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

Yue Lu^{a}, Jein-Shan Chen^{b} , and Ning Zhang^{c}

aSchool of Mathematical Sciences, Tianjin Normal University, Tianjin, China;^{b}Department of
Mathematics, National Taiwan Normal University, Taipei, Taiwan;^{c}Department 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;

G^{i}_{1}ð Þ; Gx ^{i}_{2}ð Þx

2 Lh_{i}; i ¼ 1; 2; :::; J;

(1.1)

where f : R^{n} ! R; h : R^{n} ! R^{l}, g : R^{n} ! R^{m}; G^{i}_{1}: R^{n} ! R, G^{i}_{2} : R^{n} !
R^{s}^{i}^{1} ði ¼ 1; 2; :::; JÞ are assumed to be twice continuously differentiable.

Here Lhi denotes a circular cone in R^{s}^{i} given by

Lhi :¼ ðx1; x2Þ 2 R R^{s}^{i}^{1}j jjx2jj x1tan hi

n o

(1.2)
with hi being its half-aperture angle and hi 2 ð0;^{p}_{2}Þ. From definition, it is
clear that L^{p}_{4} is the set of second-order cone K^{s}^{i}.

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 u_{i1} and the
associated another forces u_{i2}; ui3 satisfy the following condition

k uð _{i2}; ui3Þk lui1;

where jj jj represents the Euclidean norm defined in R^{n} and l denotes the
friction that depends on the angle h. If l ¼ tan h and h 6¼^{p}_{4}, 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 K^{s}, that is,

x:¼ x1

x_{2}

2 Lh () tan h 0

0 I

x1

x_{2}

2 K^{s}: (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

f^{L}^{h}ð Þx :¼ f kð 1ð Þx Þu^{ð Þ}_{x}^{1} þ f kð 2ð Þx Þu^{ð Þ}_{x}^{2};

is employed. Here f : R ! R is a given real-valued function and x ¼
ðx1; x2Þ 2 R R^{s1} has the spectral decomposition given by

x:¼ k1ð Þux ^{ð Þ}_{x}^{1} þ k2ð Þux ^{ð Þ}_{x}^{2};
where

k_{1}ð Þx :¼ x1jjx2jjcot h; k2ð Þx :¼ x1þ jjx2jj tan h
and

u^{ð Þ}_{x}^{1} :¼ 1
1 þ cot^{2}h

1 0

0 cot h I

" #

1

x2

" #

u^{ð Þ}_{x}^{2} :¼ 1
1 þ tan^{2}h

1 0

0 tan h I

x12

with x2 :¼ x2=jjx2jj if x26¼ 0 and x2 being any vector w 2 R^{s1} 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 L_{h}.

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 R^{n}, that is, distðx; XÞ :¼ infz2Xjjxzjj. L^{}_{h} is the dual
cone of a given circular cone Lh, which is defined by L^{}_{h}:¼ fv 2 R^{s}j v^{T}x
0; 8x 2 Lhg: From [2, Theorem 2.1], the structure of L^{}h can be described as

L^{}_{h}¼ ðx1; x2Þ 2 R R^{s1}j jjx_{2}jj x_{1}coth

n o

¼ L^{p}_{2}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: R^{n} ! R, the directional deriva-
tive of w at x along the direction h is denoted by w^{0}ðx; hÞ, which is given by

w^{0}ðx; 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

w^{00}ðx; h; wÞ :¼ lim

t#0

w x þ th þ ^{1}_{2}t^{2}w

w xð Þw^{0}ðx; hÞ

1

2t^{2} :

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 R^{n},
which come from Bonnans and Shaprio’s monograph [19, Definition 2.54
and Definition 3.28].

