symme y
S
S
A icle
Necessa y and Su icien Op imali y Condi ions o
Vec o Equilib ium P oblems on
Hadama d Mani olds
Gab iel Ruiz-Ga zón 1,*,†,‡ , Ra aela Osuna-Gómez 2,‡ and Jaime Ruiz-Zapa e o 3,‡
1Depa amen o de Es adís ica e I.O., Uni e sidad de Cádiz, 11405 Cádiz, Spain
2Depa amen o de Es adís ica e I.O., Uni e sidad de Se illa, 41012 Se illa, Spain
3Depa men o Physics and As onomy, Uni e si y College o London, London WC1E 6BT, UK
*Co espondence: gab iel. [email p o ec ed]
†
Cu en add ess: Depa amen o de Es adís ica e I.O., Uni e sidad de Cádiz, Campus de Je ez de la F on e a,
A da. de la Uni e sidad s/n, 11405, Je ez de la F on e a, Cádiz, Spain.
‡ These au ho s con ibu ed equally o his wo k.
Recei ed: 18 July 2019; Accep ed: 8 Augus 2019; Published: 12 Augus 2019


Abs ac :
The aim o his pape is o show he exis ence and a ainabili y o Ka ush–Kuhn–Tucke
op imali y condi ions o weakly e icien Pa e o poin s o ec o equilib ium p oblems wi h he
addi ion o cons ain s in he no el con ex o Hadama d mani olds, as opposed o he classical
examples o Banach, no med o Hausdo spaces. Mo e speci ically, classical necessa y and su icien
condi ions o weakly e icien Pa e o poin s o he cons ained ec o op imiza ion p oblem a e
p esen ed. The esul s desc ibed in his a icle gene alize esul s ob ained by Gong (2008) and Wei
and Gong (2010) and Feng and Qiu (2014) om Hausdo opological ec o spaces, eal no med
spaces, and eal Banach spaces o Hadama d mani olds, espec i ely. This is done using a no ion o
Riemannian symme ic spaces o a noncompac ype as special Hada ma d mani olds.
Keywo ds:
ec o equilib ium p oblem; gene alized con exi y; hadama d mani olds; weakly e icien
pa e o poin s
1. In oduc ion
The pu sui o equilib ium is a ubiqui ous ho izon in p ac ically all a eas o human ac i i y.
Fo example, in economics, he dynamics o o e and demand a e ypically desc ibed as equilib ium
p oblems. In he same way, physical o social phenomena such as he dis ibu ion o pa icles in
a con aine , a ic low o elecommunica ion ne wo ks can be accu a ely concep ualized in e ms
o equilib ium.
Howe e , i was no un il Fan [
1
] ha equilib ium heo y was applied in he con ex o Euclidean
spaces. Ma hema ically, he simples de ini ion o a equilib ium p oblem consis s in inding
x∈S
such ha
F(x,y)≥0, ∀y∈S
whe e
S⊆Rp
is a nonemp y closed se and
F:Rp×Rp→R
is an equilib ium bi unc ion, i.e.,
F(x,x) = 0 o all x∈S.
Some o he main ma hema ical p oblems ha can be ph ased as equilib ium p oblems a e:
Symme y 2019,11, 1037; doi:10.3390/sym11081037 www.mdpi.com/jou nal/symme y
Symme y 2019,11, 1037 2 o 12
•
The weak minimum poin o a mul iobjec i e unc ion
= ( 1
,
. . .
,
p)
o e a closed se
S⊆Rp
is
any
¯
x∈S
such ha o any
y∈S
,
∃i
such ha
i(y)− i(¯
x)≥
0. Finding a weak minimum poin
can be educed o sol ing an equilib ium p oblem by i ue o se ing
F(x,y) = max
i=1,...,p[ i(y)− i(x)].
•The S ampacchia a ia ional inequali y p oblem demands inding ¯
x∈Ssuch ha
<G(¯
x),y−¯
x>≥0, ∀y∈S
whe e
G:Rp→Rp
and
S⊆Rp
is a closed se . This p oblem is also an equilib ium p oblem whe e
F(x,y) =<G(x),y−x>.
