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# No Gap Second-Order Optimality Conditions for Circular Conic Programs

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### 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 ! Rsi1 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  Rsi1j 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

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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  Rs1 has the spectral decomposition given by

x:¼ k1ð Þux ð Þx1 þ k2ð Þux ð Þx2; where

k1ð Þx :¼ x1jjx2jjcot h; k2ð Þx :¼ x1þ jjx2jj tan h and

uð Þx1 :¼ 1 1 þ cot2h

1 0

0 cot h  I

" #

1

x2

" #

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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 Rs1 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.

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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 infz2Xjjxzjj. Lh is the dual cone of a given circular cone Lh, which is defined by Lh:¼ fv 2 Rsj vTx  0; 8x 2 Lhg: From [2, Theorem 2.1], the structure of Lh can be described as

Lh¼ ðx1; x2Þ 2 R  Rs1j jjx2jj  x1coth

n o

¼ Lp2h:

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 Þw 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 Þw0ðx; hÞ

1

2t2 :

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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].

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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 wðxÞ<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  Rs1. 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  Rs1j 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  Rs1j / xð Þ  0o

;

where /: Rs ! R is given by /ðxÞ :¼ jjx2jjx1tan 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

jjx2jjh1tan h:

Applying the relation x1tan h ¼ jjx2jj and (2.3) yield that TLhð Þ ¼x ðh1; h2Þ 2 R  Rs1j hT2x2 h1x1tan2h  0

:

Thus, the proof is complete. w

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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  Rs1 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  Rs1j 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; hT2x2h1x1tan2h 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 jjx2jjx1tan 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

jjx2jjIs1 1 jjx2jj3x2xT2 2

4

3 5 h1

h2

 

¼wT2x2

jjx2jjw1tan h þjjh2jj2

jjx2jj  hT2x2

 2 jjx2jj3

¼ 1

jjx2jjwT2x2w1x1tan2h

þjjh2jj2

jjx2jj ðh1x1tan22 x1tan h

ð Þ3

¼ 1

jjx2jjwT2x2w1x1tan2h

þjjh2jj2

jjx2jj h21tan2h x1tan h

¼ 1

jjx2jj wT2x2w1x1tan2h þ jjh2jj2h21tan2h

;

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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  Rs1j 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  Rs1. The system x 2 Lh; 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 Lh, y ¼ 0s. (c) x 2 bd Lhn f0sg, y ¼ 0s.

(d) x 2 bd Lhn f0sg, y 2 bd Lhn f0sg, and there exists r>0 such that x ¼rðHyÞ, where

H :¼ tan2h 0 0 Is1

 

:

Proof. The “sufficiency” direction is obvious from the definitions of Lh and Lh. To prove the “necessity” direction, suppose that x 2 Lh; 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 Lhn f0sg, y 2 bd Lhn f0sg, we have x2 6¼ 0s1; y26¼ 0s1; 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 Is1

 

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

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L x; l; g; C 1; C2; :::; CJ

:¼ f xð Þ þ h xð ÞTl þ g xð ÞTgXJ

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 atx, that is,

0 2 int

hð Þx gð Þx G1ð Þx

...

GJð Þx 2 66 66 66 64

3 77 77 77 75

þ

J hð Þx J gð Þx J G1ð Þx

...

J GJð Þx 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 hðxÞ; J gðxÞ and J GiðxÞ 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; Lhi像Ci? Gið Þ 2 Lx 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 ðx; 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 KðxÞ is defined by

K xð Þ :¼ l; g; C1; C2; :::; CJ

x; l; g; C1; C2; :::; CJ

satisfies the KKT condition 10ð Þ





) : (

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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 Þx gð Þx G1ð Þx

...

GJð Þx 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 ÞRx nþ TXðGð Þx Þ ¼ Y: (3.4) Analogous to [19, Definition 4.70], the constraint nondegeneracy condi- tion of (1.1) atx is defined by

J Gð ÞRx nþ lin T XðGð Þx Þ

¼ Y; (3.5)

where linfTXðGðxÞÞg denotes the linearity space of TXðGðxÞÞ, which is the largest linear space contained in TXðGðxÞÞ.

