No Gap Second-Order Optimality Conditions for Circular Conic Programs
Yue Lua, Jein-Shan Chenb , and Ning Zhangc
aSchool of Mathematical Sciences, Tianjin Normal University, Tianjin, China;bDepartment of Mathematics, National Taiwan Normal University, Taipei, Taiwan;cDepartment of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
ABSTRACT
In this article, we study the second-order optimality conditions for a class of circular conic optimization problem. First, the expli- cit expressions of the tangent cone and the second-order tan- gent set for a given circular cone are derived. Then, we establish the closed-form formulation of critical cone and calculate the
“sigma” term of the aforementioned optimization problem. At last, in light of tools of variational analysis, we present the asso- ciated no gap second-order optimality conditions. Compared to analogous results in the literature, our approach is intuitive and straightforward, which can be manipulated and verified. An example is illustrated to this end.
ARTICLE HISTORY Received 29 May 2018 Revised 23 November 2018 Accepted 23 November 2018 KEYWORDS
Circular cone; no gap second-order optimality conditions; second-order tangent set;“sigma” term;
tangent cone
1. Introduction
Consider the following general circular conic optimization problem min f xð Þ
s:t: h xð Þ ¼ 0;
g xð Þ 0;
Gi1ð Þ; Gx i2ð Þx
2 Lhi; i ¼ 1; 2; :::; J;
(1.1)
where f : Rn ! R; h : Rn ! Rl, g : Rn ! Rm; Gi1: Rn ! R, Gi2 : Rn ! Rsi1 ði ¼ 1; 2; :::; JÞ are assumed to be twice continuously differentiable.
Here Lhi denotes a circular cone in Rsi given by
Lhi :¼ ðx1; x2Þ 2 R Rsi1j jjx2jj x1tan hi
n o
(1.2) with hi being its half-aperture angle and hi 2 ð0;p2Þ. From definition, it is clear that Lp4 is the set of second-order cone Ksi.
During the past decade, optimization problems associated with circular conic constraints have become an important type of conic programing problems, which is used to modelize engineering problems. In particular, when dealing with the optimal grasping manipulation problems for
CONTACT Jein-Shan Chen jschen@math.ntnu.edu.tw Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan.
ß 2019 Taylor & Francis Group, LLC
https://doi.org/10.1080/01630563.2018.1552965
multifingered robots [1], the normal force of the ith finger ui1 and the associated another forces ui2; ui3 satisfy the following condition
k uð i2; ui3Þk lui1;
where jj jj represents the Euclidean norm defined in Rn and l denotes the friction that depends on the angle h. If l ¼ tan h and h 6¼p4, then the above problem is a typical circular cone constrained problem. At the same time, many researchers have paid attention to theoretical analysis and algorithm design for circular conic programs. Recently, some fundamental results includ- ing the spectral factorization and the metric projection onto a given circular cone Lh are established in [2–4]. On the other hand, due to the nonself-duality of circular cones, there exist very few algorithms for dealing with circular conic programs. More specifically, some algorithms including prime-dual interior- point algorithms and smoothing Newton algorithm have been proposed for circular conic programing problems, see [5–7]. In addition, for circular conic complementarity problems, some merit functions are constructed in [8].
From theoretical aspect of optimization, variational geometries including contingent cone, inner tangent cone, outer second-order tangent set and inner second-order tangent set are crucial to establishing optimality condi- tions [9–11]. Generally speaking, there have been two technical ways to obtain the aforementioned variational geometries regrading circular cone Lh. The first one follows from the methodology proposed by Zhou and Chen in their article [2], which depends on the relationship between the circular cone Lh and the second-order cone Ks, that is,
x:¼ x1
x2
2 Lh () tan h 0
0 I
x1
x2
2 Ks: (1.3) The other approach is through differential properties of vector-valued functions associated with circular cones [12–16], in which the following cir- cular cone function
fLhð Þx :¼ f kð 1ð Þx Þuð Þx1 þ f kð 2ð Þx Þuð Þx2;
is employed. Here f : R ! R is a given real-valued function and x ¼ ðx1; x2Þ 2 R Rs1 has the spectral decomposition given by
x:¼ k1ð Þux ð Þx1 þ k2ð Þux ð Þx2; where
k1ð Þx :¼ x1jjx2jjcot h; k2ð Þx :¼ x1þ jjx2jj tan h and
uð Þx1 :¼ 1 1 þ cot2h
1 0
0 cot h I
" #
1
x2
" #
uð Þx2 :¼ 1 1 þ tan2h
1 0
0 tan h I
x12
with x2 :¼ x2=jjx2jj if x26¼ 0 and x2 being any vector w 2 Rs1 satisfying jjwjj ¼ 1 if x2 ¼ 0. The tangent cone and the second-order tangent set of Lh can be characterized by the directional derivatives of circular cone func- tions, see [16, Section 4] for more details. Compared to the above two methods, in this article, we present an alternative way to obtain the explicit forms of the tangent cone and the second-order tangent set of Lh, which only relies on basic definitions of its variational geometries and an useful lemma about how to calculate these results under the case for the level set of a class of Lipschitz continuous convex functions (see Lemma 2.2 below).
