to appear in Computational Optimization and Applications, 2015

### Constructions of complementarity functions and merit functions for circular cone complementarity problem

Xin-He Miao^{1}

Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China E-mail: xinhemiao@tju.edu.cn

Shengjuan Guo Department of Mathematics

School of Science Tianjin University Tianjin 300072, P.R. China E-mail: gshengjuan@163.com

Nuo Qi

Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China

E-mail: qinuo@163.com

Jein-Shan Chen ^{2}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

March 30, 2015

1The author’s work is supported by National Young Natural Science Foundation (No. 11101302 and No. 61002027) and National Natural Science Foundation of China (No. 11471241).

2Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.

(revised on August 7, 2015)

Abstract In this paper, we consider complementarity problem associated with circular cone, which is a type of nonsymmetric cone complementarity problem. The main purpose of this paper is to show the readers how to construct complementarity functions for such nonsymmetric cone complementarity problem, and propose a few merit functions for solving such a complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also show that these merit functions provide an error bound for the circular cone complementarity problem. These results ensure that the sequence generated by descent methods has at least one accumulation point, and build up a theoretical basis for designing the merit function method for solving circular cone complementarity problem.

Keywords. circular cone complementarity problem, complementarity function, merit function, the level sets, strong coerciveness.

### 1 Motivation and Introduction

The general conic complementarity problem is to find an element x ∈ IR^{n} such that
x ∈ K, F (x) ∈ K^{∗} and hx, F (x)i = 0, (1)
where h·, ·i denotes the Euclidean inner product, F : IR^{n}→ IR^{n} is a continuously differ-
entiable mapping, K represents a closed convex cone, and K^{∗} is the dual cone of K given
by

K^{∗} := {v ∈ IR^{n}| hv, xi ≥ 0, ∀x ∈ K}.

When K is a symmetric cone, the problem (1) is called the symmetric cone complemen- tarity problem [12, 14, 18, 20]. In particular, when K is the so-called second-order cone which is defined as

K^{n}:= {(x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k ≤ x_{1}},

the problem (1) reduces to the second-order cone complementarity problem [1, 3, 4, 5, 10, 11]. In contrast to symmetric cone programming and symmetric cone complemen- tarity problem, we are not familiar with their nonsymmetric counterparts. Referring the reader to [16, 19] and the bibliographies therein, we observe that there is no any unified way to handle nonsymmetric cone constraints, and the study on each item for such prob- lems usually uses certain specific features of the nonsymmetric cones under consideration.

In this paper, we pay attention to a special nonsymmetric cone K for problem (1).

In particular, we focus on the case of K being the circular cone defined as below, which

enables the problem (1) reduce to the circular cone complementarity problem (CCCP for
short). Indeed in IR^{n}, the circular cone [7, 23] is a pointed closed convex cone having
hyper-spherical sections orthogonal to its axis of revolution about which the cone is
invariant to rotation. Let its half-aperture angle be θ with θ ∈ (0,^{π}_{2}). Then, the circular
cone denoted by L_{θ} can be expressed as

L_{θ} := x = (x1, x_{2}) ∈ IR × IR^{n−1}| kxk cos θ ≤ x_{1}

(2)

= x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k ≤ x_{1}tan θ .

When θ = ^{π}_{4}, the circular cone is exactly the second-order cone, which means the circular
cone complementarity problem is actually the second-order cone complementarity prob-
lem. Thus, the circular cone complementarity problem (CCCP) can be viewed as the
generalization of the second-order cone complementarity problem. Moreover, the CCCP
includes the KKT system of the circular programming problem [13] as a special case. For
real world applications of optimization problems involving circular cones, please refer to
[6]. Note that in [23], Zhou and Chen characterize the relation between circular cone L_{θ}
and second-order cone as follows:

L_{θ} = A^{−1}K^{n} and K^{n}= AL_{θ} with A =tan θ 0

0 I

.

In other words, for any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}, there
have

x ∈ L_{θ} ⇐⇒ Ax ∈ K^{n}, y ∈ L^{∗}_{θ} ⇐⇒ A^{−1}y ∈ K^{n}. (3)
Relation (3) indicates that after scaling the circular cone complementarity problem and
the second-order cone complementarity problem are equivalent. However, when dealing
with the circular cone complementarity problem, this approach may not be acceptable
from both theoretical and numerical viewpoints. Indeed, if the appropriate scaling is not
found or checked, some scaling step can cause undesirable numerical performance due
to round-off errors in computers, which has been confirmed by experiments. Moreover,
it usually need to exploits its associated merit functions or complementarity functions,
which plays an important role in tackling complementarity problem. To this end, we are
devoted to seeking a way to construct complementarity functions and merit functions for
the circular cone complementarity problem directly. Thus, we pay our attention to the
circular cone complementarity problem and the structure of L_{θ} mainly. There is another
relationship between the circular cone and the (nonsymmetric) matrix cone introduced
in [8, 9], where the authors study the epigraph of six different matrix norms, such as the
Frobeninus norm, the l∞ norm, l_{1} norm, the spectral or the operator norm, the nuclear
norm, the Ky Fan k-norm. If we regard a matrix as a high-dimensional vector, then the
circular cone is equivalent to the matrix cone with Frobeninus norm, see [24] for more
details.

