Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems

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by  Chao Ding

CHARACTERIZATION OF THE ROBUST ISOLATED CALMNESS
FOR A CLASS OF CONIC PROGRAMMING PROBLEMS
CHAO DING∗, DEFENG SUN†, AND LIWEI ZHANG‡
Abstract. This paper is devoted to studying the robust isolated calmness of the Karush-
Kuhn-Tucker(KKT)solutionmappingforalargeclassofinterestingconicprogrammingproblems
(includingmostcommonlyknownonesarisingfromapplications)atalocallyoptimalsolution. Under
6
theRobinsonconstraintqualification,weshowthattheKKTsolutionmappingisrobustlyisolated
1 calmifandonlyifboththestrictRobinsonconstraintqualificationandthesecondordersufficient
0 condition hold. This implies, among others, that at a locally optimal solution the second order
2 sufficientconditionisneededfortheKKTsolutionmappingtohavetheAubinproperty.

t
c Key words. stability, robust isolated calmness, C2-cone reducible sets, strict Robinson con-
O straintqualification,secondordersufficientcondition,Aubinproperty

1 AMS subject classifications. 49K40,90C31,49J53

] 1. Introduction. Let and be two finite dimensional real Euclidean spaces
C X Y
each equipped with an inner product , and its induced norm . Consider the
O h· ·i k·k
following canonically perturbed optimization problem:
.
h
min f(x) a,x
t (1) −h i
a
s.t. G(x)+b ,
m ∈K
[ where f : and G : are twice continuously differentiable functions,
isXa n→on<empty closeXd c→onYvex set, and (a,b) is the perturbation
2 K ⊂ Y ∈ X ×Y
parameter.
v
For each given (a,b) , we use X(a,b) to denote the set of all locally
8 ∈ X ×Y
1 optimal solutions of problem (1). A point x X(a,b) is said to be isolated if there
∈
4 exists an open neighborhood of x such that X(a,b) = x . Let Φ(a,b) be the
V ∩V { }
7 set of all feasible points of problem (1) with a given (a,b), i.e.,
0
. (2) Φ(a,b):= x G(x)+b , (a,b) .
1 { ∈X | ∈K} ∈X ×Y
0 Let L: be the Lagrangian function of problem (1) defined by
6 X ×Y →<
1 (3) L(x;y):=f(x)+ y,G(x) , (x,y) .
: h i ∈X ×Y
v
For any y , denote the derivative of L(;y) at x by L (x;y) and denote
i ∈ Y · ∈ X 0x
X the adjoint of L (x;y) by L(x;y). For a given perturbation parameter (a,b), the
0x ∇x
r Karush-Kuhn-Tucker(KKT)optimalityconditionforproblem(1)takesthefollowing
a form:
a= L(x;y), a= L(x;y),
x x
(4) ∇ ∇
( b G(x)+∂σ(y, ) ⇐⇒ ( y (G(x)+b),
∈− K ∈NK
∗Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing, P.R. China ([email protected]). The research of this author was
supportedbytheNationalNaturalScienceFoundationofChinaunderprojectNo. 11671387.
†DepartmentofMathematicsandRiskManagementInstitute,NationalUniversityofSingapore,
10 Lower Kent Ridge Road, Singapore ([email protected]). The research of this author was
supportedinpartbytheAcademicResearchFund(GrantNo. R-146-000-207-112).
‡School of Mathematical Sciences, Dalian University of Technology, Dalian, P.R. China.
([email protected]). The research of this author was supported by the National Natural Sci-
enceFoundationofChinaunderprojectNo. 91330206andNo. 11571059.
1
where σ(y, ) := sup y,z z is the support function of , ∂σ(y, ) is the
K {h i | ∈ K} K K
sub-differential of σ(, ) at y and (z) is the normal cone of at z in the
· K NK K ∈ K
context of convex analysis [33]. For a given (a,b) , the set of all solutions
∈ X ×Y
(x,y)totheKKTsystem(4)isdenotedbyS (a,b). WeuseX (a,b)todenote
KKT KKT
the set of all stationary points of problem (1) with (a,b), i.e.,
X (a,b):= x there exists y such that (4) holds at (x,y) .
KKT
{ ∈X | ∈Y }
The set of Lagrange multipliers associated with (x,a,b) is defined by
(5) M(x,a,b):= y (x,y) S (a,b) .