•
Nash equilib ium p oblems in a non-coope a i e game wi h
p
playe s whe e each playe
i
has a se
o possible s a egies
Ki⊆Rni
aim o minimize a loss unc ion
i:K→R
wi h
K=K1×. . . ×Kp
.
Thus, a Nash equilib ium poin is any
¯
x∈K
such ha no playe can educe i s loss by unila e ally
changing hei s a egy, i.e., any ¯
x∈Ksuch ha
i(¯
x)≤ i(¯
x(yi))
holds o any
yi∈Ki
o any
i=
1,
. . .
,
p
, wi h
¯
x(yi))
deno ing he ec o ob ained om
¯
x
by
eplacing ¯
xiwi h yi. The e o e, his p oblem amoun s o sol ing an equilib ium p oblem wi h
F(x,y) =
p
∑
i=1
[ i(x(yi)) − i(x)].
Despi e hei appa en di e si y, all he abo e-men ioned p oblems can be amed as pa icula
cases o he ec o equilib ium p oblem and hus can all be encompassed in a single ma hema ical
pic u e. Due o he powe o his o mula ion, i is o g ea in e es o ob ain and s udy he
Ka ush–Kuhn–Tucke (KKT) op imali y condi ions o he solu ion o such, mo e gene al p oblems.
Thanks o hei capaci y o p o ide such a undamen al insigh , ec o equilib ium p oblems
a e an ac i e b anch o non-linea analysis wi h plen y o publica ions being made up o his da e.
Fo example, in 2003, au ho s such as Iusem and Sosa [
2
] s udied he ela ion be ween equilib ium
p oblems and some auxilia y con ex p oblems. In addi ion, o e he pas cen u y, he ield o physics
depa ed om euclidean geome y as a space in which o alloca e i s heo ies, op ing ins ead o
mo e complex spaces also known as mani olds. A his o ical landma k ha illus a es his example is
Eins ein’s heo y o g a i y ha e ol es a ound he concep o space- ime cu a u e on a Riemannian
mani old. O he less known bu equally undamen al applica ions in he ields o physics in ol e he
appea ance o symplec ic mani olds in he ea men o Hamil onian ec o ields o Noe he ’s heo em.
Smoo h Riemannian mani olds a e spaces ha con ain cu a u e, as opposed o Euclidean spaces
which a e la e e ywhe e. This can be ma hema ically exp essed as
ax +by /∈M
,
∀x
,
y∈M
,
a
,
b∈R
,
whe e
M
is a Riemannian mani old. None heless, Riemannian geome y cons i u es a gene aliza ion o
he Euclidean case. This can be easily unde s ood by in oducing he no ion o angen planes. Fo any
poin o a smoo h cu ed space, say a 2-Sphe e, i is always possible o de ine a la angen plane
o ha poin ; i.e., a Euclidean space. We can hink o his in he same way we hink o he Ea h o
be la a local scales while o e all being sphe ical. Indeed, all cu ed mani olds locally esemble
Euclidean space, which is a i al p ope y o ou unde s anding o hem. Howe e , ca og aphy
can empi ically ell us ha la p ojec ions o cu ed su aces on o planes ails o ai h ully ep esen
he eal dimensions o he objec s ha li e on he o iginal cu ed su ace especially a la ge scales
Symme y 2019,11, 1037 3 o 12
whe e he locali y condi ion s a s weakening. Thus, me ici y is no longe i ial and measu emen s
o dis ances need o accoun o such cu a u e.
A his poin , we can al eady see how Euclidean spaces a e simply Riemannian mani olds o
which he angen plane o any o i s poin s is iden ical o he plane i sel . Thus, in Euclidean spaces,
ec o s li ing o he su ace a e equi alen o ec o s li ing on i s angen space. I is his key ea u e o
Euclidean geome y ha allows o he simple de ini ion o dis ance as he do p oduc . Thus, gi en a
ec o
u
, i alloca ed in an Euclidean space, i s leng h is gi en by
|u|2=<u
,
u>
. On he o he
hand, in non- la spaces i is necessa y o accoun o he dis o ion o he dis ances when p ojec ed
o he angen space. Riemannian mani olds a e hose equipped wi h a so called “me ic enso ”;
commonly deno ed
kij
, ha allows us o adequa ely de ine dis ances; i.e.,
|u|2=kijuiuj
. (see Sec ion 2
o mo e de ails).