In other words, our approach is intuitive and straightforward, which can be manipulated and verified. An example is illustrated to this end.
With the development of modern optimization, second-order optimality theory plays an important role in perturbation analysis [17–20], stability analysis [21–24] and numerical algorithm design [25]. Among these topics, the characterization of no gap second-order optimality condition is a very important issue, which is closely related to the quadratic growth condition.
It was shown by Drusvyatskiy and Lewis [26] recently that the quadratic growth condition has a strongly impact on establishing the metric subregu- larity and calmness of set-valued mappings, the existence of error bounds and convergence rates of numerical algorithms. From different views, the metric subregularity and the calmness of set-valued mappings are the core concepts in nonsmooth calculus and perturbation analysis of variational problems. We refer the readers to the monographs by Dontchev and Rockafellar [27], Bonnans and Shapiro [19] and references therein for a comprehensive study on both theory and applications of related subjects [28–32]. However, to our best knowledge, no results about the no gap second-order optimality conditions for the general circular conic optimization problem (1.1) have been reported.1 Hence, the purpose of this article aims to fill this gap and the contributions of our research can be summarized as follows.
1While finalizing a first version of this work, the authors became aware of an important observation made in Bonnans et al. [5], mainly focus on perturbation analysis on second-order cone programming. One possible way to obtain the results discussed in this article is to transform the circular conic constraints to the second-order cone constraints via the relation (1.3) and then adapt the conclusions based on the framework of second-order cone programming [5]. However, in this article we adopt a constructive way to deal with our mentioned issues.
We have the following two reasons: (a) Through these qualitative analysis, we can learn more details on the structure of circular cone, which plays a crucial role on developing optimization theory for nonsymmetric cones.
(b) The parameters in our discussion have an important effect on establishing the associated error bound analysis as Drusvyatskiy and Lewis [8] and consequently analyzing convergence rate of numerical algorithms such as proximal point method and its variants.
a. We propose an alternative way to derive the variational geometries of a given circular cone Lh.
b. We present explicit forms of the critical cone and the “sigma” term for the given circular conic program (1.1).
c. We establish the equivalent relationship between the no gap second- order optimality conditions and the quadratic growth condition of (1.1).
The rest of this article is organized as follows. In Section 2, we recall some frequently used concepts from variational analysis [9, 11] and explore the variational geometries (including the tangent cone and the second- order tangent set) of a given circular cone. In Section 3, we first present the closed-form of the critical cone and then calculate the “sigma” term of (1.1) directly. After these preparations, we state the no gap second-order optimality conditions for the given circular conic optimization problem.
Moreover, we illustrate an example to verify these results in Section 4.
Finally, some concluding remarks are drawn in Section 5.
1.1. Notation and terminology
In what follows, we use distðx; XÞ to denote the distance between the vector x and the given set X Rn, that is, distðx; XÞ :¼ infz2Xjjxzjj. Lh is the dual cone of a given circular cone Lh, which is defined by Lh:¼ fv 2 Rsj vTx 0; 8x 2 Lhg: From [2, Theorem 2.1], the structure of Lh can be described as
Lh¼ ðx1; x2Þ 2 R Rs1j jjx2jj x1coth
n o
¼ Lp2h:
The interior and the boundary of Lh are denoted by int Lh and bd Lh, respectively. In addition, we let ker ðAÞ and range ðAÞ denote the kernel and the range of A, respectively, that is,
ker Að Þ :¼ x j Ax ¼ 0f g; range Að Þ :¼ y j 9 x such that y ¼ Ax : For a lower semicontinuous function w: Rn ! R, the directional deriva- tive of w at x along the direction h is denoted by w0ðx; hÞ, which is given by
w0ðx; hÞ :¼ lim
t#0
w x þ thð Þw xð Þ
t :
If w is directionally differentiable at x at every direction h, we say that w is directionally differentiable at x. Moreover, the parabolic second-order directional derivative of w at x is defined by
w00ðx; h; wÞ :¼ lim
t#0
w x þ th þ 12t2w
w xð Þw0ðx; hÞ
1
2t2 :
2. Basic tools for the circular cone
As mentioned, we recall some concepts from variational analysis that will be used for subsequent analysis. First, we review the definitions of the tan- gent cone and the second-order tangent set for a given closed set X Rn, which come from Bonnans and Shaprio’s monograph [19, Definition 2.54 and Definition 3.28].