Definition 2.1. Let X R^{n} be a given closed set and x 2X. The
(Bouligand-Severi) tangent/contingent cone to X at x 2 X is defined by

T_{X}ð Þx :¼ h 2 R ^{n}j 9 t_{n} # 0; dist x þ tð nh; XÞ ¼ o tð Þ_{n}
:

Similarly, the inner tangent cone to X at x 2 X is given in the form of
T^{i}_{X}ð Þx :¼ h 2 R ^{n}j dist x þ th; Xð Þ ¼ o tð Þ; t 0

:

In addition, if h 2 T_{X}ðxÞ, the outer second-order tangent set to X at x
along the direction h is defined as

T^{2}_{X}ðx; hÞ :¼ w 2 R^{n}j 9 t_{n} # 0; dist x þ tnh þ1
2t^{2}_{n}w; X

¼ o t _{n}^{2}

:
Similarly, if h 2 T^{i}_{X}ðxÞ, the inner second-order tangent set to X at x
along the direction h is given by

T^{i;2}_{X}ðx; hÞ :¼ w 2 R^{n}j dist x þ th þ1
2t^{2}w; X

¼ o tð Þ; t 0^{2}

:
Let X R^{n} be a closed convex set and x 2X. It follows from [19,
Section 2.2.4] that the contingent cone T_{X}ðxÞ coincides with the inner tan-
gent cone T^{i}_{X}ðxÞ, that is, T_{X}ðxÞ ¼ T^{i}_{X}ðxÞ. 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) T^{2}_{X}ðx; hÞ ¼ T^{i;2}_{X}ðx; hÞ for all h 2 TXðxÞ.

(ii) For any h 2 T_{X}ðxÞ and for any sequence x þ tnh þ^{1}_{2}t_{n}^{2}h 2X such that
tnrn! 0 and

n!1lim dist r n; T^{2}_{X}ðx; 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 TL_{h}ðxÞ and the outer second-
order tangent set T^{2}_{L}_{h}ðx; 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: R^{n} ! R be a lower semicontinuous convex function.

Consider the associated level set X :¼ fx 2 R^{n}j wðxÞ 0g. Suppose that w
is Lipschitz continuous at x and wðxÞ ¼ 0. In addition, there exists x 2 R^{n}
such that wðxÞ<0 (Slater condition). Then,

T_{X}ð Þ ¼ h 2 Rx ^{n}j w^{0}ðx; hÞ 0

: (2.1.)

Moreover, for a given h 2 R^{n} satisfying w^{0}ðx; hÞ ¼ 0, the outer second-
order tangent set toX at x along the direction h can be described as

T^{2}_{X}ðx; hÞ ¼ w 2 R ^{n}j w^{00}ðx; h; wÞ 0

: (2.2)

With Lemma 2.2, we are ready to express the explicit form of the tangent
cone TL_{h}ðxÞ at any given x 2 R^{s}.

Theorem 2.3. Let x ¼ ðx1; x2Þ 2 R R^{s1}. Then, the tangent cone to Lh at
x can be written as

T_{L}_{h}ð Þ ¼x

R^{s}; if x 2 int Lh;

Lh; if x ¼ 0;

h1; h2

ð Þ 2 R R^{s1}j h^{T}_{2}x2 h1x1tan^{2}h 0

; if x 2 bd Lhn 0f g:

8<

:

Proof. The explicit form of TL_{h}ðxÞ 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 R^{s}; 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 R^{s1}j / xð Þ 0o

;

where /: R^{s} ! 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 ^{s}j /^{0}ðx; hÞ 0

: (2.3)

Note that

/^{0}ðx; hÞ ¼ r/ xð Þ^{T}h ¼ tan h x_{2}
jjx2jj

_{T}

h1

h_{2}

¼ h^{T}_{2}x_{2}

jjx2jjh1tan h:

Applying the relation x1tan h ¼ jjx2jj and (2.3) yield that
TLhð Þ ¼x ðh1; h2Þ 2 R R^{s1}j h^{T}_{2}x2 h1x1tan^{2}h 0

:

Thus, the proof is complete. ^{w}

Next theorem describes the outer second-order tangent set T^{2}_{L}_{h}ðx; hÞ at
any x 2 R^{s} and h 2 TLhðxÞ.

Theorem 2.4. Let x ¼ ðx1; x2Þ 2 R R^{s1} 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

T^{2}_{L}_{h}ðx; hÞ ¼ R^{s}; 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 TL_{h}ð Þ;x
8<

: where the set N is defined by

N :¼nðw1; w2Þ 2 R R^{s1}j w^{T}_{2}x2 w_{1}x1tan^{2}h h^{2}_{1}tan^{2}h jjh_{2}jj^{2}o
:

Proof. Again, we derive the explicit form of T^{2}_{L}_{h}ðx; hÞ by discussing
two cases.