This new de ini ion o leng h has di ec sho comings in minimiza ion and equilib ium.
The Euclidean line elemen , he sho es connec ion be ween wo poin s on a la su ace, is eplaced on
mani olds by a geodesic equa ion which plays he ole o s aigh lines in non- la spaces. This can be
seen om he ac ha geodesic cu es a e solu ions o he Eule –Lag ange equa ions which minimize
he unc ional o he Lag angian gi en by he me ic o such space,
L=kijdxidxj
, and as such desc ibe
he ajec o ies ha minimize he ac ion necessa y o mo e om A o B. Fo example, he o bi s o
plane s obey geodesics despi e clea ly no being s aigh in a Euclidean sense.
A Hadama d mani old is a simply connec ed comple e Riemannian mani old o non-posi i e
sec ional cu a u e. The mo i a ion o he s udy o Hadama d spaces is ha hey sha e some p ope ies
wi h Euclidean spaces. One o hem is he sepa a ion heo em (see Fe ei a and Oli ei a [3]).
In addi ion, o any wo poin s in
M
, he e exis s a minimal geodesic joining hese wo poin s.
In a Hadama d mani old, he geodesic be ween any wo poin s is unique and he exponen ial map a
each poin o
M
is a global di eomo phism. Mo eo e , he
exp
map is de ined on he whole angen
space ([4]).
Howe e , he minimiza ion o unc ions on a Hadama d mani old is locally equi alen o he
smoo hly cons ained op imiza ion p oblem on a Euclidean space, due o he ac ha e e y
C∞
Hadama d mani old can be isome ically embedded in an Euclidean space by i ue o John Nash’s
embedding heo em. This is consis en wi h he in ui ion we p e iously laid ou .
The s udy o op imiza ion p oblems on Hadama d mani olds is a powe ul ool. This is due o
he ac ha , gene ally, sol ing noncon ex cons ained p oblems in
Rn
wi h he Euclidean me ic can
be also amed as sol ing he uncons ained con ex minimiza ion p oblem in he Hadama d mani old
easible se wi h he a ine me ic (see [
5
]). In Colao e al. [
5
] he exis ence o solu ions o equilib ium
p oblems unde some sui able condi ions on Hadama d mani olds and hei applica ions o Nash
equilib ium o non-coope a i e games was s udied. In he same way, in Néme h [
6
] he exis ence and
uniqueness esul s o a ia ional inequali y p oblems on Hadama d mani olds we e ob ained.
Mo eo e , many op imiza ion p oblems canno be sol ed in linea spaces, o example,
con olled he monuclea usion esea ch (see [
7
]), signal p ocessing, nume ical analysis and compu e
ision (see [
8
,
9
]) equi e Hadama d mani old s uc u es o hei modeling. Also, geome ical s uc u es
hidden in da a se s o machine lea ning p oblems a e s udied in e ms o mani olds. In he ield o
medicine, Hadama d mani olds ha e been used in he analysis o magne ic esonances o quan i y he
g ow h o umo s and consequen ly deduce hei s a e o p og ession, as shown by
Fle che e al. [10]
.
The geome y necessa y o unde s and and pe o m hese echniques is bes unde s ood h ough
he use o mani olds and symme ic s uc u es. Fo example, he se o symme ic posi i e de ini e
ma ices used in magne ic esonance imaging o s udy Alzheime ’s disease [
11
] is one case in which
his ansla ion o mani olds is necessa y. In addi ion, o he p oblems in compu e ision, signal
p ocessing o lea ning algo i hms employ geodesic cu es when add essing op imiza ion p oblems.
Finally, in economics, he sea ch o Nash–S ampacchia equilib ia poin s using Hadama d mani olds
has been used by K is ály [12].
Symme y 2019,11, 1037 4 o 12
I is known ha a con ex en i onmen has good p ope ies o he sea ch o op imal poin s.