Definition 2.1. Let X Rn be a given closed set and x 2X. The (Bouligand-Severi) tangent/contingent cone to X at x 2 X is defined by
TXð Þx :¼ h 2 R nj 9 tn # 0; dist x þ tð nh; XÞ ¼ o tð Þn :
Similarly, the inner tangent cone to X at x 2 X is given in the form of TiXð Þx :¼ h 2 R nj dist x þ th; Xð Þ ¼ o tð Þ; t 0
:
In addition, if h 2 TXðxÞ, the outer second-order tangent set to X at x along the direction h is defined as
T2Xðx; hÞ :¼ w 2 Rnj 9 tn # 0; dist x þ tnh þ1 2t2nw; X
¼ o t n2
: Similarly, if h 2 TiXðxÞ, the inner second-order tangent set to X at x along the direction h is given by
Ti;2Xðx; hÞ :¼ w 2 Rnj dist x þ th þ1 2t2w; X
¼ o tð Þ; t 02
: Let X Rn be a closed convex set and x 2X. It follows from [19, Section 2.2.4] that the contingent cone TXðxÞ coincides with the inner tan- gent cone TiXðxÞ, that is, TXðxÞ ¼ TiXðxÞ. In addition, if the set X is second-order regular at x (see [19, Definition 3.85] for details), the follow- ing conditions hold at x:
(i) T2Xðx; hÞ ¼ Ti;2Xðx; hÞ for all h 2 TXðxÞ.
(ii) For any h 2 TXðxÞ and for any sequence x þ tnh þ12tn2h 2X such that tnrn! 0 and
n!1lim dist r n; T2Xðx; hÞ
¼ 0:
Moreover, from [2, Theorem 2.8], we know that the circular cone Lh is closed and second-order regular. Hence, in the sequel we only need to figure out the explicit forms for the contingent cone TLhðxÞ and the outer second- order tangent set T2Lhðx; hÞ. To this end, we need a technical lemma, which describes the tangent cone and the second-order tangent set for a level set of a given convex function. We only state it without presenting its proof because it can be found in [19, Proposition 2.61 and Proposition 3.30].
Lemma 2.2 Let w: Rn ! R be a lower semicontinuous convex function.
Consider the associated level set X :¼ fx 2 Rnj wðxÞ 0g. Suppose that w is Lipschitz continuous at x and wðxÞ ¼ 0. In addition, there exists x 2 Rn such that wðxÞ<0 (Slater condition). Then,
TXð Þ ¼ h 2 Rx nj w0ðx; hÞ 0
: (2.1.)
Moreover, for a given h 2 Rn satisfying w0ðx; hÞ ¼ 0, the outer second- order tangent set toX at x along the direction h can be described as
T2Xðx; hÞ ¼ w 2 R nj w00ðx; h; wÞ 0
: (2.2)
With Lemma 2.2, we are ready to express the explicit form of the tangent cone TLhðxÞ at any given x 2 Rs.
Theorem 2.3. Let x ¼ ðx1; x2Þ 2 R Rs1. Then, the tangent cone to Lh at x can be written as
TLhð Þ ¼x
Rs; if x 2 int Lh;
Lh; if x ¼ 0;
h1; h2
ð Þ 2 R Rs1j hT2x2 h1x1tan2h 0
; if x 2 bd Lhn 0f g:
8<
:
Proof. The explicit form of TLhðxÞ is deduced by discussing two cases.
(a) If x 2 int Lh or x ¼ 0, from Definition 2.1, we immediately obtain TLhð Þ ¼x Rs; if x 2 int Lh;
Lh; if x ¼ 0:
(b) If x 2 bd Lhn f0g, then x1tan h ¼ jjx2jj 6¼ 0. Using the definition of Lh as in (1.2), Lh can be rewritten as
Lh ¼nðx1; x2Þ 2 R Rs1j / xð Þ 0o
;
where /: Rs ! R is given by /ðxÞ :¼ jjx2jjx1tan h: It is easy to verify that / is continuously differentiable, Lipschitz continuous at x and the cor- responding Slater condition holds under this case. Hence, it follows from Lemma 2.2 that TLhðxÞ can be described as
TLhð Þ ¼ h 2 Rx sj /0ðx; hÞ 0
: (2.3)
Note that
/0ðx; hÞ ¼ r/ xð ÞTh ¼ tan h x2 jjx2jj
T
h1
h2
¼ hT2x2
jjx2jjh1tan h:
Applying the relation x1tan h ¼ jjx2jj and (2.3) yield that TLhð Þ ¼x ðh1; h2Þ 2 R Rs1j hT2x2 h1x1tan2h 0
:
Thus, the proof is complete. w
Next theorem describes the outer second-order tangent set T2Lhðx; hÞ at any x 2 Rs and h 2 TLhðxÞ.