KKT
{ ∈Y | ∈ }
For each given (a,b) , we say that the Robinson constraint qualification
∈ X × Y
(RCQ) for problem (1) holds at a feasible point x Φ(a,b) if
∈
(6) G0(x¯) + (G(x¯))= ,
X TK Y
where for any subset Ω in a finite dimensional Euclidean space , the tangent cone
E
[34, Definition 6.1] to Ω at z Ω is defined by
∈
(z):= d zk z with zk Ω and tk 0 such that (zk z)/tk d .
Ω
T ∈E |∃ → ∈ ↓ − →
For each giv(cid:8)en (a,b), it is well-known (cf. e.g., [7, Theorem 3.9 & Proposition 3(cid:9).17])
that the RCQ holds at a locally optimal solution x if and only if M(x,a,b) is a
∈X
nonempty, convex, and compact subset of .
Y
Inthispaper,wemainlyfocusonanimportantpropertyinperturbationtheoryfor
problem(1)atalocallyoptimalsolution: theKKTsolutionmappingS islocally
KKT
nonempty-valuedandisolatedcalm(seeDefinition2inSection2). Thispropertywill
be referred as the robust isolated calmness in this paper. When the set in problem
K
(1) is polyhedral, the theory on the robust isolated calmness is fairly complete even
underamoregeneralperturbationframework,e.g.,thefunctionsf andGinproblem
(1)havetheparametricforms: f(x,c)andG(x,c),wherec isanotherparameter
∈Z
inafinitedimensionalrealEuclideanspace [32,10,18]. Onenaturalquestionthat
Z
onemayaskistowhatextentonecangeneralizetheresultsfromthepolyhedralcase
tothenon-polyhedralcase. Forexample,DontchevandRockafellar[10]showthatfor
theparametricnonlinearprogrammingproblematalocallyoptimalsolution,S is
KKT
robustlyisolatedcalmifandonlyifboththestrictMangasarian-Fromovitzconstraint
qualification (MFCQ) and the second order sufficient condition (SOSC) hold. Does
thisresultstillholdinthenon-polyhedralsetting? Inparticularforthecase = n,
K S+
the cone of n by n symmetric and positive semi-definite matrices in n (PSD cone in
S
short), the Euclidean space of n by n symmetric matrices.
Before answering the above question, let us first consider the following convex
quadratic semi-definite programming (SDP) example constructed by Bonnans and
Shapiro.
Example 1. Bonnans and Shapiro [7, Example 4.54]:
min x +x2+x2
(7) 1 1 2
s.t. Diag(x)+εA 2,
∈S+
where x = (x ,x ) 2, Diag(x) is the 2 2 diagonal matrix whose i-th diagonal
1 2
∈ < ×
element is x , i=1,2, A is a non-diagonal matrix in 2, and ε is a scalar parameter.
i
S
It is clear that the Slater condition holds. When ε = 0, the optimization problem
2
(7) has the unique optimal solution x¯ = (0,0) with the unique Lagrange multiplier
1 0
Y = − . It is easy to see that for any given ε 0, problem (7) has a unique
0 0 ≥
(cid:20) (cid:21)
optimal solution X(ε)=(x¯ (ε),x¯ (ε)) with x¯ (ε) of order ε2/3 as ε 0.
1 2 2
→
Example 1 demonstrates that the KKT solution mapping S of a convex
KKT
quadraticSDPproblemwithastronglyconvexobjectivefunctionandwiththeSlater
condition being satisfied can still fail to be calm. This is completely different from
the polyhedral set case [32], where Robinson shows that if is a polyhedral cone,
K
the KKT solution mapping S always possesses the calmness property if both the
KKT
MFCQ and one strong form of the SOSC hold. Therefore, cautions must be taken
when extensions are made from the polyhedral to the non-polyhedral case. One may
argue that the unperturbed problem (7) in Example 1 is not a natural SDP problem
as the unperturbed problem reduces to a nonlinear programming problem. However,
it is not difficult to construct a non-lifted SDP example of similar behaviors (see
Example 3 in Section 4). This also indicates the following fact: under the general
non-polyhedralsetsetting,theuniquenessofLagrangemultipliersandtheSOSCeven
inthestrongformdonotimplythecalmness,letalonetheisolatedcalmness,ofS .
KKT
Of course, for nonlinear programming problems at a locally optimal solution, since
the strict MFCQ is equivalent to the uniqueness of Lagrange multipliers [21, Propo-
sition 1.1], the aforementioned result of Dontchev and Rockafellar says that S is
KKT
robustly isolated calm if and only if the uniqueness of Lagrange multipliers and the
SOSC hold.
