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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(2),634-644
PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.122067
¦)Øªåš‚5`z¯K
&6•SQP•{
444©©©###§§§‰‰‰UUUÀÀÀ
H㲌ÆÚO†êÆÆ§H&²
ÂvFϵ2023c121F¶¹^Fϵ2023c220F¶uÙFϵ2023c227F
Á‡
éØªåš‚5`z¯K¦)þ®²kéõ`z•{§~XSg5y(SQP)•{§
©K´3SQP`z•{Ä:þïÄ¿ïá˜‡ò&6••{†Ù(Ü•{))&6
•SQP•{§,‰Ñ ƒA¢~éTŽ{?1ꊢ[§¿òÙ†5SQP•{?1'
"•Ñ3·^‡e§&6•SQP•{ƒukSQP•{äk•ÐêŠJ§ê
Š(J•˜½§Ýþyù‡Ž{Œ15"
'…c
š‚5`z¯K§SQP`z•{§&6•SQP`z•{
TrustRegionSQPMethodforSolving
InequalityConstrainedNonlinear
OptimizationProblems
WenjieLiu,XidongFan
Schoolof MathematicsandStatistics,Yunnan Universityof FinanceandEconomics,KunmingYun-
nan
©ÙÚ^:4©#,‰UÀ.¦)Øªåš‚5`z¯K&6•SQP•{[J].A^êÆ?Ð,2023,12(2):
634-644.DOI:10.12677/aam.2023.122067
4©#§‰UÀ
Received:Jan.21
st
,2023;accepted:Feb.20
th
,2023;published:Feb.27
th
,2023
Abstract
Therehavebeen manyoptimizationmethodsforsolving inequalityconstrainednonlin-
earoptimizationproblems,suchassequential quadraticprogramming (SQP)method.
Inthispaper,amethodcombiningconfidenceregionmethodwithSQPmethodis
studiedandestablishedonthebasisofSQPoptimizationmethod-confidenceregion
SQPmethod.Thenthecorrespondingexamplesaregiventosimulatethealgorithm
andcompareitwiththeoriginalSQPmethod.Finally,underappropriateconditions,
the confidence region SQPmethod hasbetter numerical results than the original SQP
method,andthenumericalresultsalsoverifythefeasibilityofthisalgorithmtoa
certainextent.
Keywords
NonlinearOptimizationProblems,SQPOptimizationMethod,SQPOptimizationMethod
ofTrustRegion
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.122067636A^êÆ?Ð
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k
) = ∇
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m
X
j=1
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k
)
j
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x
C
j
(x
k
) = g
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T
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)
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(A
k
δ
x
+C
k
)
j
= 0forj= 0,1,2,3···m(11)
DOI:10.12677/aam.2023.122067637A^êÆ?Ð
4©#§‰UÀ
Ù¥C
k
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k
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k
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b3S“(x
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1
2
δ
T
Y
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g
k
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k
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äNS“Ž{Ú½Xe:
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0
,µ
0
),Âñ°Ýε,˜k= 0;
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,A
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,g
k
,C
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Cq)(x
k+1
,µ
k+1
);
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2.2.&6•SQP•{
2.2.1.ÄgŽ
•ÄXeØªå`z¯K
min
p
f
k
+∇f
T
k
p+
1
2
p
T
∇
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xx
L
k
p(14a)
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i
(x
k
)
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p+c
i
(x
k
) = 0,i∈ε,(14b)
∇c
i
(x
k
)
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i
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k
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kpk≤∆
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DOI:10.12677/aam.2023.122067638A^êÆ?Ð
4©#§‰UÀ
,˜«•\Ünw{´,·‚vknd‡z˜ÚÑ°(÷v÷v‚5å,A ´3z˜
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þߎ.