Theorem 2.4. Let x ¼ ðx1; x2Þ 2 R Rs1 and h ¼ ðh1; h2Þ 2 TLhðxÞ. The outer second-order tangent set to Lh at x along the direction h can be described as
T2Lhðx; hÞ ¼ Rs; if h 2 int TLhð Þ;x TLhð Þ; if x ¼ 0;h
N; if x 2 bd Lhn 0f g; h 2 bd TLhð Þ;x 8<
: where the set N is defined by
N :¼nðw1; w2Þ 2 R Rs1j wT2x2 w1x1tan2h h21tan2h jjh2jj2o :
Proof. Again, we derive the explicit form of T2Lhðx; hÞ by discussing two cases.
(a) If h 2 int TLh or x ¼ 0, from Definition 2.1, we have T2Lhðx; hÞ ¼ Rs; if h 2 int TLh;
TLhð Þ; if x ¼ 0:h
(b) If x 2 bd Lhn f0g and h 2 bd TLhðxÞ, we have
0 6¼ jjx2jj ¼ x1tan h; hT2x2h1x1tan2h ¼ 0: (2.4) Then, it follows from Lemma 2.2 that the second-order tangent set T2Lhðx; hÞ has the form of
T2Lhðx; hÞ ¼ w 2 R sj /00ðx; h; wÞ 0
; (2.5)
where /ðxÞ:¼ jjx2jjx1tan h. Note that /00ðx; h; wÞ
¼ r/ xð ÞTw þ hTr2/ xð Þh
¼ tan hjjxx2
2jj
h iT w1
w2
þ h½ 1h2 T 0 0
0 1
jjx2jjIs1 1 jjx2jj3x2xT2 2
4
3 5 h1
h2
¼wT2x2
jjx2jjw1tan h þjjh2jj2
jjx2jj hT2x2
2 jjx2jj3
¼ 1
jjx2jjwT2x2w1x1tan2h
þjjh2jj2
jjx2jj ðh1x1tan2hÞ2 x1tan h
ð Þ3
¼ 1
jjx2jjwT2x2w1x1tan2h
þjjh2jj2
jjx2jj h21tan2h x1tan h
¼ 1
jjx2jj wT2x2w1x1tan2h þ jjh2jj2h21tan2h
;
where the last two equalities are due to (2.4). Hence, under this case, the above expression together with (2.5) imply that
T2Lhðx; hÞ ¼nðw1; w2Þ 2 R Rs1j wT2x2 w1x1tan2h h21tan2h jjh2jj2o :
Thus, the proof is complete. w
To end this section, we introduce an useful complementarity property of the circular cone Lh, which plays a major role in the analysis of the Karush-Kuhn-Tucker (KKT) condition for (1.1).
Theorem 2.5. For any x ¼ ðx1; x2Þ and y ¼ ðy1; y2Þ in R Rs1. The system x 2 Lh; y 2 Lh; xTy ¼ 0
has at least one solution if and only if one of the following cases holds.
(a) x ¼ 0s, y 2 Lh. (b) x 2 int Lh, y ¼ 0s. (c) x 2 bd Lhn f0sg, y ¼ 0s.
(d) x 2 bd Lhn f0sg, y 2 bd Lhn f0sg, and there exists r>0 such that x ¼rðHyÞ, where
H :¼ tan2h 0 0 Is1
:
Proof. The “sufficiency” direction is obvious from the definitions of Lh and Lh. To prove the “necessity” direction, suppose that x 2 Lh; y 2 Lh; xTy ¼ 0. Then, from definitions, the cases (a)-(c) are triv- ial and we only need to verify the case (d). Taking x 2 bd Lhn f0sg, y 2 bd Lhn f0sg, we have x2 6¼ 0s1; y26¼ 0s1; jjx2jj ¼ x1cot h and jjy2jj ¼ y1tan h. In addition, the relation xTy ¼ 0 yields that x1y1þ xT2y2 ¼ 0, which implies xT2y2¼ jjx2jj jjy2jj and there exists r>0 such that x2¼ ry2; x1y1¼ rjjy2jj2; y16¼ 0 and
x ¼ x1
x2
¼ r jjy2jj2 y1
y2
2 4
3
5 ¼ r y1tan2h
y2
¼ r tan2h 0 0 Is1
y1
y2
¼ r Hyð Þ;
where the third equality is due to jjy2jj ¼ y1tan h. Thus, the proof is
complete. w
3. Optimality conditions
This section aims to establish optimality conditions for the circular conic optimization problem (1.1). First of all, the Lagrangian function of (1.1) is defined as
L x; l; g; C 1; C2; :::; CJ
:¼ f xð Þ þ h xð ÞTl þ g xð ÞTgXJ
i¼1
Gið Þx TCi; (3.1) where l 2 Rl; g 2 Rm. For simplicity, we write the vectors GiðxÞ and Ci 2 Rsi ði ¼ 1; 2 ; JÞ into the following form, respectively,
Gið Þx :¼ Gi1ð Þx Gi2ð Þx
" #
; Cið Þx :¼ Ci1ð Þx Ci2ð Þx
" # :
Let x 2 Rn be a local minimizer of (1.1) and Robinson’s constraint quali- fication (RCQ) holds atx, that is,
0 2 int
hð Þx gð Þx G1ð Þx
...