In Fe ei a [
13
], he au ho gi es necessa y and su icien condi ions o con ex unc ions on Hadama d
mani olds. A signi ican gene aliza ion o he con ex unc ions a e he in ex unc ions, in oduced
by Hanson [
14
], whe e he x-y ec o is eplaced by any unc ion
η(x
,
y)
. The main esul o
in ex unc ions s a es ha a scala unc ion is in ex i and only i e e y c i ical poin is a global
minimum solu ion. This p ope y is essen ial o ob ain op imal poin s h ough algo i hms, due o
he coincidence o c i ical poin s and solu ions being always assu ed. In Ba ani and Pou yayeli [
15
]
and Hosseini and Pou yaye ali [
16
], he ela ion be ween in exi y and mono onici y using he mean
alue heo em is s udied. Ruiz-Ga zón e al. [
17
] showed ha in exi y can be cha ac e ized in he
con ex o Riemannian mani olds o bo h scala and ec o cases, in a simila way o Euclidean spaces.
Recen ly, in Ahmad e al.
[18]
he au ho s in oduced he log-p ein ex and log-in ex unc ions on
Riemannian mani olds and he mean alue heo em on Ca an-Hadama d mani olds.
In he same way, se e al au ho s ha e s udied ec o equilib ium p oblems. Ansa i and
Flo es-Bazán [
19
] we e capable o p o iding a heo em o exis ence o solu ions o ec o
quasi-equilib ium p oblems. Fu he mo e, a cha ac e iza ion o a weakly e icien Pa e o poin o he
ec o equilib ium p oblems wi h cons ain s unde con exi y condi ions on eal Hausdo opological
ec o spaces we e p esen ed by Gong [
20
]. In he ollowing yea s, scala iza ion esul s o he solu ions
o he ec o equilib ium p oblems we e also gi en by Gong [
21
]. La e , op imali y condi ions o
weakly e icien Pa e o poin s o ec o equilib ium p oblems wi h cons ain s in eal no med spaces
we e in es iga ed by Wei and Gong [
22
]. Also, su icien condi ions o weakly e icien Pa e o poin s
on eal Banach spaces o ec o equilib ium and ec o op imiza ion p oblems wi h cons ain s unde
gene alized in exi y we e ob ained by Feng and Qiu [23].
Mo i a ed by Gong’s wo ks men ioned abo e, ou objec i e will ocus on ex ending he KKT
necessa y and su icien condi ions o cons ained ec o equilib ium p oblems ob ained in opological
o no med spaces o o he en i onmen s like he Hadama d mani olds, no p esen in he li e a u e up
o da e o publica ion. Hence, we p opose a gene aliza ion ha ex ends he linea space de ini ion o
Hadama d mani olds, by i ue o subs i u ing line segmen s by geodesic a cs. We will see ha he
KKT classic condi ions o cons ained ec o op imiza ion a e a pa icula case o he ones ob ained
o cons ained ec o equilib ium p oblem.
The o ganiza ion o he pape is as ollows: In Sec ion 2, we discuss no a ion, di e en ials and
in ex unc ion concep s on Hadama d mani olds. Sec ion 3is de o ed o p o ing he main esul s
ob ained in his pape , and s udying he necessa y and su icien op imali y condi ions o weakly
e icien poin s o he cons ained ec o equilib ium p oblem. Sec ion 4dwells on how he p e ious
esul s can be educed o classical KKT condi ions o cons ained ec o op imiza ion p oblems,
i s ob ained by William Ka ush [
24
] and edisco e ed by Ha old Kuhn and Albe Tucke [
25
].
Finally, an example is p esen ed as well as he inal conclusions.
2. P elimina ies
Le
M
be a
C∞
-mani old modeled on a Hilbe space
H
endowed wi h a Riemannian me ic
gx
on
a angen space
TxM
. We deno e by
TxM
he angen space o
M
a
x
, by
TM =Sx∈MTxM
he angen
bundle o
M
, by
¯
TM
an open neighbo hood o he submani old
M
o
TM
. The co esponding no m is
deno ed by k.kxand he leng h o a piecewise C1cu e α:[a,b]→Mis de ined by
L(α) = Zb
akα0( )kα( )d .
We de ine das he dis ance which induces he o iginal opology on Msuch ha
d(x,y) = in {L(α)|αis a piecewise C1cu e joining xand y∀x,y∈M}.
