On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

Abstract

In this paper, we considered the mathematical programs with vanishing constraints or MPVC. We proved that an MPVC-tailored penalty function, introduced in [5], is still exact under a very weak and new constraint qualification. Most importantly, this constraint qualification is shown to be strictly stronger than MPVC-Abadie constraint qualification.

Full text

minx∈Rnf(x)
s.t. gi(x)60∀i= 1,2, ..., m,
hj(x)=0∀j= 1,2, ..., l,
Hi(x)>0∀i= 1,2, ..., q,
Gi(x)Hi(x)60∀i= 1,2, ..., q.
f:Rn→R, gi:Rn→R, hi:Rn→R, Gi:Rn→
R, Hi:Rn→R
Gi(x)60
Hi(x)=0 C
Pα(x) = f(x)+α[
m
X
i=1
max{0, gi(x)}+
l
X
j=1
|hj(x)|+
q
X
i=1
max{0,−Hi(x),min{Gi(x), Hi(x)}}]
l1−g h
P1
α(x) = f(x) + α
q
X
i=1
max{−Hi(x),0}+α
q
X
i=1
max{Gi(x)Hi(x),0}.
l1

l1
I00
l1
Pα(x)
x∗
Ig:= {i|gi(x∗)=0},
I+:= {i|Hi(x∗)>0}, I0:= {i|Hi(x∗)=0},
I+0 := {i|Hi(x∗)>0, Gi(x∗)=0}, I+−:= {i|Hi(x∗)>0Gi(x∗)<0},
I0+ := {i|Hi(x∗)=0, Gi(x∗)>0}, I0−:= {i|Hi(x∗)=0, Gi(x∗)<0},
I00 := {i|Hi(x∗) = 0 , Gi(x∗)=0}.

C⊂Rnx∗∈C
C x∗
TC(x∗) := {d∈Rn| ∃{xk} →Cx∗,{tk} ↓ 0 : xk−x∗
tk
→d}
:= {d∈Rn| ∃{dk} → d, {tk} ↓ 0 : x∗+tkdk∈C∀k∈N},
{xk} →Cx∗{xk}x∗
xk∈C∀k∈N
d∈TC(x∗)C x∗
C⊂Rnx∗∈C
C x∗
NF
C(x∗) := TC(x∗)◦.
C⊂Rnx∗∈C
C x∗
NC(x∗) := {d∈Rn| ∃{xk} →Cx∗, dk∈NF
C(xk) : dk→d}.
Φ : Rn⇒RmgphΦ := {(x, y)|y∈
Φ(x)}x∈Rnδ > 0B(x, δ) := {y∈Rn| ky−xk< δ}
k.kl1
x∗∈ C
{∇gi(x∗)|i∈Ig} ∪ {∇hi(x∗)|i= 1, ..., p} ∪ {∇Gi(x∗)|i∈I+0 ∪I00}
∪ {∇Hi(x∗)|i∈I0}
x∗∈ C
∇hi(x∗) (i= 1, ..., p),∇Hi(x∗) (i∈I0+ ∪I00)
d∈Rn
∇hi(x∗)d= 0 (i= 1, ..., p),∇Hi(x∗)Td= 0 (i∈I0+ ∪I00),
∇gi(x∗)Td < 0 (i∈Ig),∇Hi(x∗)Td > 0 (i∈I0−),
∇Gi(x∗)Td < 0 (i∈I+0 ∪I00).
x∗∈ C
(λ, µ, ηH, ηG)6= 0
(i) Pm
i=1 λi∇gi(x∗)+Pl
i=1 µi∇hi(x∗)+Pq
i=1 ηG
i∇Gi(x∗)−Pq
i=1 ηH
i∇Hi(x∗)=0