GJð Þx 2 66 66 66 64
3 77 77 77 75
þ
J hð Þx J gð Þx J G1ð Þx
...
J GJð Þx 2
66 66 66 64
3 77 77 77 75
Rn 0l
Rm Lh1
...
LhJ
2 66 66 66 66 4
3 77 77 77 77 5 8>
>>
>>
>>
>>
<
>>
>>
>>
>>
>:
9>
>>
>>
>>
>>
=
>>
>>
>>
>>
>;
;
where J hðxÞ; J gðxÞ and J GiðxÞ denote the derivatives of hðxÞ; gðxÞ and J GiðxÞ at x, respectively. Then, there exist l 2 Rl; g 2 Rm, Ci2 Rsi ði ¼ 1; 2; :::; JÞ satisfying the KKT condition
rxL x; l; g; C1; C2; :::; CJ
¼ 0; h xð Þ ¼ 0l; Rmþ像g ? g xð Þ 2 Rm; Lhi像Ci? Gið Þ 2 Lx hi; i ¼ 1; 2; :::; J;
8<
: (3.2)
where “a ? b” means that aTb ¼ 0.
It is easy to see that the condition (3.2) is a special form of mathematical programing with equilibrium constraints (MPEC in brief). During the past two decades, MPECs have been drawn much attention not only in multiple applications such as engineering design and economics but also in the the- oretical analysis themselves, we refer to the monographs [33, 34] and the references therein for more details.
In the sequel, if ðx; l; g; C1; C2; :::; CJÞ satisfies the above system (3.2), we call x a stationary point of (1.1). In addition, the set of the associated Lagrangian multipliers KðxÞ is defined by
K xð Þ :¼ l; g; C1; C2; :::; CJ
x; l; g; C1; C2; :::; CJ
satisfies the KKT condition 10ð Þ
) : (
For convenience, let us denote
X :¼ 0l Rm Lh1
L...hJ
2 66 66 66 4
3 77 77 77 5
; G xð Þ :¼
hð Þx gð Þx G1ð Þx
...
GJð Þx 2 66 66 64
3 77 77
75; Y :¼ Rl Rm Rs1 R...sJ 2 66 66 64
3 77 77
75: (3.3)
Then, the above RCQ can be rewritten as
J Gð ÞRx nþ TXðGð Þx Þ ¼ Y: (3.4) Analogous to [19, Definition 4.70], the constraint nondegeneracy condi- tion of (1.1) atx is defined by
J Gð ÞRx nþ lin T XðGð Þx Þ
¼ Y; (3.5)
where linfTXðGðxÞÞg denotes the linearity space of TXðGðxÞÞ, which is the largest linear space contained in TXðGðxÞÞ.
To understand the constraint nondegeneracy condition intuitively, we define the following index sets:
Iþð Þ :¼ i j gx ið Þ ¼ 0; gx i>0; i ¼ 1; 2; :::; m
; I0ð Þ :¼ i j gx ið Þ ¼ 0; gx i ¼ 0; i ¼ 1; 2; :::; m
; Ið Þ :¼ i j gx ið Þ<0; gx i ¼ 0; i ¼ 1; 2; :::; m
; IGð Þ :¼ i j Gx ið Þ 2 int Lx hi; i ¼ 1; 2; :::; J
; ZGð Þ :¼ i j Gx ið Þ ¼ 0x si; i ¼ 1; 2; :::; J
;
BGð Þ :¼ i j Gx n ið Þ 2 bd Lx hjn 0f g; i ¼ 1; 2; :::; Jsi o :
Theorem 3.1. Let x be a stationary point of (1.1). Then, the following condi- tions are equivalent:
(a) The constraint nondegeneracy condition holds atx.