Symme y 2019,11, 1037 5 o 12
I
d
is he dis ance induced by he Riemannian me ic
kij
hen any Riemannian mani old
(M
,
kij)
can
be con e ed in o a me ic space
(M
,
d)
. The de i a i es o he cu es a a poin
x
on he mani old lies
in a ec o space TxM. Wha e e pa h αjoining xand yin Msuch ha L(α) = d(x,y)is a geodesic.
Le
exp : ¯
TM →M
be he Riemannian exponen ial map de ined as
expx(V) = αV(
1
)
o e e y
V∈¯
TM, whe e αVis he geodesic s a ing a xwi h eloci y V(i.e., α(0) = x,α0(0) = V).
Assume now ha ηis a map η:M×M→TM de ined on he p oduc mani old such ha
η(x,y)∈TyM,∀x,y∈M.
De ini ion 1.
[
26
] A subse
S1
o
M
is conside ed o ally con ex i
S1
con ains e e y geodesic
αx,y
o
M
whose
endpoin s x and y belong o S1.
On a Hadama d mani old
M
, we can de ine he unc ion
η
as
η(x
,
y) = α0
x,y(
0
)
o all
x
,
y∈M
.
This unc ion plays he same ole o
x−y∈Rn
. He e
αx,y
is he unique minimal geodesic joining
y
o
xas ollows
αx,y=expy(λexp−1
yx)∀λ∈[0, 1].
Example 1.
Le
M=R++ ={y∈R:y>
0
}
endowed wi h he Riemannian me ic de ined by
g(y) = y−2
be a Hadama d mani old. Hype bolic spaces and geodesic spaces, mo e p ecisely, a Busemann non-posi i e
cu a u e (NPC) space a e examples o Hada ma d mani olds.
We will need an adequa e concep o he di e en ial:
De ini ion 2.
[
27
] A mapping
i:M→R
is said o be a di e en ial map along he geodesic
αx,y
a
y∈M
i
and only i he limi
0
i(y) = lim
λ→0
i(expy(λη(x,y))) − i(y)
λkη(x,y)k
exis s.
The g adien o a eal- alued
C∞
unc ion
= ( 1
,
. . . p):S1⊆M→Rn
on
M
in
x
, deno ed by
g ad x= ( 0
1(x)
,
0
2(x)
,
. . .
,
0
n(x))
, is he unique ec o in
TxM
such ha
d x(X) = hg ad x
,
Xi
o all
X
in
TxM is he di e en ial o a ¯
x o X.
Rema k 1.
The di e en ial o
a
¯
x
o
X
is simila o he de ini ion o di ec ional de i a i e in he
Euclidean space.
Le
S1⊂M
be a nonemp y open o ally con ex subse and le
F:S1×S1→Rp
,
g:S1→Rp
be mappings.
De ini ion 3.
We de ine he cons ain se
S={x∈S1:g(x)∈ −Rp
+}
and conside he ec o equilib ium
p oblem wi h cons ain s (VEPC): ind x ∈S such ha
F(x,y)/∈ −Rp
+ {0},∀y∈S
whe e Rp
+is he non-nega i e o han o Rp.
We ecall he classical concep :
De ini ion 4.
A ec o
x∈S
sa is ying
F(x
,
y)/∈ −in Rp
+
,
∀y∈S
is called a weakly e icien Pa e o poin
o he VEPC.
No a ion 1. We deno e as Hx(y) = F(x,y),∀y∈S1,gi en x ∈S, whe e H :S1→Rpis a mapping.

Symme y 2019,11, 1037 6 o 12
Inspi ed by he concep o con exi y on a linea space, he no ion o in exi y unc ion concep on
Hadama d mani olds has become a success ul ool in ec o op imiza ion. This gene alized de ini ion
was no ably p o ided by Hanson in [14].
De ini ion 5.
Le
S1
be a nonemp y open o ally con ex subse o a Hadama d mani old
M
. A di e en iable
h:S1→Rp
unc ion is said o be a
Rp
+
-in ex a
¯
x∈S1
espec o
η:M×M→TM
i he e exis
η(x,¯
x)∈T¯
xM such ha
h(x)−h(¯
x)−dh ¯
x(η(x,¯
x)) ∈Rp
+.
Using he p e iously s a ed de ini ions, we can ob ain he su icien condi ions o op imali y by
i ue o he assump ion o in exi y o he unc ions o he p oblem.