The literature on perturbation analysis of general optimization problems is enor-
mous, and even a short summary about the most important results achieved would
befarbeyondourreach. Forrecentworkscoveringmanytopicsinperturbationanal-
ysis, one may refer to [6, 7, 11, 13, 19, 25, 34] and references therein. Here, we only
touch those results that are mostly relevant to the research conducted in this paper.
When the non-polyhedral set is the second-order cone (SOC) or the PSD cone,
K
the characterizations of the strong regularity of the KKT point (see Robinson [31]
for the definition) are established in [5] and [36], respectively. In [5], Bonnans and
Ram´ırez show that for a locally optimal solution of the nonlinear SOC programming
problem, under the RCQ, the KKT solution mapping is strongly regular if and only
if the strong SOSC and the constraint non-degeneracy hold. In [36], Sun shows that
for a locally optimal solution of the nonlinear SDP problem, the strong SOSC and
the constraint non-degeneracy, the non-singularity of Clarke’s Jacobian of the KKT
system and the strong regularity are all equivalent under the RCQ. The strong reg-
ularity always implies the robust isolated calmness, but not vice versa. So, in order
for characterizing the robust isolated calmness, we need weaker conditions than the
strong SOSC and the constraint non-degeneracy.
FromKingandRockafellar[17]andLevy[22],weknowthattheisolatedcalmness
of S for problem (1) with the non-polyhedral set can be characterized by the
KKT
K
non-singularity of its graphical derivative. Under the assumption that the constraint
non-degeneracy(see(12)forthedefinition)holdsatastationarypoint,notnecessarily
a locally optimal solution, to problem (1), Mordukhovich et al. [26, 27] establish an
explicitformulaforthegraphicalderivativeofS andthusacharacterizationofthe
KKT
isolated calmness of S in terms of the non-singularity of its graphical derivative.
KKT
The constraint non-degeneracy assumption is crucial for the analysis conducted in
[26, 27], in particular [26, Lemma 3.1] and [27, Proposition 3.1].
Ingeneral,thenon-singularityofthegraphicalderivativeofS maybedifficult
KKT
3
to verify. However, recently there are some good progresses in addressing this non-
singularity issue if the concerned point is a locally optimal solution. In [37], Zhang
and Zhang show that the strict Robinson constraint qualification (SRCQ) (see (11)
forthedefinition)andtheSOSCconditionimplythenon-singularityofthegraphical
derivative of S for the nonlinear SDP problem at a locally optimal solution. Han
KKT
et al. [16] show that for the nonlinear SDP problem, the isolated calmness of S
KKT
implies the SRCQ. Moreover, complete characterizations of the isolated calmness of
S are provided for convex composite quadratic SDP problems [16, Theorem 5.1].
KKT
In[23],LiuandPanextendseveralresultsin[16]and[37]tothematrixoptimization
problem constrained by the epigraph cone of the Ky Fan matrix k-norm. For the
nonlinear SOC programming problem at a locally optimal solution, Zhang et al. [38]
show that the isolated calmness of S is equivalent to the SRCQ and the SOSC.
KKT
In this paper, assuming that the set in problem (1) belongs to the class of C2-
K
cone reducible sets (Definition 6), we study the continuity of X and the robust
KKT
isolatedcalmnessofS . ItisworthmentioningthattheclassofC2-conereducible
KKT
setsisrich. Itincludesnotablyallthepolyhedralconvexsetsandmanynon-polyhedral
sets such as the SOC, the PSD cone [35, 7] and the epigraph cone of the Ky Fan
matrix k-norm [8]. Moreover, the Cartesian product of C2-cone reducible sets is also
C2-cone reducible (see, e.g., [7, 35]). Under the C2-cone reducibility assumption, we
willprovideasufficientconditionforthestationarymappingX ofthecanonically
KKT
perturbed problem (1) to be continuous and isolated. Furthermore, for the C2-cone
reducible case, under the RCQ, we present a complete characterization of the robust
isolated calmness of the KKT solution mapping S , namely, the robust isolated
KKT
calmness of S holds at a locally optimal solution if and only if the corresponding
KKT
SRCQ and the SOSC hold. Additionally, we construct an example (Example 4) to
show that our conditions for characterizing the robust isolated calmness of S
KKT
studied in this paper are strictly weaker than those for characterizing the Aubin
property (see Definition 3 in Section 2) of the KKT system (4). Thus, by combining
with [20, Theorem 1] established recently by Klatte and Kummer, we conclude that
undertheC2-conereducibleassumption,atalocallyoptimalsolutiontoproblem(1),
the constraint non-degeneracy and the SOSC are both necessary for S to have
KKT
the Aubin property.
