-t'btu .MU'/-tU.My.M ifjy'/-t'IC'U;UU
f : -X ~lR '/-ta .M'/-to::»eecm8e Q c X, '/-teo6xoau.Mo, "tmo6'bt a.l/,Jl, 'lCa::»eaoao hE Q'(xO)nker f'(xO)
8'bmO.l/,'/-tJl,.I/,UC'b YC.l/,08UJl,
f'(XO)W
inf
(wn)ES~Q(:z;o ,
h, dirw)
h) ~ 0 a.l/,Jl, 8cex W E QI/(xO, h),
(lim inf f'(XO)w n )
+ f"(XO)[h,
(3)
[5-7]),
XO,ll,HMbIM YCJlOBHeM BToporo nOp.H,ll,Ka B TeopeMe
2
(3)
H
I1MeHHO ero H He,ll,OCTaBaJIO, 'fTo6bI, 3aMeHHB B
2
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(4)
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+ fl/(xO)[h,
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H nOJly'fHTb B KOHe'fHOMepHOM cJIy'fae ,ll,OCTaTO'fHbIe YCJlOBH.H JlOKaJIb­
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3
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.M'/-to::»eecm8e Q c X, aocmamo"t'/-to, "tmo6'bt
37
(i)
f'(xO)h ~ 0 a.ft.R, 8cexh E Q'(xo);
f=. 0, 8'btnO.ft1-t.R,.ftUC'b YC.ft08U.R,
(ii) a.ft.R, 'lCaaICaOeO hE Q'(xO) n ker 1'(xO), h
(5)
MO)KHO IIOKa3aTb, paccY)K,l1;aJI OT IIPOTHBHoro, 'ITO H3 YCJ10BH}I
f'(xO)w ~
06paTHO, eCJIH ,n;JI}I HeKOToporo
liminfn~oo f'(xO)w n
O,n;JI}I Bcex
(4)
CJIe,n;yeT yCJIOBHe
w E Q~(xO, h).
w E Q~ (xO, h)
BbIlIOJIHeHO YCJIOBHe
= +00 ,n;Jl}I JII060H IIOCJIe,n;OBaTeJIbHOCTH (w n )
f' (XO)w > 0,
E S~Q(xO, h,
TO
dirw) H,
CJIe,n;OBaTeJIbHO,
inf
,
(W,,)ES~Q(xO,h,
113
(lim inf
dirw)
3TOrO 3aKJII09aeM, 'ITO yCJIOBHe
n~oo
(4)
l' (xO)w n ) + f" (xO)[h,
h] >
o.
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YCJIOBHH:
H
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B
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h] ~ 0
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f'(xO)w ~ 0 ,n;JI}I
Bcex
w E Q~(xO, h) H
inf
(w,,)ES~Q(xO,h,
dirw)
(lim inf
n-+oo
I' (xO)w n ) + 1"{XO)[h, h] > 0
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J.13
(6a)
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(6).
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[8,10].
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KOTopble paHee He3aBHCHMO OT YCJIOBHH
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(4)
+ 1"(XO)[h, h]
H
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~ 0
6bIJIH ,n;OKa3aHbI B pa60Tax
a.ft.R, 8cexw E Q"(XO, h)
U
(7) 38
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ljjy'H'lCV,U.R. f : X ~ ]R aea:JICa'b£ auifxjjepe'Hv,upyeMa no rIJpewe e mo"t'ICe xO E Q c X. A//,.R.
mOlO "tm06'bt mo"t'ICa xO E Q aocmae//,.R.//,a //,o'ICa//,'b'H'b£ii. MU'HUMYM ljjy'H'lCV,UU I : X ~ ]R 'Ha
M'HO:JICeCmee Q eX, aocmamo"t'HO, "tm06'b£
(i) f'(xO)h ;:;: 0 a//,.R. ecex hE QI(XO); (ii) a//,.R. 'lCa:JICaOlO h E QI (xO)
n ker f' (xO),
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1
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E intQ.
KpoMe Toro, B TaKMX
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(7)
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0 1'1,
'-IT06hl 3TO IIpO,l1;eMOHCTpMpoBaTh, 3aMeTMM,
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I, JIe)Ka­
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f'(xO)
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h)
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IIepBOrO IIop.H,l1;Ka.
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(0,0)
MHO)KeCTBe
,l1;OCTaBJI.HeT JIOKaJIhHhrH cTporMH MMHMMYM <PYHKIJ,HM
Q :=
{(Xl,
X2) E ]R21
xi
= x~,
Xl ;:;:
I(XI,X2)
= Xl
-
x~
Ha
o}.
39
HenOCpe,l1,CTBeHHO yCTaHaBJUmaeTCR, 'ITO Q'(O) = {h = (0, h 2) E ]R2 I h2 ~ O},
Q"(O, h) = 0 H Q~(O, h) = {(WI, W2) E ]R21 WI ~ OJ. TaK KaK f'(O)h = hI ,D;JIR Bcex
h = (hI, h2) E ]R2, TO Q'(O) C ker 1'(0) H, CJIe,l1,OBaTeJIbHO, YCJIOBHe (1) BbInOJIHeHO. TIo­
CKOJIbKY Q"(O, h) = 0, TO YCJIOBHe (5) CqHTaeTCR BbInOJIHeHHbIM TPHBHaJIbHO. TIpoBepHM
BbInOJIHelme yCJIOBHR (6).