3. Main Resul s
Nex , we will ob ain a cha ac e iza ion o he weakly e icien poin s o VEPC h ough he
applica ion o necessa y and su icien op imali y condi ions. We s a wi h he necessa y condi ions:
Theo em 1.
[Necessa y KKT-condi ions] Le
S1
be a nonemp y open o ally con ex subse o a Hadama d
mani old
M
and le
F:S1×S1→Rp
,
g:S1→Rp
,
η:M×M→TM
be mappings. Le
F(¯
x
,
¯
x) =
H¯
x(¯
x) =
0. Assume ha
H
and
g
a e di e en iable a
¯
x∈S
. Fu he mo e, assume ha he e exis s
x1∈S1
such ha
g(¯
x) + dg¯
x(η(x1
,
¯
x)) ∈ −in Rp
+
. I
¯
x
is a weakly e icien Pa e o poin o he VEPC, hen he e
exis s ∈Rp
+ {0}, u ∈Rp
+such ha
dH¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1(1)
ug(¯
x) = 0. (2)
P oo . Le he e be ¯
x∈Sas a weakly e icien Pa e o poin o he VEPC. We deno e by
W={(y,z)∈Rp×Rp: he e exis s x∈S1, such ha y−dH¯
x(η(x,¯
x)) ∈in Rp
+,
z−[g(¯
x) + dg¯
x(η(x,¯
x))] ∈in Rp
+}.
I may be no ed ha W is a nonemp y open o ally con ex se . This p oo can be di ided in o
i e s eps:
S ep 1.
We ha e o p o e ha
(
0, 0
)/∈W
. By educ ion ad absu dum, i
(
0, 0
)∈W⇒ ∃x0∈S1
,
such ha
dH¯
x(η(x0,¯
x)) ∈ −in Rp
+,g(¯
x) + dg¯
x(η(x0,¯
x)) ∈ −in Rp
+. (3)
F om he di e en iabili y we ob ain ha
dH¯
x(η(x0,¯
x)) = lim
λ→0
1
λ[H¯
x(exp ¯
x(λη(x0,¯
x)) −H¯
x(¯
x)] ∈ −in Rp
+(4)
g(¯
x) + dg¯
x(η(x0,¯
x)) = g(¯
x) + lim
λ→0
1
λ[g(exp ¯
x(λη(x0,¯
x))) −g(¯
x)] ∈ −in Rp
+. (5)
As −in Rp
+is an open se , hen ∃λ0, 0 <λ0<1 such ha
1
λ0
[H¯
x(exp ¯
x(λ0η(x0,¯
x))) −H¯
x(¯
x)] ∈ −in Rp
+(6)
g(¯
x) + 1
λ0
[g(exp ¯
x(λ0η(x0,¯
x))) −g(¯
x)] ∈ −in Rp
+. (7)
Symme y 2019,11, 1037 7 o 12
By hypo hesis, om g(¯
x)∈ −Rp
+,F(¯
x,¯
x) = H¯
x(¯
x) = 0, and 1
λ0
>1, hen
H¯
x[exp ¯
x(λ0η(x0,¯
x))] ∈ −in Rp
+and g(exp ¯
x(λ0η(x0,¯
x)))∈ −in Rp
+. (8)
As S1is a o ally con ex se we ha e ha
exp ¯
x(λ0η(x0,¯
x)) ∈S1,F(¯
x,exp ¯
x(λ0η(x0,¯
x)))∈ −in Rp
+(9)
and
g(exp ¯
x(λ0η(x0,¯
x)))∈ −in Rp
+(10)
s ands in con adic ion wi h
¯
x∈S
as a weakly e icien Pa e o poin o he VEPC, consequen ly
(0, 0)/∈W.
S ep 2.
We will p o e ha he e exis s a mul iplie
∈Rp
+
. As
W
is an open se and he sepa a ion
heo em holds (see Theo em 2.13 and Rema k 2.14 in [
28
]) o [
3
]), he e exis s
(
,
u)6= (
0, 0
)∈Rp×Rp
such ha
y +uz >0, ∀(y,z)∈W. (11)
Le (y,z)∈Wbe a poin hen ∃x∈S1such ha
y−dH¯
x(η(x,¯
x)) ∈in Rp
+,z−[g(¯
x) + dg¯
x(η(x,¯
x))] ∈in Rp
+. (12)
Fo any ∈in Rp
+,s∈in Rp
+, 0, 00 >0, we ha e (y+ 0 ,z)∈Wand (y,z+ 00s)∈W.