3S“:x
k
,·‚ÏL¦)±ef¯KOŽSQPŽ{S“Ú:
min
p
f
k
+∇f
T
k
p+
1
2
p
T
∇
2
xx
L
k
p(15a)
subjecttoA
k
p+c
k
= r
k
(15b)
kpk
2
≤∆
k
(15c)
·‚8I´À•¦åƒNr
k
§=¦)±ef¯K
min
p
kA
k
p+c
k
k
2
2
(16a)
subjecttokvk
2
≤0.8∆
k
(16b)
Pdf¯K)•v
k
·‚½Â
r
k
= A
k
v
k
+c
k
(17)
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k+1
=x
k
+p
k
,2^•¦úª#¦fOλ
k+1
,
2|^±eª
p
B
= −A
T
k
[A
k
A
T
k
]
−1
c
k
(18)
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B
,=8I¼êÃå4:.
·^ud•{˜‡dŠ¼ê•š1wL
2
¼ê§=φ
2
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2
·‚òÙCq
•(¢Sþ•´l&6•f¯K)
q
µ
(p) = f
k
+∇f
T
k
p+
1
2
p
T
∇
2
xx
L
k
p+µm(p)(19)
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m(p) = kc
k
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pk
2
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p
k
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k
p
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(1−ρ)kc
k
k
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k
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ρ=
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k
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φ
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(x
k
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2
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k
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k
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q
µ
(0)−q
µ
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k
)
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DOI:10.12677/aam.2023.122067639A^êÆ?Ð
4©#§‰UÀ
2.2.2.&6•SQPŽ{Ú½
äNS“Ž{Ú½Xe:
[1]Щz.?À°ÝÚη,γ∈(0,1);ЩŠx
0
,&6•Žfδ
0
>0;
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k
,c
k
,∇f
k
,A
k
;
[3]|^
ˆ
λ
k
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k
A
T
k
)
−1
A
k
∇f
k

ˆ
λ
k
¶XJk∇f
k
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<;ÊŽ$1¿ÑÑdžx
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k
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[5]|^þ©J•¦{úª(18)ŽÑp
k
eρ
k
>η,Kx
k+1
= x
k
+p
k
;∆
k+1
= 2∆
k
;
ÄK-x
k+1
= x
k
;é÷v∆
k+1
≤γkp
k
k;2‘\±þÌ‚¥†÷v^‡.
3.ꊢ
3c¡Ù!¥·‚®²0•`znØ¥ü«`zOŽ•{,!·‚ò‰ÑA‡¢
~f5ySg5y{Ú&6•SQP`z•{`5.
3.1.êŠ~f9(J
3e¡¢¥¤¦^ÎÒ¹Â:
x:š‚5å`z¯KgCþ;
f:š‚5å`z¯K¼êŠ;
cpu:š‚5å`z¯KS“žm,ü •t/s;
T:š‚5å`z¯KS“gêk;
y3·‚•ÄXeA‡š‚5`z¯K:
Table1.NumericalresultsoftrustregionSQPalgorithm
L1.&6•SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
5.23963.7460−79.80780.521611
Table2.NumericalresultsofSQPalgorithm
L2.SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
5.23933.7455−79.80630.859229
DOI:10.12677/aam.2023.122067640A^êÆ?Ð
4©#§‰UÀ
~f1:
minf(x) =x
2
1
+x
2
2
−16x
1
−10x
2
subjectto−x
2
1
+6x
1
−4x
2
+11 >0
x
2
1
−3x
2
−e
x
1
−3
+1 >0
x
1
>0
x
2
>0
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Šx=(4,4)
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L1ÚL2¤«.
~f2:
minf(x) =(x
1
−3)
2
+(x
2
−1)
2
subjectto−x
2
1
+x
2
>0
2x
1
+x
2
−3 = 0
·‚ éþªØªå`z~f2¯K¦^9ÏOŽóäMATLAB?1ꊢ,þª¯K
Œ•°()•x=(1,1),f=4,ùpЩŠ·‚x=(0,0)
T
,“ è´3MATLAB7.1‚¸e$1
.CPU´C(R)2.19GHZS•.¢(JXeL3ÚL4¤«.