(a) If h 2 int TLh or x ¼ 0, from Definition 2.1, we have
T^{2}_{L}_{h}ðx; hÞ ¼ R^{s}; 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; h^{T}_{2}x2h1x1tan^{2}h ¼ 0: (2.4)
Then, it follows from Lemma 2.2 that the second-order tangent set
T^{2}_{L}_{h}ðx; hÞ has the form of

T^{2}_{L}_{h}ðx; hÞ ¼ w 2 R ^{s}j /^{00}ðx; h; wÞ 0

; (2.5)

where /ðxÞ:¼ jjx2jjx_{1}tan h. Note that
/^{00}ðx; h; wÞ

¼ r/ xð Þ^{T}w þ h^{T}r^{2}/ xð Þh

¼ tan h_{jjx}^{x}^{2}

2jj

h i_{T} w_{1}

w2

þ h½ 1h2 ^{T} 0 0

0 1

jjx2jjI_{s1} 1
jjx_{2}jj^{3}x_{2}x^{T}_{2}
2

4

3
5 h^{1}

h2

¼w^{T}_{2}x2

jjx2jjw1tan h þjjh2jj^{2}

jjx2jj h^{T}_{2}x2

_{2}
jjx2jj^{3}

¼ 1

jjx_{2}jjw^{T}_{2}x2w1x1tan^{2}h

þjjh2jj^{2}

jjx_{2}jj ðh1x1tan^{2}hÞ^{2}
x_{1}tan h

ð Þ^{3}

¼ 1

jjx_{2}jjw^{T}_{2}x_{2}w_{1}x_{1}tan^{2}h

þjjh2jj^{2}

jjx_{2}jj h^{2}_{1}tan^{2}h
x_{1}tan h

¼ 1

jjx2jjw^{T}_{2}x2w1x1tan^{2}h þ jjh2jj^{2}h^{2}_{1}tan^{2}h

;

where the last two equalities are due to (2.4). Hence, under this case, the above expression together with (2.5) imply that

T^{2}_{L}_{h}ðx; hÞ ¼nðw1; w2Þ 2 R R^{s1}j w^{T}_{2}x2 w1x1tan^{2}h h^{2}_{1}tan^{2}h jjh2jj^{2}o
:

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 ðx_{1}; x2Þ and y ¼ ðy_{1}; y2Þ in R R^{s1}. The system
x 2 L^{}_{h}; y 2 Lh; x^{T}y ¼ 0

has at least one solution if and only if one of the following cases holds.

(a) x ¼ 0s, y 2 L_{h}.
(b) x 2 int L^{}_{h}, y ¼ 0s.
(c) x 2 bd L^{}_{h}n f0sg, y ¼ 0s.

(d) x 2 bd L^{}_{h}n f0sg, y 2 bd L_{h}n f0sg, and there exists r>0 such that
x ¼rðHyÞ, where

H :¼ tan^{2}h 0
0 Is1

:

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; x^{T}y 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^{}_{h}n 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 x^{T}y ¼ 0 yields that
x1y1þ x^{T}_{2}y2 ¼ 0, which implies x^{T}_{2}y2¼ jjx2jj jjy2jj and there exists r>0
such that x2¼ ry2; x1y1¼ rjjy2jj^{2}; y16¼ 0 and

x ¼ x1

x2

¼ r jjy2jj^{2}
y1

y2

2 4

3

5 ¼ r y^{1}tan^{2}h

y2

¼ r tan^{2}h 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}; C^{2}; :::; C^{J}

:¼ f xð Þ þ h xð Þ^{T}l þ g xð Þ^{T}gX^{J}

i¼1

G^{i}ð Þx ^{T}C^{i}; (3.1)
where l 2 R^{l}; g 2 R^{m}. For simplicity, we write the vectors G^{i}ðxÞ and C^{i} 2
R^{s}^{i} ði ¼ 1; 2 ; JÞ into the following form, respectively,

G^{i}ð Þx :¼ G^{i}_{1}ð Þx
G^{i}_{2}ð Þx

" #

; C^{i}ð Þx :¼ C^{i}_{1}ð Þx
C^{i}_{2}ð Þx

" # :

Let x 2 R^{n} 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
G^{1}ð Þx

...

G^{J}ð Þx
2
66
66
66
64

3 77 77 77 75

þ

J hð Þx
J gð Þx
J G^{1}ð Þx

...