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W = (wi, 'W2) E Q~(O, h), W i- 0, H paCCMOTpHM npOH3BOJIbHYlO nOCJIe,l1,OBaTeJIbHOCTb
(WIn, W2n) E 8~Q(xO, h, dir W). TIo Onpe,l1,eJIeHHIO CeMeHCTBa 8~Q(xO, h, dir W) CYllI,eCT­
ByeT nOCJIe,l1,OBaTeJIbHOCTb (tn) E 8(0) TaKaR, 'ITO tn(Wl n ,W2n) -t 0 H xO + tnh + ~t2 Wn =
(~t;Wln' tnh2 + ~t;W2n) E Q ,l1,JIR BCeX n. TIoCJIe,l1,uee BKJIlOqelme 3KBHBaJIeHTHO YC;OBHIO
I
2 _ (h
)3
6
_ 4(h2 + tnW2n)3
WIn ~ 0 H paBeUCTBY 4tnWIn 2 + tnW2n . TaKHM 0 Pa30M, WIn ,a
tnWln
nOCKOJIbKY tnWln -t 0 H tnW2n -t 0, TO WIn -t +00.
"yqHTbIBaR, 'ITO f" (0) [h, h] = 0 ,l1,JIR BCeX h E ]R2, nOJIyqaeM, 'ITO ,l1,JIR JIlO60H nOCJIe,l1,O­
BaTeJIbHOCTH (W n ):;;: (Wl n ,W2n) E 8~Q(xO, h, dirw) BbInOJIHReTCR paBeHCTBO
liminf f'(xO)w n
n-too
+ f"(XO)[h, h] = n-too
lim WIn = +00.
1/lTaK, yCJIOBHe (6) TaK)Ke BbInOJIUeHO H, CJIe,l1,OBaTeJIbHO, TOqKa XO = (0, 0) ,l1,OCTaBJIReT
JIOKaJIbHbIH CTPOrHH MHHHMYM <PYHKI.\HH f(XI, X2) = Xl -X~ ua MHO)KeCTBe Q := {(Xl, X2) E
]R2
Ixi = X~,
Xl
~
O}.
Summary
Gorokhovik V. V. Second order asymptotically tangent cone to sets and optimality conditions
for optimization problems with constraints.
To improve second order local approximations of sets Penot J. P. (SIAM J. Control and Op­
timization. 1998. Vol. 37, N 1. P . 303-318) supplemented second order tangent vectors with the
second order asymptotically tangent vectors. In this paper we give a new definition of second order
asymptotically tangent vectors based on extending a normed space X with "direction points" form­
ing a set called the horizon of X (see: Rockafellar R . T ., Wets R. J.-B. Variational analysis. Berlin:
Springer-Verlag, 1998) . The application of such an extended second order local approximation to
the problem of minimizing a smooth function f : X -t lR over an abstract subset Q C X enables
to prove a new necessary condition for constrained local minimum that is of the form of inequality
depending on second order asymptotically tangent vectors to Q. It is especially important that in
finite-dimensional settings the collection consisting of (known and new) necessary conditions of the
first and second order may be coupled with the sufficient conditions for the local a strict minimum
that differs from the necessary ones by replacing inequalities with strict inequalities in the second
order conditions.
JIHTepaTypa
1. /I,y606utpWU A. E., MUAwmuH A. A. 3a,l1,a'llf Ha 3KcTpeMyM IIpH HaJIH'lHH OrpaHH'leHHH / /
)KYPH. Bbl'lHCJI. MaTeMaTHKH H MaT. <pH3HKH. 1965. T. 5, .N~ 3. C. 395-453.
2. /I,e.M:bJI/H,06 ,B. <P., Py6UH06 A. M. I1pH6JIH)KeHHble MeTO,l1,b1 pemeHH5I 3KCTpeMaJIbHbIX 3a,n;a'l.
JI.: 113,n;-Bo JIeHHHrp. YH-Ta, 1968. 180 c.
3. Aubin J.-P., Frankowska H. Set-valued analysis. Boston: Birkhauser Boston, Inc., 1990.
461 p.
4. Rockafellar R . T., Wets R. J.-B. Variational analysis. Berlin: Springer-Verlag, 1998. 733 p .
5. Bonnans J . F., Cominetti R., Shapiro A. Second order optimality conditions based on par­
abolic second order tangent sets / / SIAM J. Optimization. 1999. Vol. 9, N 2. P. 466- 492.
6. Bonnans J. F ., Shapiro A. Perturbation analysis of optimization problems. Berlin: Springer,
2000. 601 p.
7. rOpOX08UK; B. B., Pa"{K;o8cK;UU H. H. YCJIOBHH rrepBoro H BToporo rropH,n;Ka JIOKaJIhHOH C06CT­
BeHliOH MHHHMaJIbHOCTH B 3a,n;aQax BeKTopHoH orrTHMH3aIJ,HH. IIperrpHHT MH-Ta MaTeMaTHKH AH
SCCP, N~ 50(450). MHHCK, 1990. 27 c.
8. Penot J. P. Second order conditions for optimization problems with constraints / / SIAM J.
Control and Optimization. 1998. Vol. 37, N 1. P. 303-318.
.
9. rOpOX08UK; B . B . BbIIIYKJIble H HerJIa,n;KHe 3a,n;a'IH BeKTopHoH OIlTHMH3aIJ,HH. MHHCK: HayKa
D TeXHHKa, 1990. 239 c.
10. Cambini A., Martein L ., Vlach M. Second order tangent sets and optimality conditions / /
Math. Japonica. 1999. Vol. 49, N 3. P . 451- 461.
11. JfS.MaU/I,08 A . lP., Tpem'bJl,K;08 A. A. <DaKTOp-aHaJIH3 HeJIHHeHHbIX oTo6pa)KeHHH. M. : Hay­
Ita, 1994. 336 c.
OtaTbR IIOCTYIIHJIa B peAaK~HIO
24
HOR6pR
2005
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