The remaining parts of this paper are organized as follows. In Section 2, we
introduce some definitions and preliminary results on variational analysis. In Section
3, we study the continuity and the isolatedness of the stationary mapping X of
KKT
problem(1). ThecharacterizationoftherobustisolatedcalmnessoftheKKTsolution
mapping S and its implication for the Aubin property are provided in Section 4.
KKT
We conclude our paper in Section 5.
2. Preliminaries. Firstly, let us recall some common notions and definitions
related to set-valued mappings. Let be a finite dimensional real Euclidean space.
E
Let and be two finite dimensional real Euclidean spaces and Ψ : ⇒ be a
E F E F
set-valuedmappingwith(p¯,q¯) gphΨ,i.e.,q¯ Ψ(p¯),wheregphΨdenotesthegraph
∈ ∈
of Ψ. Let B be the unit ball in . The set-valued mapping Ψ is said to be lower
F F
semi-continuous at p¯for q¯if for any open neighborhood of q¯there exists an open
V
neighborhood of p¯such that
U
=Ψ(p) p .
∅6 ∩V ∀ ∈U
The mapping Ψ is said to be upper semi-continuous (in Berge’s sense [2]) at p¯ if
for any open set Ψ(p¯) there exists an open neighborhood such that for any
O ⊃ U
4
p , Ψ(p) . Furthermore, if Ψ is lower semi-continuous at (p¯,q¯) and is upper
∈ U ⊂ O
semi-continuous at p¯, then Ψ is said to be continuous at (p¯,q¯) gphΨ.
∈
For the set-valued mapping Ψ, we are interested in the following three Lipschitz-
like properties: the calmness, isolated calmness and Aubin property.
Definition 1. The set-valued mapping Ψ: ⇒ is said to be calm at p¯if there
E F
exist a constant κ>0 and an open neighborhood of p¯such that
U
Ψ(p) Ψ(p¯)+κ p p¯ B p .
⊂ k − k F ∀ ∈U
Definition 2. The set-valued mapping Ψ: ⇒ is said to be isolated calm at
E F
p¯for q¯if there exist a constant κ>0 and open neighborhoods of p¯and of q¯such
U V
that
(8) Ψ(p) q¯ +κ p p¯ B p .
∩V ⊂{ } k − k F ∀ ∈U
Moreover, Ψ is said to be robustly isolated calm1 at p¯for q¯if (8) holds and for each
p , Ψ(p) = .
∈U ∩V 6 ∅
Definition 3. The set-valued mapping Ψ : ⇒ has the Aubin property at p¯
E F
for q¯if there exist a constant κ > 0 and open neighborhoods of p¯and of q¯such
U V
that
Ψ(p) Ψ(p0)+κ p p0 B p,p0 .
∩V ⊂ k − k F ∀ ∈U
The calmnessfor theset-valuedmappingΨ given inDefinition1 comesfrom [34,
9(30)]anditwascalled“upperLipschitzian”byRobinson[30]. Theisolatedcalmness
for the set-valued mapping Ψ was called differently in the literature, e.g., the local
upper Lipschitz continuity in [10, 22], to distinguish it from Robinson’s definition of
upper Lipschitz continuity [30]. The property defined by Definition 3 was designated
“pseudo-Lipschitzian” by Aubin [1].
Remark 1. Itis worthnotingthat theset-valuedmapping Ψisisolatedcalm atp¯
forq¯doesnotimpliesΨisrobustisolatedcalmatp¯forq¯ingeneral(see, forinstance,
[27, Example 6.4]). However, if Ψ is lower semi-continuous at (p¯,q¯), then it follows
from the definition that the robust isolated calmness of Ψ also holds.
The graphical derivative [34, Definition 8.33] of Ψ at p¯for q¯is a set-valued map-
ping DΨ(p¯q¯): ⇒ defined by
| E F
DΨ(p¯q¯)(u):= v (u,v) (p¯,q¯) ,
gphΨ
| { ∈F | ∈T }
which is a convenient tool for investigating the isolated calmness property. In fact,
wehavethefollowingbasiccharacterizationoftheisolatedcalmnessoftheset-valued
mapping Ψ at p¯for q¯.
Lemma 4 (King and Rockafellar [17], Levy [22]). Let (p¯,q¯) gphΨ. Then Ψ is
∈
isolated calm at p¯for q¯if and only if 0 =DΨ(p¯q¯)(0).
{ } |
In order to study the relationship between the isolated calmness and the Aubin
property of the set-valued mappings, we need the following result.