J G^{J}ð Þx
2

66 66 66 64

3 77 77 77 75

R^{n}
0l

R^{m}_{}
Lh1

...

Lh_{J}

2 66 66 66 66 4

3 77 77 77 77 5 8>

>>

>>

>>

>>

<

>>

>>

>>

>>

>:

9>

>>

>>

>>

>>

=

>>

>>

>>

>>

>;

;

where J hðxÞ; J gðxÞ and J G^{i}ðxÞ denote the derivatives of hðxÞ; gðxÞ and
J G^{i}ðxÞ at x, respectively. Then, there exist l 2 R^{l}; g 2 R^{m}, C^{i}2 R^{s}^{i} ði ¼
1; 2; :::; JÞ satisfying the KKT condition

rxLx; l; g; C^{1}; C^{2}; :::; C^{J}

¼ 0; h xð Þ ¼ 0l; R^{m}þ像g ? g xð Þ 2 R^{m}_{};
L^{}_{h}_{i}像C^{i}? G^{i}ð Þ 2 Lx _{h}_{i}; i ¼ 1; 2; :::; J;

8<

: (3.2)

where “a ? b” means that a^{T}b 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; C^{1}; C^{2}; :::; C^{J}Þ 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; C^{1}; C^{2}; :::; C^{J}

x; l; g; C^{1}; C^{2}; :::; C^{J}

satisfies the KKT condition 10ð Þ

) : (

For convenience, let us denote

X :¼
0_{l}
R^{m}_{}
Lh1

L...hJ

2 66 66 66 4

3 77 77 77 5

; G xð Þ :¼

hð Þx
gð Þx
G^{1}ð Þx

...

G^{J}ð Þx
2
66
66
64

3 77 77

75; Y :¼
R^{l}
R^{m}
R^{s}^{1}
R...^{s}^{J}
2
66
66
64

3 77 77

75: (3.3)

Then, the above RCQ can be rewritten as

J Gð ÞRx ^{n}þ T_{X}ðGð 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 ^{n}þ lin T _{X}ðGð Þx Þ

¼ Y; (3.5)

where linfT_{X}ðGðxÞÞg denotes the linearity space of TXðGðxÞÞ, which is the
largest linear space contained in T_{X}ðGðxÞÞ.

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 ^{i}ð Þ 2 int Lx hi; i ¼ 1; 2; :::; J

;
ZGð Þ :¼ i j Gx ^{i}ð Þ ¼ 0x si; i ¼ 1; 2; :::; J

;

BGð Þ :¼ i j Gx n ^{i}ð 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 h^{1}ð Þx ^{T}; :::; J h^{l}ð Þx ^{T};
J g^{i}ð Þx ^{T}; i 2 Iþð Þ [ Ix 0ð Þ;x
J G^{i}ð Þx ^{T}HhiG^{i}ð Þ; i 2 Bx Gð Þ;x

J G^{i}ð Þx ^{T}e^{j}si; j ¼ 1; 2; :::; si; i 2 ZGð Þx

are linearly independent, where e^{j}si denotes the jth column vector of the iden-
tity matrix I_{s}_{i} and Hhi is defined by

Hhi :¼ tan^{2}h_{i} 0
0 Isi1

:

Proof. Without loss of generality, we assume that

I_{G}ð Þ :¼ 1; 2; :::; Jx f 1g; ZGð Þ :¼ Jx f1þ 1; J1þ 2; :::; J2g;

B_{G}ð 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
P_{i2I}_{G}ð Þ_{x} J G^{i}ð Þx
Pi2ZGð Þx J G^{i}ð Þx
Pi2BGð Þx J G^{i}ð Þx
2

66 66 64

3 77 77

75R^{n}þ lin

0_{l}
f g
TR^{m}_{}gð Þx

Pi2IGð Þx R^{s}^{i}
P_{i2Z}_{G}ð Þ_{x} Lhi

P_{i2B}_{G}ð Þ_{x} TL_{hi}G^{i}ð Þx
2

66 66 66 4

3 77 77 77 5

¼

R^{l}
R^{m}
P_{i2I}_{G}ð Þ_{x} R^{s}^{i}
Pi2ZGð Þx R^{s}^{i}
Pi2BGð Þx R^{s}^{i}
2

66 66 64

3 77 77 75;

(3.6) where

Pi2IGð Þx J G^{i}ð Þ :¼x

J G^{1}ð Þx
J G^{2}ð Þx

...