(ii) λi>0∀i∈Ig, λi= 0 ∀i /∈Ig
and ηG
i= 0 ∀i∈I+−∪I0−∪I0+, ηG
i>0∀i∈I+0 ∪I00,
ηH
i= 0 ∀i∈I+, ηH
i>0∀i∈I0−and ηH
iis free ∀i∈I0+,
ηH
iηG
i= 0 ∀i∈I00.
x∗∈ C
TC(x∗) = LMP V C (x∗)
LMP V C (x∗)
LMP V C (x∗) = {d∈Rn| ∇gi(x∗)Td≤0∀i∈Ig,
∇hi(x∗)Td= 0 ∀i= 1, ..., p,
∇Hi(x∗)Td= 0 ∀i∈I0+,
∇Hi(x∗)Td≥0∀i∈I00 ∪I0−,
∇Gi(x∗)Td≤0∀i∈I+0}.
x∗∈ C
(λ, µ, ηH, ηG)6= 0
(i) Pm
i=1 λi∇gi(x∗)+Pl
i=1 µi∇hi(x∗)+Pq
i=1 ηG
i∇Gi(x∗)−Pq
i=1 ηH
i∇Hi(x∗)=0
(ii) λi>0∀i∈Ig, λi= 0 ∀i /∈Ig
and ηG
i= 0 ∀i∈I+−∪I0−∪I0+, ηG
i>0∀i∈I+0 ∪I00,
ηH
i= 0 ∀i∈I+, ηH
i>0∀i∈I0−and ηH
iis free ∀i∈I0+,
ηH
iηG
i= 0 ∀i∈I00.
(iii) {xk} → x∗k∈N
m
X
i=1
λigi(xk) +
p
X
i=1
µihi(xk) +
q
X
i=1
ηG
iGi(xk)−
q
X
i=1
ηH
iHi(xk)>0.
x∗∈ C
(λ, µ, ηH, ηG)6= 0
(i) Pm
i=1 λi∇gi(x∗)+Pl
i=1 µi∇hi(x∗)+Pq
i=1 ηG
i∇Gi(x∗)−Pq
i=1 ηH
i∇Hi(x∗)=0
(ii) λi>0∀i∈Ig, λi= 0 ∀i /∈Ig
and ηG
i= 0 ∀i∈I+−∪I0−∪I0+, ηG
i>0∀i∈I+0 ∪I00,
ηH
i= 0 ∀i∈I+, ηH
i>0∀i∈I0−and ηH
iis free ∀i∈I0+,
ηH
iηG
i= 0 ∀i∈I00.

(iii) {xk} → x∗∀k∈N
λi>0⇒λigi(xk)>0{i= 1, ..., m},
µi6= 0 ⇒µihi(xk)>0{i= 1, ..., p},
ηH
i6= 0 ⇒ηH
iHi(xk)<0{i= 1, ..., q},
ηG
i>0⇒ηG
iGi(xk)>0{i= 1, ..., q}.
⇒ ⇒
⇒ ⇒
min f(x)
g(x) = x1−x260,
H(x) = x1>0,
G(x)H(x) = x1x260,
x∗= (0,0) (0,0)
x∗= (0,0) ∇H(x∗) = 1
0
d= (d1, d2)T∈R2
∇g(x∗)Td=1−1d1
d2<0,
∇H(x∗)Td=1 0 d1
d2= 0,
∇G(x∗)Td=0 1 d1
d2<0.
d2>0d2<0
λ1
−1+ηG0
1−ηH1
0=0
0,
λ>0, ηG>0ηHηG= 0 λ=ηG=ηH= 0

R2
min x2
1+x2
2
g(x) = x1≤0,
H(x) = x2≥0,
G(x)H(x) = −x1x2≤0.
x∗= (0,0) x∗
x∗(λ, ηG, ηH)6= 0
λ1
0+ηG−1
0−ηH0
1=0
0,
λ>0, ηG>0ηHηG= 0
(λ, ηG, ηH) = c(1,1,0) c > 0
x∗
λxk
1+ηG(−xk
1)−ηHxk
2=cxk
1−cxk
1−0=0,
{xk} → x∗
min f(x) s.t. F(x)∈∆,
F(x) := 



gi(x)i=1,...,m
hi(x)i=1,...,l
Gi(x)
Hi(x)i=1,...,q




∆ := 

(−∞,0]m
{0}l
Ωq

Ω := {(a, b)∈R2|b≥0, ab ≤0}.

Pα(x) := f(x) + αdist∆(F(x))
Pα(x) := f(x)+α

m
X
i=1
dist(−∞,0](gi(x)) +
l
X
j=1
dist{0}(hj(x)) +
q
X
i=1
distΩ(Gl(x), Hl(x))
,
Pα(x) := f(x) + α ||g+(x)||1+||h(x)||1+
q
X
i=1
distΩ(Gl(x), Hl(x))!,
distS(x)l1x S g+(x) = max{0, g(x)}
g+
Pα(x) = f(x)+α

m
X
i=1
|g+
i(x)|+
l
X
j=1
|hj(x)|+
q
X
i=1
max{0,−Hi(x),min{Gi(x), Hi(x)}}
.
x∗∈ C
x∗δ, c > 0
distC(x)6c ||h(x)||1+||g+(x)||1+
q
X
i=1
distΩ(Gl(x), Hl(x))!
x∈B(x∗, δ/2)
x∗
f x∗L > 0
x∗Pα
x∗
δand c
distC(x)≤cdist∆(F(x)),
x∈B(x∗, δ) > 0 2 < δ f
x∗B(x∗,2)∩ C f x∗
L f B(x∗,2)
xB(x∗, )
xπ∈ΠC(x) = {z∈ C | distC(x) = ||z−c||1}ΠC(x)
xC
||xπ−x||1≤ ||x∗−x||1≤⇒ ||xπ−x∗||1≤ ||xπ−x||1+||x−x∗||1≤2,