Lemma 5 (Fusek [14]). Suppose that F : is locally Lipschitz continuous
E → E
near q¯ and that F is directionally differentiable at q¯. If F 1 has the Aubin
−
∈ E
property at p¯:= F(q¯) for q¯, then there exists an open neighborhood of q¯such that
V
F 1(p¯) = q¯ .
−
∩V { }
1If Ψ=Θ−1 is the inverse of a given set-valued mapping Θ:Y ⇒X, then the robust isolated
calmnessofΨatp¯forq¯isequivalenttotheupperregularity[19]ofΘatq¯forp¯.
5
It then follows from Lemma 5 that for a function F : satisfying the
E → E
assumptions in this lemma, the Aubin property of F 1 implies the isolated calmness
−
of F 1.
−
Suppose that x¯ is a feasible solution to problem (1) with (a,b) = (0,0). The
critical cone (x¯) of (1) with (a,b)=(0,0) at x¯ is defined by
C
(9) (x¯):= d G0(x¯)d (G(x¯)), f0(x¯)d 0 .
C { ∈X | ∈TK ≤ }
If x¯ is a stationary point of problem (1) with (a,b)=(0,0) and y¯ M(x¯,0,0), then
∈
(x¯)= d G0(x¯)d (G(x¯)), f0(x¯)d=0
C { ∈X | ∈TK }
= d G0(x¯)d (G(x¯),y¯) ,
{ ∈X | ∈CK }
where for any A , (A,B) is the critical cone of at A with respect to B
∈ K CK K ∈
(A) defined by
NK
(10) (A,B):= (A) B⊥,
CK TK ∩
and for any s , s := z z,s = 0 . Since is convex, it is clear that for
⊥
∈ Y { ∈ Y | h i } K
each y¯ M(x¯,0,0), the critical cone (G(x¯),y¯) is indeed a closed convex cone.
∈ CK
The SRCQ is said to hold for problem (1) with (a,b) = (0,0) at x¯ with respect
to y¯ M(x¯,0,0)= if
∈ 6 ∅
(11) G0(x¯) + (G(x¯)) y¯⊥ = .
X TK ∩ Y
Itfollowsfrom[7,Proposition4.50]thatthesetofLagrangemultipliersM(x¯,0,0)isa
singletoniftheSRCQholds. Theconstraintnon-degeneracy,introducedbyRobinson
[31], is said to hold at x¯ if
(12) G0(x¯) +lin( (G(x¯)))= ,
X TK Y
where lin( (G(x¯))) is the lineality space of (G(x¯)), i.e., the largest linear space
TK TK
in (G(x¯)). It is well-known (cf. [7, Proposition 4.73]) that the constraint non-
TK
degeneracy is stronger than the SRCQ since lin( (G(x¯))) (G(x¯)) y¯ .
⊥
TK ⊂TK ∩
Through the whole paper, we always assume that the set has the following
K
C2-cone reducibility.
Definition 6 ([7, Definition 3.135]). The closed convex set is said to be C2-
K
cone reducible at A , if there exist a open neighborhood of A, a pointed
∈ K W ⊂ Y
closed convex cone (a cone is said to be pointed if and only if its lineality space
Q
is the origin) in a finite dimensional space and a twice continuously differentiable
Z
mapping Ξ : such that: (i) Ξ(A) = 0 ; (ii) the derivative mapping
W → Z ∈ Z
Ξ(A) : is onto; (iii) = A Ξ(A) . We say that is
0
Y → Z K∩W { ∈ W | ∈ Q} K
C2-cone reducible if is C2-cone reducible at every A .
K ∈K
Let Z be a closed set in . Recall that the inner and outer second order tangent
Y
sets ([7, (3.49) and (3.50)]) to the given closed set Z in the direction h can be
∈ Y
defined, respectively, by
1
i,2(z,h):= w dist(z+th+ t2w,Z)=o(t2), t 0
TZ { ∈Y | 2 ≥ }
and
2(z,h):= w t 0 such that dist(z+t h+ 1t2w,Z)=o(t2) ,
TZ { ∈Y |∃ k ↓ k 2 k k }
6
where for any s , dist(s,Z) := inf s z z Z . Note that in general,
i,2(z,h) = 2(z,∈h)Yeven if Z is convex ({[7k, S−ectkio|n 3∈.3]).}However, it follows from
TZ 6 TZ
[7, Proposition 3.136] that if Z is a C2-cone reducible convex set, then the equality
alwaysholds. Inthiscase, 2(z,h)willbesimplycalledthesecondordertangentset
TZ
to Z at z Z in the direction h .