(b) The vectors
J h1ð Þx T; :::; J hlð Þx T; J gið Þx T; i 2 Iþð Þ [ Ix 0ð Þ;x J Gið Þx THhiGið Þ; i 2 Bx Gð Þ;x
J Gið Þx Tejsi; j ¼ 1; 2; :::; si; i 2 ZGð Þx
are linearly independent, where ejsi denotes the jth column vector of the iden- tity matrix Isi and Hhi is defined by
Hhi :¼ tan2hi 0 0 Isi1
:
Proof. Without loss of generality, we assume that
IGð Þ :¼ 1; 2; :::; Jx f 1g; ZGð Þ :¼ Jx f1þ 1; J1þ 2; :::; J2g;
BGð Þ :¼ Jx f2þ 1; J2þ 2; :::; Jg:
It follows from Theorem 2.3 and (3.3) that the constraint nondegeneracy condition (3.5) can be described as
J hð Þx J gð Þx Pi2IGð Þx J Gið Þx Pi2ZGð Þx J Gið Þx Pi2BGð Þx J Gið Þx 2
66 66 64
3 77 77
75Rnþ lin
0l f g TRmgð Þx
Pi2IGð Þx Rsi Pi2ZGð Þx Lhi
Pi2BGð Þx TLhiGið Þx 2
66 66 66 4
3 77 77 77 5
¼
Rl Rm Pi2IGð Þx Rsi Pi2ZGð Þx Rsi Pi2BGð Þx Rsi 2
66 66 64
3 77 77 75;
(3.6) where
Pi2IGð Þx J Gið Þ :¼x
J G1ð Þx J G2ð Þx
...
J GJ1ð Þx 2
66 66 4
3 77 77
5; Pi2IGð Þx Rsi :¼ Rs1 Rs2 ...
RsJ1 2 66 66 4
3 77 77 5;
Pi2ZGð Þx J Gið Þ :¼x
J GJ1þ1ð Þx J GJ1þ2ð Þx
...
J GJ2ð Þx 2
66 66 4
3 77 77
5; Pi2ZGð Þx Rsi :¼
RsJ1þ1 RsJ1þ1
...
RsJ2 2 66 66 4
3 77 77 5;
Pi2BGð Þx J Gið Þ :¼x
J GJ2þ1ð Þx J GJ2þ2ð Þx
...
J GJð Þx 2
66 66 4
3 77 77
5; Pi2BGð Þx Rsi :¼
RsJ2þ1 RsJ2þ1
...
RsJ 2 66 66 4
3 77 77 5;
Pi2ZGð Þx Lhi :¼
LhJ1þ1
LhJ1þ2
...
LhJ2
2 66 66 4
3 77 77
5; Pi2BGð Þx TLhiGið Þx :¼
TLhJ
2þ1GJ2þ1ð Þx TLhJ
2þ2GJ2þ2ð Þx ...
TLhJGJð Þx 2
66 66 66 4
3 77 77 77 5 :
Notice that
lin Tn Rmgð Þx o
:¼ g 2 R mj gi ¼ 0; i 2 Iþð Þ [ Ix 0ð Þx
;
lin Lf hig:¼ 0f g; i 2 Zsi Gð Þ:x
Taking i 2 BGðxÞ, the explicit description of TLhiðGiðxÞÞ implies that lin Tn LhiGið Þx o
¼ Ci1; Ci2
j Ci1ð ÞGx i1ð Þ tanx 2hi C i2 T
Gi2ð Þ ¼ 0x
n o
¼ Ci1; Ci2
Gi1ð Þx Gi2ð Þx
" #T
tan2hi 0 0 Isi1
" #
Ci1 Ci2
" #
¼ 0 8<
:
9=
;
¼ ker Gið Þx THhi
:
Hence, the equality (3.6) is equivalent to J hð Þx
J gð Þx Pi2ZGð Þx J Gið Þx Pi2BGð Þx J Gið Þx 2
66 66 4
3 77 77 5Rnþ
0l f g linTRmgð Þx
Pi2ZGð Þx 0si
Pi2BGð Þx ker Gið Þx THhi
2 66 66 64
3 77 77 75
¼
Rl Rm Pi2ZGð Þx Rsi Pi2BGð Þx Rsi 2
66 66 4
3 77 77 5:
By taking the orthogonal complements for both sides of the above equality, we obtain
ker J hh ð Þx T J gð Þx T J GJ1þ1ð Þx T J GJ2ð Þx T J GJ2þ1ð Þx T J GJð Þx Ti
\ Rl g 2 R mj gi ¼ 0; i 2 Ið Þx
RsJ1þ1 RsJ2
range HTh
J2þ1GJ2þ1ð Þx
range H ThJGJð Þx
¼ 0l 0m 0sJ1þ1 0sJ2 0sJ2þ1 0sJ:
(3.7) Let l ¼ ðl1; :::; llÞT; gi 2 R; i 2 IþðxÞ [ I0ðxÞ, Ci 2 Rsi; i 2 ZGðxÞ; pi 2 R; i 2 BGðxÞ satisfying
J hð Þx Tl þ X
i2Iþð Þ[Ix 0ð Þx
J gið Þx Tgiþ X
i2ZGð Þx
J Gið Þx TCiþ X
i2BGð Þx
J Gið Þx THThiGið Þpx i¼ 0:
This together with (3.7) yields