f(x∗)≤f(xπ)≤f(x) + L||xπ−x||1
=f(x) + LdistC(x)
=f(x) + cL dist∆F(x)
Pα¯α=cL
min x2
1+x2
2
g(x) = x1≤0,
H(x) = x2≥0,
G(x)H(x) = −x1x2≤0.
x∗= (0,0) x∗
Pα(x) = x2
1+x2
2+α[max{0, g(x)}+ max{0,−H(x),min{G(x), H (x)}}]
x∗= (0,0) α>0Pα(x)
x∗

min f(x) s.t. F (x)∈∆
f F
min f(x) s.t. F (x) + p∈∆
p∈Rt, t =m+l+q
M(p) := {x∈Rn|F(x) + p∈∆}
C=F−1(∆) = M(0).
Φ : Rp⇒
Rq(u, v)∈gphΦ
Φ (u, v)U u V v
L≥0
Φ(u0)∩V⊆Φ(u) + L||u−u0||B∀u0∈U
B:= B(0,1)
x∗∈M(0)
M(0, x∗)∈gphM.
δ > 0c > 0
distF−1(∆)(x)6cdist∆(F(x))
x∈B(x∗, δ)
x∗
x∗
F0(x∗)Tλ= 0 λ∈N∆(F(x∗))⇒λ= 0.
NΩ(a, b) = 










ξ
ζ
ξ=0=ζ;if a > 0, b < 0
ξ= 0, ζ >0 ; if a > 0, b = 0
ζ>0, ξ ·ζ= 0 ; if a =0=b
ξ60, ζ = 0 ; if a = 0, b < 0
ξ∈R, ζ = 0 ; if a = 0, b > 0











,

N(−∞,0](a) = 


{0};a < 0
[0,∞) ; a= 0
φ;a > 0


,
N{0}(0) = R.
N∆(F(x∗))
N∆(F(x∗)) =
m
Y
i=1
N(−∞,0](gi(x∗)) ×
p
Y
i=1
N{0}(hi(x∗)) ×
q
Y
i=1
NΩ(Gi(x∗, Hi(x∗)).
m
X
i=1
λi∇gi(x∗) +
p
X
i=1
µi∇hi(x∗) +
q
X
i=1
ηG
i∇Gi(x∗)−
q
X
i=1
ηH
i∇Hi(x∗)=0,
λi∈N(−∞,0](gi(x∗)) ∀i= 1, ..., m,
µi∈N{0}(hi(x∗)) ∀i= 1, ..., p,
(ηG
i,−ηH
i)∈ −NΩ(Gi(x∗), Hi(x∗)) ∀i= 1, ..., q,
=⇒(λ, µ, ηG, ηH) = 0,
M(0, x∗)∈gphM
x∗∈M(0)
∆
x∗
T∆(F(x∗)) =
m
Y
i=1
T(−∞,0](gi(x∗)) ×
p
Y
i=1
T{0}(hi(x∗)) ×
q
Y
i=1
TΩ(Gi(x∗), Hi(x∗)).
⊇ ⊆
dgi∈T(−∞,0](gi(x∗)), dhi∈T{0}(hi(x∗))
(dGi, dHi)∈TΩ(Gi(x∗), Hi(x∗))
d:= (dgi, i=1,...,m, dhi, i=1,...,p ,(dGi, dHi)i=1,...,q).