To understand the constraint nondegeneracy condition intuitively, we define the following index sets:

Iþð Þ :¼ i j gx  ið Þ ¼ 0; gx i>0; i ¼ 1; 2; :::; m

; I0ð Þ :¼ i j gx  ið Þ ¼ 0; gx i ¼ 0; i ¼ 1; 2; :::; m

; Ið Þ :¼ i j gx  ið Þ<0; gx i ¼ 0; i ¼ 1; 2; :::; m

; IGð Þ :¼ i j Gx  ið Þ 2 int Lx hi; i ¼ 1; 2; :::; J

; ZGð Þ :¼ i j Gx  ið Þ ¼ 0x si; i ¼ 1; 2; :::; J

;

BGð Þ :¼ i j Gx n ið Þ 2 bd Lx 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 atx.

(b) The vectors

J h1ð Þx T; :::; J hlð Þx T; J gið Þx T; i 2 Iþð Þ [ Ix 0ð Þ;x J Gið Þx THhiGið Þ; i 2 Bx Gð Þ;x

J Gið Þx Tejsi; j ¼ 1; 2; :::; si; i 2 ZGð Þx

are linearly independent, where ejsi denotes the jth column vector of the iden- tity matrix Isi and Hhi is defined by

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Hhi :¼ tan2hi 0 0 Isi1

 

:

Proof. Without loss of generality, we assume that

IGð Þ :¼ 1; 2; :::; Jx f 1g; ZGð Þ :¼ Jx f1þ 1; J1þ 2; :::; J2g;

BGð Þ :¼ Jx 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 Þx J gð Þx Pi2IGð Þx J Gið Þx Pi2ZGð Þx J Gið Þx Pi2BGð Þx J Gið Þx 2

66 66 64

3 77 77

75Rnþ lin

0l f g TRmgð Þx 

Pi2IGð Þx Rsi Pi2ZGð Þx Lhi

Pi2BGð Þx TLhiGið Þx  2

66 66 66 4

3 77 77 77 5

¼

Rl Rm Pi2IGð Þx Rsi Pi2ZGð Þx Rsi Pi2BGð Þx Rsi 2

66 66 64

3 77 77 75;

(3.6) where

Pi2IGð Þx J Gið Þ :¼x

J G1ð Þx J G2ð Þx

...

J GJ1ð Þx 2

66 66 4

3 77 77

5; Pi2IGð Þx Rsi :¼ Rs1 Rs2 ...

RsJ1 2 66 66 4

3 77 77 5;

Pi2ZGð Þx J Gið Þ :¼x

J GJ1þ1ð Þx J GJ1þ2ð Þx

...

J GJ2ð Þx 2

66 66 4

3 77 77

5; Pi2ZGð Þx Rsi

RsJ1þ1 RsJ1þ1

...

RsJ2 2 66 66 4

3 77 77 5;

Pi2BGð Þx J Gið Þ :¼x

J GJ2þ1ð Þx J GJ2þ2ð Þx

...

J GJð Þx 2

66 66 4

3 77 77

5; Pi2BGð Þx Rsi

RsJ2þ1 RsJ2þ1

...

RsJ 2 66 66 4

3 77 77 5;

Pi2ZGð Þx Lhi

LhJ1þ1

LhJ1þ2

...

LhJ2

2 66 66 4

3 77 77

5; Pi2BGð Þx TLhiGið Þx  :¼

TLhJ

2þ1GJ2þ1ð Þx  TLhJ

2þ2GJ2þ2ð Þx  ...

TLhJGJð Þx  2

66 66 66 4

3 77 77 77 5 :

(12)

Notice that

lin Tn Rmgð Þx o

:¼ g 2 R mj gi ¼ 0; i 2 Iþð Þ [ Ix 0ð Þx 

;

lin Lf hig:¼ 0f g; i 2 Zsi Gð Þ:x

Taking i 2 BGðxÞ, the explicit description of TLhiðGiðxÞÞ implies that lin Tn LhiGið Þx o

¼ Ci1; Ci2

j Ci1ð ÞGx i1ð Þ tanx 2hi C i2 T

Gi2ð Þ ¼ 0x

n o

¼ Ci1; Ci2



Gi1ð Þx Gi2ð Þx

" #T

tan2hi 0 0 Isi1

" #

Ci1 Ci2

" #

¼ 0 8<

:

9=

;

¼ ker Gið Þx THhi

:

Hence, the equality (3.6) is equivalent to J hð Þx

J gð Þx Pi2ZGð Þx J Gið Þx Pi2BGð Þx J Gið Þx 2

66 66 4

3 77 77 5Rnþ

0l f g linTRmgð Þx 

Pi2ZGð Þx 0si

Pi2BGð Þx ker Gið Þx THhi

2 66 66 64

3 77 77 75

¼

Rl Rm Pi2ZGð Þx Rsi Pi2BGð Þx 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 Þx T J gð Þx T J GJ1þ1ð Þx T    J GJ2ð Þx T J GJ2þ1ð Þx T    J GJð Þx Ti

\ Rl g 2 R mj gi ¼ 0; i 2 Ið Þx 

 RsJ1þ1      RsJ2

 range HTh

J2þ1GJ2þ1ð Þx

     range H ThJGJð Þx

¼ 0l 0m 0sJ1þ1     0sJ2  0sJ2þ1    0sJ:

(3.7) Let l ¼ ðl1; :::; llÞT; gi 2 R; i 2 IþðxÞ [ I0ðxÞ, Ci 2 Rsi; i 2 ZGðxÞ; pi 2 R; i 2 BGðxÞ satisfying

J hð Þx Tl þ X

i2Iþð Þ[Ix 0ð Þx

J gið Þx Tgiþ X

i2ZGð Þx

J Gið Þx TCiþ X

i2BGð Þx

J Gið Þx THThiGið Þpx i¼ 0:

(13)

This together with (3.7) yields

l ¼ 0l; gi ¼ 0; i 2 Iþð Þ [ Ix 0ð Þ;x Ci ¼ 0si; i 2 ZGð Þ; px 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 Þx T; :::; J hlð Þx T; J gið Þx T; i 2 Iþð Þ [ Ix 0ð Þ;x

J Gið Þx Tejsi; j ¼ 1; 2; :::; si; i 2 ZGð Þ; J Gx ið Þx THhiGið Þ; i 2 Bx Gð Þx 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 KðxÞ is nonempty, convex and compact. Furthermore, if the con- straint nondegeneracy condition (3.5) holds at x, the set KðxÞ 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 Rx n J hð Þd ¼ 0x l; rf xð ÞTd ¼ 0;

J gð Þd 2 Tx Rmgð Þx 

; J Gið Þd 2 Tx LhiGið Þx 

; i ¼ 1; 2; :::; J





9>

=

>;: 8>

><

>>

:

If KðxÞ is nonempty, then there exist l 2 Rl, g 2 Rmþ and Ci 2 Lhi ði ¼ 1; 2; :::; JÞ such that CðxÞ can be rewritten as

Cð Þ ¼ d 2 Rx n

J hð Þdx J gð Þdx J G1ð Þdx J GJ...ð Þdx 2

66 66 64

3 77 77 75

2

0l

TRf gmgð Þx  TLh1G1ð Þx 

TLhJG... Jð Þx  2

66 66 66 4

3 77 77 77 5

\

lg

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 CðxÞ.

Theorem 3.3. Let x be a stationary point of (1.1), w :¼ ðl; g; C1; :::; CJÞ 2 Rl Rm Rs1     RsJ and w 2 KðxÞ. Then, the critical cone CðxÞ can be described as

(14)

Cð Þ ¼ d 2 Rx n

J hð Þdx

ð Þk¼ 0; k ¼ 1; 2; :::; l;

J gð Þdx

 

i¼ 0; i 2 Iþð Þ;x J gð Þdx

 

i 0; i 2 I0ð Þ;x J Gjð Þd 2 Tx LhjGjð Þx 

; Cj¼ 0sj; J Gjð Þd ¼ 0;x Cj2 int Lhj; J Gjð Þd 2 Rx þ HhjCj

; Cj2 bd Lhjn 0sj

 ; Gjð Þ ¼ 0;x J Gjð Þdx

 TCj¼ 0; Cj2 bd Lhjn 0 sj ; Gjð Þ 2 bd Lx hjn 0 sj :











9>

>>

>>

>>

>>

>>

>=

>>

>>

>>

>>

>>

>>

;

; 8>

>>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>>

>:

(3.9) where the set RþðHhjCjÞ is defined by

Rþ HhjCj

:¼ rHn hjCjj r  0o :

Proof. From the equality (3.8), we have Cð Þ ¼ d 2 Rx n

J hð Þdx

ð Þk¼ 0; k ¼ 1; 2; :::; l;

J gð Þdx

 

i 0; J g x ð Þd

igi¼ 0; i 2 Iþð Þ [ Ix 0ð Þ;x J Gjð Þd 2 Tx LhjGjð Þx

; J G jð Þdx TCj¼ 0; j ¼ 1; 2; :::; J:





9>

=

>;: 8>

><

>>

:

(3.10) By the definitions of IþðxÞ and I0ðxÞ, we notice that the equalities in the second row of (2.15) are equivalent to

J gð Þdx

 

i ¼ 0; i 2 Iþð Þ;x J gð Þdx

 

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 GiðxÞd 2 TLhjðGjðxÞÞ.

Case (2): If Cj 2 int Lhj, from the KKT condition (3.2), then GjðxÞ ¼ 0sj. The explicit form of TLhjðGjðxÞÞ defined in Theorem 2.3 implies that TLhjðGjðxÞÞ ¼ Lhj. From the last row of (3.10), we obtain J GjðxÞd 2 Lhj. It follows from Theorem 2.5 and ðJ GjðxÞdÞTCj¼ 0 that J GjðxÞd ¼ 0sj:

Case (3): If Cj 2 bd Lhj n f0sjg and GjðxÞ ¼ 0sj, then TLhjðGjðxÞÞ ¼ Lhj

and J GjðxÞd 2 Lhj\ ðCjÞ?. It follows from Theorem 2.5 that J GjðxÞd ¼ 0sj or there exists r>0 such that J GjðxÞd ¼ rHhjCj. Hence, we have J GjðxÞd 2 RþðHhjCjÞ.

Case (4): If Cj 2 bd Lhj n f0sjg; GjðxÞ 2 bdLhjn f0sjg, we have TLhjGjð Þx 

¼nðh1; h2Þ j hT2Gj2ð Þhx 1Gj1ð Þ tanx 2hj  0o :

(15)

Combining the above equality with the fact J GjðxÞd 2 TLhjðGjðxÞÞ \ ðCjÞ? as in (3.10), we obtain

J Gj2ð Þdx

T

Gj2ð Þ J Gx j1ð Þdx

Gj1ð Þ tanx 2hj  0;

J Gj2ð Þdx

T

Cj2þ J G j1ð Þdx

Cj1 ¼ 0: (3.11)

From the KKT condition (3.2), we know ðCjÞTGjðxÞ ¼ 0. Because

Cj 2 bd Lhjn f0sjg; GjðxÞ 2 bd Lhj n f0sjg, by the case (d) in Theorem 2.5, there exists r>0 such that Cj¼ rHhjGjðxÞ and ðJ Gj2ðxÞdÞTGj2ðxÞ

ðJ Gj1ðxÞdÞGj1ðxÞ tan2hj ¼ 0: Under this case, the equality (3.11) reduces to ðJ GjðxÞdÞTCj 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 ðl; g;

C1; :::; CJÞ 2 KðxÞ  Rl Rm Rs1      RsJ, d 2 CðxÞ, and RCQ (3.4) holds at x. Denote the sigma term of (1.1) by

! l; g; C 1; :::; CJ

; T2XðGð Þ; J G xx 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 xx ð ÞdÞ

¼ dT XJ

j¼1

Aj x; l; g; Cj 0

@

where the matrix Ajðx; l; g; CjÞ is defined by Aj x; l; g; Cj

Cj1

Gj1ð Þx cot2hjJ Gjð Þx T

HhjJ Gjð Þ; if Gx jð Þ 2 bd Lx 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 xx ð ÞdÞ

¼ ! l; T 2f g0l ðhð Þ; J h xx ð ÞdÞ

þ ! g; T 2Rmgð Þ; J g xx ð Þd þXJ

j¼1

! C j; T2LhjGjð Þ; J Gx jð Þdx  :

(3.12)

By modifying the proof of Theorem 1.7, we can also establish the second inequality in (iii).. W EN -C HING L IEN Advanced

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