∈ ∈Y
For the C2-cone reducible set , we have the following “no gap” second order
K
conditions for problem (1) with (a,b) = (0,0) (see [4, Theorem 3.1 and Theorem
4.1]).
Theorem 7. Suppose that x¯ is a locally optimal solution to problem (1) with
(a,b) = (0,0) and the RCQ holds at x¯. Then the following second-order necessary
condition holds
(13) y Msu(x¯p,0,0) d,∇2xxL(x¯;y)d −σ y,TK2(G(x¯),G0(x¯)d) ≥0 ∀d∈C(x¯),
∈ (cid:8)(cid:10) (cid:11) (cid:0) (cid:1)(cid:9)
where for any y , 2 L(x¯;y) is the Hessian of L(;y) at x¯. Conversely, suppose
∈ Y ∇xx ·
x¯ is a stationary point of problem (1) with (a,b) = (0,0) and the RCQ holds at x¯.
Then the following second order sufficient condition (SOSC)
(14) y Msu(x¯p,0,0) d,∇2xxL(x¯;y)d −σ y,TK2(G(x¯),G0(x¯)d) >0 ∀d∈C(x¯)\{0}
∈ (cid:8)(cid:10) (cid:11) (cid:0) (cid:1)(cid:9)
isnecessaryandsufficientforthequadraticgrowthconditionatthepointx¯forproblem
(1) with respect to (a,b)=(0,0).
Next,welistsomeresultsthatareneededforoursubsequentdiscussionsfromthe
standard reduction approach. For more details on the reduction approach, one may
refer to [7, Section 3.4.4]. The following results on the representations of the normal
cone and the “sigma term” of the C2-cone reducible set are stated in [7, (3.266)
K
and (3.274)].
Lemma 8. LetA begiven. Then,thereexistanopenneighborhood of
∈K W ⊂Y
A, a pointed closed convex cone in a finite dimensional space and a twice contin-
Q Z
uously differentiable function Ξ : satisfying conditions (i)-(iii) in Definition
W → Z
6 such that for all A sufficiently close to A,
∈W
(15) (A)=Ξ0(A)∗ (Ξ(A)),
NK NQ
whereΞ(A) : istheadjointofΞ(A). Inparticular,foranyB (A),there
0 ∗ 0
is a unique elemZen→t uYin (Ξ(A)) such that B =Ξ(A) u, denoted by∈(NΞK(A) ) 1B.
0 ∗ 0 ∗ −
NQ
Furthermore, we have for any D (A,B),
∈CK
(16) σ(B, 2(A,D))= (Ξ0(A)∗)−1B,Ξ00(A)(D,D) .
TK −
For a feasible solution x¯ to prob(cid:10)lem (1) with (a,b) = (0,0)(cid:11), let , and Ξ
W Q
be the open neighborhood of G(x¯), the pointed closed convex cone and the twice
continuously differentiable function defined in Lemma 8, respectively, with respect to
G(x¯) . Since G is continuous, we know that there exist open neighborhoods of
∈K U
the origin in and of x¯ such that G(x)+b for any (x,a,b) .
X ×Y V ∈ W ∈ V ×U
Consequently, problem (1) is locally equivalent to the following reduced problem:
min f(x) a,x
(17) −h i
s.t. (x,b) ,
G ∈Q
where (x,b) := Ξ(G(x)+b) for each (x,a,b) , in the sense that the sets
G ∈ V ×U
of optimal solutions to (1) and (17) restricted to are the same. Moreover, for
V
7
(a,b)=(0,0), it is known from [7, Section 3.4.4] that the RCQ for problem (1) holds
at the feasible point x¯ if and only if the RCQ for problem (17) holds at x¯.
Next, we shall present some useful results about the directional derivative of the
metric projection operator over the C2-cone reducible set . Suppose that B
K ∈
(A). LetC :=A+B. ThenwehaveA=Π (C),whereΠ : isthemetric
NK K K Y →Y
projection operator over , i.e., for any C ,
K ∈Y
1
Π (C):=argmin Y C 2 Y .