Table3.NumericalresultsoftrustregionSQPalgorithm
L3.&6•SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
5.23963.7460−79.80780.521611
Table4.NumericalresultsofSQPalgorithm
L4.SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
5.23933.7455−79.80630.859229
~f3:
minf(x) =1000(x
2
−x
2
1
)
2
+(1−x
1
)
2
subjectto3x
2
1
−5x
2
≤0
3x
1
−x
2
−2 ≤0
DOI:10.12677/aam.2023.122067641A^êÆ?Ð
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·‚ éþªØªå`z~f3¯K¦^9ÏOŽóäMATLAB?1ꊢ,þª¯K
Œ•°()•x=(1,1),f=0,ùpЩŠ·‚x=(0,0)
T
,“ è´3MATLAB7.1‚¸e$1
.CPU´C(R)2.19GHZS•.¢(JXeL5ÚL6¤«.
Table5.NumericalresultsoftrustregionSQPalgorithm
L5.&6•SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
115.8129e−210.12005844
Table6.NumericalresultsofSQPalgorithm
L6.SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
111.9737e−90.71916548
~f4:
minf(x) =x
2
1
+x
2
2
−4x
1
+4
subjectto−x
2
1
+x
2
−1 >0
x
1
−x
2
+2 >0
x
1
>0
x
2
>0
·‚ éþªØªå`z~f4¯K¦^9ÏOŽóäMATLAB?1ꊢ,·‚Ð
©Šx=(0,1)
T
,“è´3MATLAB7.1‚¸e$1.CPU´C(R)2.19GHZS•.¢(JX
eL7ÚL8¤«.
Table7.NumericalresultsoftrustregionSQPalgorithm
L7.&6•SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
0.55361.30643.79890.075097
3.2.Ù"(
éØªš‚5`z¯K¦)·‚ÏL±þo ‡~f êŠ(J?1©Û,~f1êŠ
(J¥&6•SQP•{¦)Ú©SQP•{¦)Ø4ž,©•{¤¦)¤s
OŽžm'&6•SQP•{žm•.~f2Ú~f3¥êŠ(JL²,3®²•°()
œ¹e,·‚¤¦^©SQP•{Ú&6•SQP•{ÑU¦¤I‡),3()(œ
¹e·‚©Ûü«•{¤sOŽžm uy,&6•SQP•{3ˆÓOŽ(Jœ¹eOŽž
DOI:10.12677/aam.2023.122067642A^êÆ?Ð
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Table8.NumericalresultsofSQPalgorithm
L8.SQPŽ{êŠ(J
x
1
x
2
f(x)cpuT
0.55361.30643.79890.517327
máu©SQP•{.~f1!~f29~f3êŠ(J¥·‚„Œ±uy &6•SQP•{
uSQP•{S“gê.•3~f4ꊢ(J¥,&6•SQP•{ÚSQP•{¤¦
)ƒÓ,¼êŠƒÓ,¿…S“gꕃÓœ¹e,·‚é'&6•SQP•{Ú©SQP•{O
Ž¤Ižm,Œ±uy&6•SQP•{u©SQP•{•¯.nþA‡~fꊢ(J©Û
•,&6•SQP•{3˜½^‡e,´k¿…pÇŽ{.
4.o(†Ð"
3Ø©¥·‚{ü0•`z{¤±9uÐµ,,Vã¦)š‚5Øªå`z
¯K•~^˜«SQP•{,¿ïáïÄ©̇gŽ:¦^&6 ••{†SQP•{ƒ(Ü
Ž{5¦)š‚5Øªå`z¯K.•31nÙ¥ÏLo‡š‚5Øªå`z¯Kê
Š~féü«•{?1Á,¿¥y‹·‚ýŽ˜—êŠ(J.ÏLùg¢(J©Û•,3Œ
±ˆƒÓêŠ)œ¹e,&6•SQP•{©SQP`z•{•k`³,äk°Ýp,OŽ
ǯA:,Ïd3,E,š‚5Øªå`z¯Kþ·‚Œ±æ^&6•SQP`z•{
5¦),¬ŒŒ/!Ž$Žžm.
©3¦^&6•SQP`z•{¦)š‚5Øªå`z¯Kþ•Ä˜¯K,XJ3
˜•E,AÏ•§¥¦^&6•SQP`z•{5¦),§´Ä„UkÓ`³.Ïd,
e5óŠ·‚¬UYé&6•SQP`z•{?1ƒ'•¡ïÄ.
ë•©z
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