l ¼ 0l; gi ¼ 0; i 2 Iþð Þ [ Ix 0ð Þ;x Ci ¼ 0si; i 2 ZGð Þ; px i ¼ 0; i 2 BGð Þ;x
which means that the constraint nondegeneracy condition holds at x if and only if the vectors
J h1ð Þx T; :::; J hlð Þx T; J gið Þx T; i 2 Iþð Þ [ Ix 0ð Þ;x
J Gið Þx Tejsi; j ¼ 1; 2; :::; si; i 2 ZGð Þ; J Gx ið Þx THhiGið Þ; i 2 Bx Gð Þx are linearly independent. Thus, the proof is complete. w
Similar to [19, Theorem 3.9], we establish the first-order optimality con- dition of (1.1) in the following theorem.
Theorem 3.2. Let x be a local minimizer of (1.1) and RCQ (3.4) holds at x.
Then the set KðxÞ is nonempty, convex and compact. Furthermore, if the con- straint nondegeneracy condition (3.5) holds at x, the set KðxÞ is a singleton.
Let x be a stationary point of (1.1), the corresponding critical cone at x is defined by
Cð Þ :¼ d 2 Rx n J hð Þd ¼ 0x l; rf xð ÞTd ¼ 0;
J gð Þd 2 Tx Rmgð Þx
; J Gið Þd 2 Tx LhiGið Þx
; i ¼ 1; 2; :::; J
9>
=
>;: 8>
><
>>
:
If KðxÞ is nonempty, then there exist l 2 Rl, g 2 Rmþ and Ci 2 Lhi ði ¼ 1; 2; :::; JÞ such that CðxÞ can be rewritten as
Cð Þ ¼ d 2 Rx n
J hð Þdx J gð Þdx J G1ð Þdx J GJ...ð Þdx 2
66 66 64
3 77 77 75
2
0l
TRf gmgð Þx TLh1G1ð Þx
TLhJG... Jð Þx 2
66 66 66 4
3 77 77 77 5
\
lg
C1
C...J 2 66 66 64
3 77 77 75
?
9>
>>
>>
>=
>>
>>
>>
; : 8>
>>
>>
>>
<
>>
>>
>>
>:
(3.8)
With Theorem 2.5, the following theorem shows the explicit expression of CðxÞ.
Theorem 3.3. Let x be a stationary point of (1.1), w :¼ ðl; g; C1; :::; CJÞ 2 Rl Rm Rs1 RsJ and w 2 KðxÞ. Then, the critical cone CðxÞ can be described as
Cð Þ ¼ d 2 Rx n
J hð Þdx
ð Þk¼ 0; k ¼ 1; 2; :::; l;
J gð Þdx
i¼ 0; i 2 Iþð Þ;x J gð Þdx
i 0; i 2 I0ð Þ;x J Gjð Þd 2 Tx LhjGjð Þx
; Cj¼ 0sj; J Gjð Þd ¼ 0;x Cj2 int Lhj; J Gjð Þd 2 Rx þ HhjCj
; Cj2 bd Lhjn 0sj
; Gjð Þ ¼ 0;x J Gjð Þdx
TCj¼ 0; Cj2 bd Lhjn 0 sj ; Gjð Þ 2 bd Lx hjn 0 sj :
9>
>>
>>
>>
>>
>>
>=
>>
>>
>>
>>
>>
>>
;
; 8>
>>
>>
>>
>>
>>
>>
<
>>
>>
>>
>>
>>
>>
>:
(3.9) where the set RþðHhjCjÞ is defined by
Rþ HhjCj
:¼ rHn hjCjj r 0o :
Proof. From the equality (3.8), we have Cð Þ ¼ d 2 Rx n
J hð Þdx
ð Þk¼ 0; k ¼ 1; 2; :::; l;
J gð Þdx
i 0; J g x ð Þd
igi¼ 0; i 2 Iþð Þ [ Ix 0ð Þ;x J Gjð Þd 2 Tx LhjGjð Þx
; J G jð Þdx TCj¼ 0; j ¼ 1; 2; :::; J:
9>
=
>;: 8>
><
>>
:
(3.10) By the definitions of IþðxÞ and I0ðxÞ, we notice that the equalities in the second row of (2.15) are equivalent to
J gð Þdx
i ¼ 0; i 2 Iþð Þ;x J gð Þdx
i 0; i 2 I0ð Þ:x
To proceed, we analyze the remain part of the theorem by discussing four cases:
Case (1): If Cj ¼ 0sj, then the third row of (3.10) becomes J GiðxÞd 2 TLhjðGjðxÞÞ.