K 2k − k | ∈K
(cid:26) (cid:27)
Since isC2-conereducible, weknowfrom[3, Theorem7.2]thatΠ isdirectionally
K K
differentiable at C and the directional derivative Π (C;H) for any direction H
0
istheuniqueoptimalsolutiontothefollowingstronKglyconvexoptimizationprobl∈emY:
(18) min D H 2 σ(B, 2(A,D)) D (A,B) ,
k − k − TK | ∈CK
(cid:8) (cid:9)
where (A,B) is the critical cone of at A with respect to B defined by (10). It
CK K
followsfrom(16)inLemma8thatthereexistsaself-adjointlinearoperator :
H Y →Y
such that
(19) Υ(D):= D, (D) = σ(B, 2(A,D)) 0 D (A,B),
h H i − TK ≥ ∀ ∈CK
which means that that is co-positive on the cone (A,B). However, this does
H CK
not mean that is positive semi-definite on the whole space . To overcome this
H Y
difficulty, let us define h: ( , ] by
Y → −∞ ∞
(20) h(D):=Υ(D)+δ (D), D ,
(A,B) ∈Y
CK
where δ () is the indicator function of the critical cone (A,B).
ProCpKo(As,iBt)io·n 9. The function h : ( , ] defineCdKby (20) is a closed
Y → −∞ ∞
proper convex function and the sub-differential ∂h(D) of h at any D (A,B) is
∈ CK
given by
∂h(D)= Υ(D)+ (D).
∇ N (A,B)
CK
Proof. First, by using (19) and (20), we know that
h(D)= σ(B, 2(A,D))+δ (D) D .
− TK CK(A,B) ∀ ∈Y
Since δ (D) = 0 for any D (A,B) and the function σ(B, 2(A, )) is
a closedCKp(rAo,Bpe)r convex function [4,∈LeCmKma 4.1], it follows that h(−) is alTsKo a cl·osed
·
proper convex function on . Moreover, we know from [34, Proposition 8.12] (or [25,
Y
Theorem 1.93]) that ∂h(D) = ∂ h(D) for any D (A,B), where ∂ h(D) is the
L L
∈ CK
limiting sub-differential of h at D (cf. e.g., [25, Definition 1.77]). Thus, it follows
from the sum rule [25, Proposition 1.107 (ii)] that
∂h(D)=∂ h(D)= Υ(D)+ (D) D (A,B).
L ∇ NCK(A,B) ∀ ∈CK
This completes the proof of this proposition.
For any given nonempty convex cone K , we use K to denote the polar
◦
⊂ Y
of K, i.e., K := z z,s 0 s K . The following simple lemma is a
◦
{ ∈ Y | h i ≤ ∀ ∈ }
generalization of [16, Lemmas 4.1 & 4.3].
Lemma 10. Let C , A=Π (C) and B =C A.
∈Y K −
8
(i) Let A, B . A Π (C; A+ B)=0 if and only if
0
4 4 ∈Y 4 − K 4 4
A (A,B),
4 ∈CK
(21) B 1 Υ( A) [ (A,B)] ,
 4 − 2∇ 4 ∈ CK ◦
 A, B = σ(B, 2(A, A)),
h4 4 i − TK 4
where Υ() is the quadratic function defined by (19).
·
(ii) Let : be a linear operator. Then, the following two statements are
A X →Y
equivalent:
(a) B is a solution to the following system of equations
4 ∈Y
B =0,
∗
A 4
( Π0 (C; B)=0;
K 4
(b) B + (A) B⊥ ◦.
4 ∈ AX TK ∩
Proof. We firsthprove part (i). Letih : ( , ] be defined by (20). From
Y → −∞ ∞
Proposition9weknowthath()isaclosedproperconvexfunctionandD (A,B)
· ∈CK
is the unique optimal solution to problem (18) with H =∆A+∆B if and only if D
is the unique optimal solution to the following strongly convex optimization problem
min D (∆A+∆B) 2+h(D) ,
k − k
or equivalently, (cid:8) (cid:9)
0 2(D ( A+ B))+ Υ(D)+ (D).
∈ − 4 4 ∇ N (A,B)
CK
Therefore, by using the fact that (A,B) is a closed convex cone, we can see that
CK
A=Π (C; A+ B) if and only if
0
4 K 4 4
A (A,B),
4 ∈CK
B 1 Υ( A) [ (A,B)] ,
 4 − 2∇ 4 ∈ CK ◦
 A, B 1 Υ( A) =0,
h4 4 − 2∇ 4 i
which, together with thefact that for each A (A,B),
4 ∈CK
A, Υ( A) =2Υ( A)= 2σ(B, 2(A, A)),
h4 ∇ 4 i 4 − TK 4
show that (21) holds.
By noting 0 (A,B) and taking A = 0 in part (i), we obtain part (ii)
∈ CK 4
immediately.