J G^{J}^{1}ð Þx
2

66 66 4

3 77 77

5; Pi2IGð Þx R^{s}^{i} :¼
R^{s}^{1}
R^{s}^{2}
...

R^{s}^{J1}
2
66
66
4

3 77 77 5;

Pi2ZGð Þx J G^{i}ð Þ :¼x

J G^{J}^{1}^{þ1}ð Þx
J G^{J}^{1}^{þ2}ð Þx

...

J G^{J}^{2}ð Þx
2

66 66 4

3 77 77

5; Pi2ZGð Þx R^{s}^{i} :¼

R^{s}^{J1þ1}
R^{s}^{J1þ1}

...

R^{s}^{J2}
2
66
66
4

3 77 77 5;

Pi2BGð Þx J G^{i}ð Þ :¼x

J G^{J}^{2}^{þ1}ð Þx
J G^{J}^{2}^{þ2}ð Þx

...

J G^{J}ð Þx
2

66 66 4

3 77 77

5; Pi2BGð Þx R^{s}^{i} :¼

R^{s}^{J2þ1}
R^{s}^{J2þ1}

...

R^{s}^{J}
2
66
66
4

3 77 77 5;

P_{i2Z}_{G}ð Þ_{x} Lhi :¼

Lh_{J1þ1}

Lh_{J1þ2}

...

Lh_{J2}

2 66 66 4

3 77 77

5; P_{i2B}_{G}ð Þ_{x} TL_{hi}G^{i}ð Þx
:¼

TL_{hJ}

2þ1G^{J}^{2}^{þ1}ð Þx
TL_{hJ}

2þ2G^{J}^{2}^{þ2}ð Þx
...

TL_{hJ}G^{J}ð Þx
2

66 66 66 4

3 77 77 77 5 :

Notice that

lin Tn R^{m}_{}gð Þx o

:¼ g 2 R ^{m}j g_{i} 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 TL_{hi}ðG^{i}ðxÞÞ implies that
lin Tn L_{hi}G^{i}ð Þx o

¼ C^{i}_{1}; C^{i}_{2}

j C^{i}_{1}ð ÞGx ^{i}_{1}ð Þ tanx ^{2}h_{i} C ^{i}_{2} _{T}

G^{i}_{2}ð Þ ¼ 0x

n o

¼ C^{i}_{1}; C^{i}_{2}

G^{i}_{1}ð Þx
G^{i}_{2}ð Þx

" #T

tan^{2}h_{i} 0
0 I_{s}_{i}_{1}

" #

C^{i}_{1}
C^{i}_{2}

" #

¼ 0 8<

:

9=

;

¼ ker G^{i}ð Þx ^{T}Hhi

:

Hence, the equality (3.6) is equivalent to J hð Þx

J gð Þx
Pi2ZGð Þx J G^{i}ð Þx
Pi2BGð Þx J G^{i}ð Þx
2

66 66 4

3
77
77
5R^{n}þ

0_{l}
f g
linTR^{m}_{}gð Þx

P_{i2Z}_{G}ð Þ_{x} 0si

P_{i2B}_{G}ð Þ_{x} ker G^{i}ð Þx ^{T}Hhi

2 66 66 64

3 77 77 75

¼

R^{l}
R^{m}
Pi2ZGð Þx R^{s}^{i}
Pi2BGð Þx R^{s}^{i}
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 G^{J}^{1}^{þ1}ð Þx ^{T} J G^{J}^{2}ð Þx ^{T} J G^{J}^{2}^{þ1}ð Þx ^{T} J G^{J}ð Þx ^{T}i

\ R^{l} g 2 R ^{m}j g_{i} ¼ 0; i 2 Ið Þx

R^{s}^{J1þ1} R^{s}^{J2}

range H^{T}_{h}

J2þ1G^{J}^{2}^{þ1}ð Þx

range H ^{T}_{h}_{J}G^{J}ð Þx

¼ 0l 0m 0s_{J1þ1} 0s_{J2} 0s_{J2þ1} 0sJ:

(3.7)
Let l ¼ ðl_{1}; :::; llÞ^{T}; gi 2 R; i 2 IþðxÞ [ I0ðxÞ, C^{i} 2 R^{s}^{i}; i 2 ZGðxÞ; pi 2
R; i 2 BGðxÞ satisfying

J hð Þx ^{T}l þ X

i2Iþð Þ[Ix 0ð Þx

J g^{i}ð Þx ^{T}g_{i}þ X

i2ZGð Þx

J G^{i}ð Þx ^{T}C^{i}þ X

i2BGð Þx

J G^{i}ð Þx ^{T}H^{T}_{h}_{i}G^{i}ð Þpx i¼ 0:

This together with (3.7) yields

l ¼ 0_{l}; g_{i} ¼ 0; i 2 Iþð Þ [ Ix _{0}ð Þ;x
C^{i} 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 h^{1}ð Þx ^{T}; :::; J h^{l}ð Þx ^{T}; J g^{i}ð Þx ^{T}; i 2 Iþð Þ [ Ix 0ð Þ;x

J G^{i}ð Þx ^{T}e^{j}_{s}_{i}; j ¼ 1; 2; :::; si; i 2 ZGð Þ; J Gx ^{i}ð Þx ^{T}HhiG^{i}ð 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 Þ^{T}d ¼ 0;

J gð Þd 2 Tx R^{m}_{}gð Þx

;
J G^{i}ð Þd 2 Tx L_{hi}G^{i}ð Þx

; i ¼ 1; 2; :::; J

9>

=

>;: 8>

><

>>

:

If KðxÞ is nonempty, then there exist l 2 R^{l}, g 2 R^{m}_{þ} and C^{i} 2 L^{}_{h}_{i} 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 G^{1}ð Þdx
J G^{J}...ð Þdx
2

66 66 64

3 77 77 75

2

0l

TRf g^{m}_{}gð Þx
TL_{h1}G^{1}ð Þx

TL_{hJ}G... ^{J}ð Þx
2

66 66 66 4

3 77 77 77 5

\

lg

C^{1}

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; C^{1}; :::; C^{J}Þ 2
R^{l} R^{m} R^{s}^{1} R^{s}^{J} and w 2 KðxÞ. Then, the critical cone CðxÞ can
be described as

Cð Þ ¼ d 2 Rx ^{n}

J hð Þdx

ð Þ_{k}¼ 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 G^{j}ð Þd 2 Tx L_{hj}G^{j}ð Þx

; C^{j}¼ 0_{s}_{j};
J G^{j}ð Þd ¼ 0;x C^{j}2 int L^{}_{h}_{j};
J G^{j}ð Þd 2 Rx þHhjC^{j}

; C^{j}2 bd L^{}_{h}_{j}n 0sj

; G^{j}ð Þ ¼ 0;x
J G^{j}ð Þdx

TC^{j}¼ 0; C^{j}2 bd L^{}_{h}_{j}n 0 _{s}_{j} ; G^{j}ð Þ 2 bd Lx hjn 0 _{s}_{j} :

9>

>>

>>

>>

>>

>>

>=

>>

>>

>>

>>

>>

>>

;

; 8>

>>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>>

>:

(3.9)
where the set RþðHhjC^{j}Þ is defined by

RþHhjC^{j}

:¼ rHn hjC^{j}j r 0o
:

Proof. From the equality (3.8), we have
Cð Þ ¼ d 2 Rx ^{n}

J hð Þdx

ð Þ_{k}¼ 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 G^{j}ð Þd 2 Tx L_{hj}G^{j}ð Þx

; J G ^{j}ð Þdx TC^{j}¼ 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 C^{j} ¼ 0_{s}_{j}, then the third row of (3.10)
becomes J G^{i}ðxÞd 2 TL_{hj}ðG^{j}ðxÞÞ.