F om Equa ion (11) we ha e ha
(y+ 0 ) + u(z)>0, ∀ ∈in Rp
+, 0>0. (13)
Then
>−uz − y
0. (14)
Le ing
0→∞
we ge
≥
0,
∀ ∈in Rp
+
and he e o e
≥
0 o all
∈Rp
+
, ha is
∈Rp
+
.
In he same way, we can show ha u∈Rp
+.
S ep 3.
We will p o e ha
6=
0, hus is,
∈Rp
+ {
0
}
. By educ ion ad absu dum, i
=
0,
om Equa ion (11) we ge
uz >0, ∀(y,z)∈W. (15)
Acco ding o he hypo hesis,
∃x1∈S1
such ha
g(¯
x) + dg¯
x(η(x1
,
¯
x)) ∈ −in Rp
+
; hen, we ob ain
(dH¯
x(η(x1,¯
x)) + ,g(¯
x) + dg¯
x(η(x1,¯
x)) + s)∈W,∀ ∈in Rp
+∀s∈in Rp
+. (16)
The e o e, om Equa ion (11) we ha e ha
u[g(¯
x) + dg¯
x(η(x1,¯
x)) + s]>0, ∀s∈in Rp
+(17)
us >−u[g(¯
x) + dg¯
x(η(x1,¯
x))]. (18)
As
[g(¯
x) + dg¯
x(η(x1
,
¯
x))] ∈ −in Rp
+
, and i
s=
0, we ge
u·
0
=
0
>
0, which implies a
con adic ion, hus 6=0.
S ep 4. We will p o e he i s KKT condi ion.
Since
(dH¯
x(η(x,¯
x)) + ,g(¯
x) + dg¯
x(η(x,¯
x)) + s)∈W,x∈S1, ∈in Rp
+,s∈in Rp
+. (19)
Symme y 2019,11, 1037 8 o 12
F om Equa ion (11) we ge
[dH¯
x(η(x,¯
x)) + ] + u[g(¯
x) + dg¯
x(η(x,¯
x)) + s]>0, ∀x∈S1, ∈in Rp
+,s∈in Rp
+. (20)
Le ing →0, s→0, we ob ain
dH¯
x(η(x,¯
x)) + u[g(¯
x) + dg¯
x(η(x,¯
x))] ≥0, ∀x∈S1. (21)
S ep 5. We will p o e he second KKT condi ion. As
dH¯
x(η(¯
x,¯
x)) + 0 ,g(¯
x) + dg¯
x(η(¯
x,¯
x)) + 0s∈W,∀ ∈in Rp
+,s∈in Rp
+, 0>0. (22)
F om Equa ion (11) we ha e ha
[dH¯
x(η(¯
x,¯
x)) + 0 ] + u[g(¯
x) + dg¯
x(η(¯
x,¯
x)) + 0s] = 0 +ug(¯
x) + 0us >0. (23)
Le ing
0→
0, we ob ain
ug(¯
x)≥
0. No ing ha
g(¯
x)∈ −Rp
+
and
u∈Rp
+
, we ha e ha
ug(¯
x)≤0, in consequence
ug(¯
x) = 0 (24)
and he e o e ∃ ∈Rp
+ {0},u∈Rp
+such ha KKT condi ions
dH¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1(25)
ug(¯
x) = 0 (26)
hold.
Le us see now he ecip ocal o he p e ious heo em. To ob ain i we i s need condi ions
o in exi y.
Theo em 2.
[Su icien KKT-condi ions] Le
S1
be a nonemp y open o ally con ex subse o Hadama d mani old
M
and le
F:S1×S1→Rp
,
g:S1→Rp
be mappings. Le
F(¯
x
,
¯
x) = H(¯
x) =
0. Assume ha
H
and
g
a e
di e en iable a
¯
x∈S
.