3. The continuity and isolatedness of X . Let x be a feasible solu-
KKT
∈X
tion to problem (1) with (a,b)=(0,0), i.e., x Φ(0,0). It is well-known (cf. e.g., [7,
∈
Themorem 2.87] and [34, Theorem 9.43]) that the feasible point mapping Φ defined
by (2) has the Aubin property at (0,0) for x if the RCQ (6) holds at x. Therefore,
the following result on the existence of local minimizers of problem (1) extends [10,
Lemma 2.5] for nonlinear programming.
Lemma 11. Suppose that x¯ is an isolated locally optimal solution of problem (1)
with (a,b) = (0,0). If the RCQ (6) holds at x¯, then the locally optimal solution
mapping X is lower semi-continuous at (0,0,x¯) gphX.
∈
9
Proof. SincethefeasiblesolutionmappingΦhastheAubinpropertyat(0,0)for
x¯, we know that there exist ε >0, ε >0 and κ>0 such that for any (a,b) <ε ,
1 2 1
k k
(a,b) <ε ,
0 0 1
k k
(22) Φ(a,b) x x x¯ <ε2 Φ(a0,b0)+κ (a,b) (a0,b0) B .
∩{ ∈X |k − k }⊂ k − k Y
Let be an arbitrary open neighborhood of x¯. Since x¯ is an isolated locally optimal
V
solutionofproblem(1)with(a,b)=(0,0),weareabletochooseτ (0,ε )sufficiently
2
∈
small such that x x x¯ <τ and x¯ is the unique minimizer of (1) with
{ ∈X |k − k }⊂V
(a,b) = (0,0) in x x x¯ < τ . For the fixed τ and for any (a,b) < ε ,
1
{ ∈ X | k − k } k k
define the following set-valued mapping
Φ (a,b):=Φ(a,b) x x x¯ τ +κ (a,b) .
τ
∩{ ∈X |k − k≤ k k}
Note that the graph of Φ is closed, which means that Φ is closed. Also, for
τ τ
any (a,b) <ε , Φ (a,b) belongs the compact subset x x x¯ ε +κε .
1 τ 2 1
k k { ∈X |k − k≤ }
Therefore, we know (cf. e.g., [7, Lemma 4.3 (i)]) that Φ is upper semi-continuous at
τ
any (a,b) <ε .
1
k k
On the other hand, for any x Φ (0,0) = Φ(0,0) x x x¯ τ , we
τ
∈ ∩{ ∈X |k − k≤ }
know from (22) that if (a,b) is sufficiently close to (0,0), there exists xˆ Φ (a,b)
τ
∈
such that x xˆ κ (a,b) . Therefore, we have
k − k≤ k k
xˆ x¯ xˆ x + x x¯ κ (a,b) +τ,
k − k≤k − k k − k≤ k k
which implies xˆ Φ (a,b). By noting that xˆ x as (a,b) (0,0), we conclude that
τ
∈ → →
the set-valued mapping Φ is also lower semi-continuous at (0,0).
τ
Next, for any (a,b) sufficiently close to (0,0), consider the following optimization
problem
min f(x) a,x
(23) −h i
s.t. x Φ (a,b).
τ
∈
It follows from the Berge theorem (cf. [12, Chapter 9, Theorem 3]) that the solution
mapping X of (23) is nonempty around (0,0) and upper semi-continuous at (0,0).
τ
Thus,bynotingthatX (0,0)=X(0,0)= x¯ ,wehaveforanygiven0<ε <τ there
τ { } 02
exists 0<ε <ε such that for any (a,b) <ε , X (a,b) x x x¯ <ε ,
01 1 k k 01 τ ⊆{ ∈X |k − k 02}
and for any solution x X (a,b) of (23),
τ
∈
kx−x¯k≤ε02 <τ +κk(a,b)k.
Therefore, for any (a,b) < ε , the constraint x x¯ τ +κ (a,b) of (23) is
k k 01 k − k ≤ k k
inactive at any solution point x X (a,b), which implies that
τ
∈
∅=6 Xτ(a,b)⊂X(a,b)∩{x∈X |kx−x¯k<ε02}⊂X(a,b)∩V.
Thus, we know that X is lower semi-continuous at (0,0,x¯) gphX. The proof is
∈
completed.
For any given open set , we use X to denote the set-valued
KKT
O ⊂ X ∩ O
mapping XKKT : ⇒ defined by (XKKT )(a,b) = XKKT(a,b)
∩O X ×Y X ∩O ∩O
for all (a,b) . By using the reduction approach, we can extend Robinson’s
∈ X ×Y
classical result [32, Theorem 2.3] to problem (1) easily. Note that we are not able
to apply [32, Theorem 2.3] to problem (1) directly, since the closed convex set in
K
problem (1) may not be a cone.
10