dk
gi→dgi, tk
gi↓0 with gi(x∗) + tk
gidk
gi≤0,
dk
hi→dhi, tk
hi↓0 with hi(x∗) + tk
hidk
gi= 0,
(dk
Gi, dk
Hi)→(dGi, dHi), tk
GiHi↓0 with (Hi(x∗) + tk
GiHidk
Hi)≥0
and (Gi(x∗) + tk
GiHidk
Gi)(Hi(x∗) + tk
GiHidk
Hi)≤0,
∀k∈N
dk:= dk
gi, i=1,...,m, dk
hi, i=1,...,p,(dk
Gi, dk
Hi)i=1,...,q→d.
d∈T∆(F(x∗))
tk↓0F(x∗) + tkdk∈∆,∀k∈N
tk:= min{tk
gi,i=1,...,m, tk
GiHi,1,...,q},
∀k∈Ntk↓0F(x∗) + tkdk∈∆,∀k∈N
k∈Nx∗
i= 1, ..., m dk
gi<0dk
gi≥0
dk
gi<0
gi(x∗) + tkdk
gi< gi(x∗)≤0,
dk
gi≥0
gi(x∗) + tkdk
gi≤gi(x∗) + tk
gidk
gi≤0.
hi(x∗) = 0 tk
hi>0,∀i= 1, ..., p dk
hi= 0
hi(x∗) + tkdk
hi= 0.
Hi(x∗)>0Gi(x∗)=0 Gi(x∗)<0
Gi(x∗)=0 i∈I+0 dk
Hi→dHitk
GiHi↓0
Hi(x∗) + tk
GiHidk
Hi>0 ; ∀k∈Nsufficiently large.
Hi(x∗) + tkdk
Hi>0k∈N
Gi(x∗) + tk
GiHidk
Gi≤0 ; ∀k∈N,
dk
Gi≤0
Gi(x∗) + tkdk
Gi≤0 ; ∀k∈Nsufficiently large,
⇒Hi(x∗) + tkdk
HiGi(x∗) + tkdk
Gi≤0,

F(x∗) + tkdk∈∆k∈N
Gi(x∗)<0i∈I+−Hi(x∗) + tkdk
Hi>0k
Hi(x∗) + tk
GiHidk
Hi>0
(Gi(x∗) + tk
GiHidk
Hi)≤0,by eq (11),
(Gi(x∗) + tkdk
Hi)≤0 ; for all sufficiently large k.
Hi(x∗) + tkdk
HiGi(x∗) + tkdk
Gi≤0 ; for all sufficiently large k,
i∈I+−F(x∗) + tkdk∈∆k∈N
Hi(x∗) = 0 dk
Hi≥0Hi(x∗) +
tkdk
Hi≥0k∈NGi(x∗)
dk
Hi
dk
Hi>0
Gi(x∗) + tk
GiHidk
Gi≤0 ; ∀i∈I0+ ∪I0−∪I00 ,
Gi(x∗) + tkdk
Gi≤0 ; for sufficiently large k∈N,
Hi(x∗) + tkdk
HiGi(x∗) + tkdk
Gi≤0,
i∈I0+ ∪I0−∪I00
dk
Hi= 0 Hi(x∗) + tkdk
Hi= 0
Hi(x∗) + tkdk
HiGi(x∗) + tkdk
Gi= 0,
i∈I0+ ∪I0−∪I00 F(x∗)+ tkdk∈∆
k∈N
x∗
x∗
x∗

M(p) (0, x∗)F
TC(x∗) = LC(x∗),
LC(x∗)Cx∗
LC(x∗) = {d∈Rn| ∇F(x∗)Td∈T∆(F(x∗))}.
T∆(F(x∗)) =
m
Y
i=1
T(−∞,0](gi(x∗)) ×
p
Y
i=1
T{0}(hi(x∗)) ×
q
Y
i=1
TΩ(Gi(x∗, Hi(x∗)).
LC(x∗)
LC(x∗) = {d∈Rn| ∇gi(x∗)Td∈T(−∞,0](gi(x∗)) ∀i= 1, ..., m,
∇hi(x∗)Td∈T{0}(hi(x∗)) ∀i= 1, ..., p,
(∇Gi(x∗)Td, ∇Hi(x∗)Td)∈TΩ(Gi(x∗), Hi(x∗)) ∀i= 1, ..., q}
={d∈Rn| ∇gi(x∗)Td≤0∀i∈Ig,
∇hi(x∗)Td= 0 ∀i= 1, ..., p,
∇Hi(x∗)Td= 0 ∀i∈I0+,
∇Hi(x∗)Td≥0∀i∈I00 ∪I0−,
∇Gi(x∗)Td≤0∀i∈I+0}
=LMP V C (x∗).
LMP V C
TC(x∗) = LC(x∗) = LMP V C (x∗)
x∗
min f(x) = |x1|+|x2|
g(x) = x1+x2≤0,
H(x) = x1≥0,
G(x)H(x) = x1(x2
1−x2
2)≤0.

x∗= (0,0) x∗
x∗TC(x∗) = C=LMP V C (x∗)C
Pα(x)x∗= (0,0)
x∗
x∗
M P V C −MF C Q
⇓
M P V C −GMF C Q
⇓
M P V C −generalized pseudonormality
⇓
M P V C −generalized quasinormality
⇓
M P V C −ACQ ⇐=Calmness of M (p)at (0, x∗) =⇒exactness of Pα.
Pα
l1

l1