H
and
g
a e
Rp
+
-in ex a
¯
x
espec o
η
on
S1
. I he e exis
∈Rp
+ {
0
}
and
u∈Rp
+
such ha
dH¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1(27)
ug(¯
x) = 0 (28)
hen ¯
x is a weakly e icien Pa e o poin o he VEPC.
P oo . On he assump ion ha Hand ga e Rp
+-in ex a ¯
x espec o ηon S1 hen
dH¯
x(η(x,¯
x)) ∈H¯
x(x)−H¯
x(¯
x)−Rp
+=H¯
x(x)−Rp
+,∀x∈S1(29)
dg¯
x(η(x,¯
x)) ∈g(x)−g(¯
x)−Rp
+,∀x∈S1. (30)
F om ∈Rp
+ {0},u∈Rp
+and (27) we ob ain ha
H¯
x(x) + u(g(x)−g(¯
x)) = dH¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1. (31)
F om hypo hesis (28), we ge on he one hand ha :
H¯
x(x) + ug(x)≥0, ∀x∈S1. (32)
Symme y 2019,11, 1037 9 o 12
On he o he hand, we will show ha
¯
x
is a weakly e icien Pa e o poin o he VEPC. I no ,
consequen ly by de ini ion ∃y0∈Ssuch ha
F(¯
x,y0)∈ −in Rp
+. (33)
F om ∈Rp
+ {0} ⇒ F(¯
x,y0)<0.
Since y0∈S, we ha e g(y0)∈ −Rp
+, so ug(y0)≤0 because o u∈Rp
+and hen
F(¯
x,y0) + ug(y0)<0 (34)
s ands in con adic ion wi h (32) and he e o e ¯
xis a weakly e icien Pa e o poin o he VEPC.
Rema k 2.
Theo em 3.1 in [
20
] on eal Hausdo opological ec o spaces and Theo em 3.2 and Theo em 3.4
in [
22
] on eal no med spaces a e pa icula cases o Theo ems 1and 2ob ained in his pape on Hadama d
mani olds. The same is ue o Theo ems 3.1 and 3.3 in [23] on eal Banach spaces.
To sum up, we ob ain he KKT op imali y condi ions o weakly e icien Pa e o poin s o he
ec o equilib ium p oblems wi h cons ain s. These esul s a e no only necessa y bu also su icien .
4. Applica ion
As a pa icula case o he esul s ob ained in he p e ious sec ion, we will ob ain he op imali y
condi ions o KKT o cons ained ec o op imiza ion p oblems.
Le us conside he cons ained mul iobjec i e p og amming (CVOP) de ined as:
(CVOP) min (x)
subjec o:
g(x)≤0
x∈X⊆M
whe e
= ( 1
,
. . . p):X⊆M→Rp
,
g= (g1
,
. . .
,
gm):X⊆M→Rm
a e di e en iable
mul iobjec i e unc ions on he open se X⊆Mand le Mbe a Hadama d mani old.
As a consequence o he p e ious heo ems and conside ing CVOP as a pa icula case o VEPC
we ha e he KKT classical condi ions.
Co olla y 1.
Le
S1
be a nonemp y open o ally con ex subse o Hadama d mani old
M
and le
,
g:S1→Rp
be mappings. Assume ha
and
g
a e di e en iable a
¯
x∈S
. Fu he mo e, assume ha he e exis s
x1∈S1
such ha
g(¯
x) + dg¯
x(η(x1
,
¯
x)) ∈ −in Rp
+
. I
¯
x
is a weakly e icien Pa e o poin o he CVOP, hen he e exis
∈Rp
+ {0}, u ∈Rp
+such ha
d ¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1(35)
ug(¯
x) = 0. (36)
Co olla y 2.
Le
S1
be a nonemp y open o ally con ex subse o Hadama d mani old
M
and le
,
g:S1→Rp
be mappings. Assume ha
and
g
a e di e en iable a
¯
x∈S
. Assume ha
and
g
a e di e en iable a
¯
x∈S
and and g a e Rp
+-in ex espec a ¯
x o ηon S1. I he e exis ∈Rp
+ {0}, u ∈Rp
+such ha
d ¯
x(η(x,¯
x)) + udg ¯
x(η(x,¯
x)) ≥0, ∀x∈S1(37)
ug(¯
x) = 0 (38)
hen ¯
x is a weakly e icien Pa e o poin o he CVOP.