Case (2): If C^{j} 2 int L^{}_{h}_{j}, from the KKT condition (3.2), then G^{j}ðxÞ ¼ 0sj.
The explicit form of TL_{hj}ðG^{j}ðxÞÞ defined in Theorem 2.3 implies that
TL_{hj}ðG^{j}ðxÞÞ ¼ Lhj. From the last row of (3.10), we obtain J G^{j}ðxÞd 2 Lhj. It
follows from Theorem 2.5 and ðJ G^{j}ðxÞdÞ^{T}C^{j}¼ 0 that J G^{j}ðxÞd ¼ 0sj:

Case (3): If C^{j} 2 bd L^{}_{h}_{j} n f0_{s}_{j}g and G^{j}ðxÞ ¼ 0sj, then TL_{hj}ðG^{j}ðxÞÞ ¼ Lhj

and J G^{j}ðxÞd 2 Lhj\ ðC^{j}Þ^{?}. It follows from Theorem 2.5 that J G^{j}ðxÞd ¼
0_{s}_{j} or there exists r>0 such that J G^{j}ðxÞd ¼ rHhjC^{j}. Hence, we
have J G^{j}ðxÞd 2 RþðHhjC^{j}Þ.

Case (4): If C^{j} 2 bd L^{}_{h}_{j} n f0_{s}_{j}g; G^{j}ðxÞ 2 bdLhjn f0_{s}_{j}g, we have
TL_{hj}G^{j}ð Þx

¼nðh1; h2Þ j h^{T}_{2}G^{j}_{2}ð Þhx 1G^{j}_{1}ð Þ tanx ^{2}h_{j} 0o
:

Combining the above equality with the fact J G^{j}ðxÞd 2 TL_{hj}ðG^{j}ðxÞÞ \
ðC^{j}Þ^{?} as in (3.10), we obtain

J G^{j}_{2}ð Þdx

_{T}

G^{j}_{2}ð Þ J Gx ^{j}_{1}ð Þdx

G^{j}_{1}ð Þ tanx ^{2}h_{j} 0;

J G^{j}_{2}ð Þdx

_{T}

C^{j}_{2}þ J G ^{j}_{1}ð Þdx

C^{j}_{1} ¼ 0: (3.11)

From the KKT condition (3.2), we know ðC^{j}Þ^{T}G^{j}ðxÞ ¼ 0. Because

C^{j} 2 bd L^{}_{h}_{j}n f0sjg; G^{j}ðxÞ 2 bd Lhj n f0sjg, by the case (d) in Theorem 2.5,
there exists r>0 such that C^{j}¼ rHhjG^{j}ðxÞ and ðJ G^{j}_{2}ðxÞdÞ^{T}G^{j}_{2}ðxÞ

ðJ G^{j}_{1}ðxÞdÞG^{j}_{1}ðxÞ tan^{2}h_{j} ¼ 0: Under this case, the equality (3.11) reduces
to ðJ G^{j}ðxÞdÞ^{T}C^{j} 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;

C^{1}; :::; C^{J}Þ 2 KðxÞ R^{l} R^{m} R^{s}^{1} R^{s}^{J}, d 2 CðxÞ, and RCQ (3.4)
holds at x. Denote the sigma term of (1.1) by

! l; g; C ^{1}; :::; C^{J}

; T^{2}_{X}ðGð Þ; J G xx ð ÞdÞ

;

where !ð; T^{2}_{X}ð; ÞÞ means the support function of the second-order tangent
set T^{2}_{X}ð; Þ. Then, we have

! l; g; C ^{1}; :::; C^{J}

; T^{2}_{X}ðGð Þ; J G xx ð ÞdÞ

¼ d^{T} X^{J}

j¼1

A^{j}x; l; g; C^{j}
0

@

1 Ad;

where the matrix A^{j}ðx; l; g; C^{j}Þ is defined by
A^{j}x; l; g; C^{j}

:¼ C^{j}1

G^{j}_{1}ð Þx cot^{2}h_{j}J G^{j}ð Þx T

Hh_{j}J G^{j}ð Þ; if Gx ^{j}ð Þ 2 bd Lx h_{j}n 0 _{s}_{j} ;

0; otherwise:

8>

<

>:

Proof. From the definitions of X and GðxÞ, we have

! l; g; C ^{1}; :::; C^{J}

; T^{2}_{X}ðGð Þ; J G xx ð ÞdÞ

¼ ! l; T ^{2}_{f g}_{0}_{l} ðhð Þ; J h xx ð ÞdÞ

þ ! g; T ^{2}_{R}^{m}_{}gð Þ; J g xx ð Þd
þX^{J}

j¼1

! C ^{j}; T^{2}_{L}_{hj}G^{j}ð Þ; J Gx ^{j}ð Þdx
:

(3.12)