Case (2): If Cj 2 int Lhj, from the KKT condition (3.2), then GjðxÞ ¼ 0sj. The explicit form of TLhjðGjðxÞÞ defined in Theorem 2.3 implies that TLhjðGjðxÞÞ ¼ Lhj. From the last row of (3.10), we obtain J GjðxÞd 2 Lhj. It follows from Theorem 2.5 and ðJ GjðxÞdÞTCj¼ 0 that J GjðxÞd ¼ 0sj:
Case (3): If Cj 2 bd Lhj n f0sjg and GjðxÞ ¼ 0sj, then TLhjðGjðxÞÞ ¼ Lhj
and J GjðxÞd 2 Lhj\ ðCjÞ?. It follows from Theorem 2.5 that J GjðxÞd ¼ 0sj or there exists r>0 such that J GjðxÞd ¼ rHhjCj. Hence, we have J GjðxÞd 2 RþðHhjCjÞ.
Case (4): If Cj 2 bd Lhj n f0sjg; GjðxÞ 2 bdLhjn f0sjg, we have TLhjGjð Þx
¼nðh1; h2Þ j hT2Gj2ð Þhx 1Gj1ð Þ tanx 2hj 0o :
Combining the above equality with the fact J GjðxÞd 2 TLhjðGjðxÞÞ \ ðCjÞ? as in (3.10), we obtain
J Gj2ð Þdx
T
Gj2ð Þ J Gx j1ð Þdx
Gj1ð Þ tanx 2hj 0;
J Gj2ð Þdx
T
Cj2þ J G j1ð Þdx
Cj1 ¼ 0: (3.11)
From the KKT condition (3.2), we know ðCjÞTGjðxÞ ¼ 0. Because
Cj 2 bd Lhjn f0sjg; GjðxÞ 2 bd Lhj n f0sjg, by the case (d) in Theorem 2.5, there exists r>0 such that Cj¼ rHhjGjðxÞ and ðJ Gj2ðxÞdÞTGj2ðxÞ
ðJ Gj1ðxÞdÞGj1ðxÞ tan2hj ¼ 0: Under this case, the equality (3.11) reduces to ðJ GjðxÞdÞTCj ¼ 0.
From the above discussions, the conclusion holds at the given stationary
point x. Thus, the proof is complete. w
Next, we calculate the “sigma” term of the optimization problem (1.1) in the below lemma, which plays an important role in describing the second- order optimality conditions for (1.1).
Lemma 3.4. Let x be a stationary point of (1.1), w :¼ ðl; g;
C1; :::; CJÞ 2 KðxÞ Rl Rm Rs1 RsJ, d 2 CðxÞ, and RCQ (3.4) holds at x. Denote the sigma term of (1.1) by
! l; g; C 1; :::; CJ
; T2XðGð Þ; J G xx ð ÞdÞ
;
where !ð; T2Xð; ÞÞ means the support function of the second-order tangent set T2Xð; Þ. Then, we have
! l; g; C 1; :::; CJ
; T2XðGð Þ; J G xx ð ÞdÞ
¼ dT XJ
j¼1
Aj x; l; g; Cj 0
@
1 Ad;
where the matrix Ajðx; l; g; CjÞ is defined by Aj x; l; g; Cj
:¼ Cj1
Gj1ð Þx cot2hjJ Gjð Þx T
HhjJ Gjð Þ; if Gx jð Þ 2 bd Lx hjn 0 sj ;
0; otherwise:
8>
<
>:
Proof. From the definitions of X and GðxÞ, we have
! l; g; C 1; :::; CJ
; T2XðGð Þ; J G xx ð ÞdÞ
¼ ! l; T 2f g0l ðhð Þ; J h xx ð ÞdÞ
þ ! g; T 2Rmgð Þ; J g xx ð Þd þXJ
j¼1
! C j; T2LhjGjð Þ; J Gx jð Þdx :
(3.12)