While there have been much attention to the symmetric cone complementarity prob- lem and the second-order cone complementarity problem, the study about nonsymmetric cone complementarity problem is very limited. The main difficulty is that the idea for constructing complementarity functions (C-functions for short) and merit functions is not clear. Hence, The main goal of this paper is showing the readers how to construct C-functions and merit functions for such complementarity problem, and studying the properties of these merit functions. To our best knowledge, the idea is new and we be- lieve that it will help in analyzing other types of nonsymmetric cone complementarity problems.

Recall that for solving the problem (1), a popular approach is to reformulate it as
an unconstrained smooth minimization problem or a system of nonsmooth equations. In
this category of methods, it is important to adapt a merit function. Officially, a merit
function for the circular cone complementarity problem is a function h : IR^{n}→ [0, +∞),
provided that

h(x) = 0 ⇐⇒ x solves the CCCP (1).

Hence, solving the problem (1) is equivalent to handling the unconstrained minimization problem

x∈IRmin^{n}h(x)

with the optimal value zero. For constructing the merit functions in finite dimensional
vector space, please refer to [17]. Until now, for solving symmetric cone complementarity
problem, a number of merit functions have been proposed. Among them, one of the most
popular merit functions is the natural residual (NR) merit function ΨN R : IR^{n} → IR,
which is defined as

ΨN R(x) := 1

2kφ_{NR}(x, F (x))k^{2} = 1

2kx − (x − F (x))+k^{2},

where (·)_{+} denotes the projection onto the symmetric cone K. It is well known that
Ψ_{N R}(x) = 0 if and only if x is a solution to the symmetric cone complementarity prob-
lem. In this paper, we present two classes of complementarity functions and four types of
merit functions for the circular cone complementarity problem. Moreover, we investigate
the properties of these proposed merit functions, and study conditions under which these
merit functions provide bounded level sets. Note that such properties will guarantee
that the sequence generated by descent methods has at least one accumulation point,
and build up a theoretical basis for designing the merit function method for solving cir-
cular cone complementarity problem.

### 2 Preliminaries

In this section, we briefly review some basic concepts and background materials about the circular cone and second-order cone, which will be extensively used in subsequent analysis.

As defined in (2), the circular cone L_{θ} is a pointed closed convex cone and has a
revolution axis which is the ray generated by the canonical vector e_{1} := (1, 0, · · · , 0)^{T} ∈
IR^{n}. Its dual cone denoted by L^{∗}_{θ} is given as

L^{∗}_{θ} := {y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}| kyk sin θ ≤ y_{1}}.

Note that the circular cone L_{θ} is not a self-dual cone when θ 6= ^{π}_{4}, that is, L^{∗}_{θ} 6= L_{θ},
whenever θ 6= 45^{◦}. Hence, L_{θ} is not a symmetric cone for θ ∈ 0,^{π}_{2} \{^{π}_{4}}. It is also
known from [23] that the dual cone of L_{θ} can be expressed as

L^{∗}_{θ} = {y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}| ky_{2}k ≤ y_{1}cot θ} = L^{π}

2−θ.

Now, we talk about the projection onto L_{θ} and L^{∗}_{θ}. To this end, we let x_{+} denote
the projection of x onto the circular cone L_{θ}, and x− be the projection of −x onto the
dual cone L^{∗}_{θ}. With these notations, for any x ∈ IR^{n}, it can be verified that x = x+− x−.
Moreover, due to the special structure of the circular cone L_{θ}, the explicit formula of
projection of x ∈ IR^{n} onto L_{θ} is obtained in [23] as below:

x+=

x if x ∈ L_{θ},
0 if x ∈ −L^{∗}_{θ},
u otherwise,

(4)

where

u =

x_{1} + kx_{2}k tan θ
1 + tan^{2}θ

x_{1}+ kx_{2}k tan θ
1 + tan^{2}θ tan θ

x_{2}
kx_{2}k

. Similarly, we can obtain the expression of x− as below:

x_{−}=

0 if x ∈ L_{θ},

−x if x ∈ −L^{∗}_{θ},
w otherwise,

(5)

where

w =

−x_{1}− kx_{2}k cot θ
1 + cot^{2}θ

x_{1}− kx2k cot θ
1 + cot^{2}θ cot θ

x2

kx_{2}k

.

From the expressions (4)-(5) for x_{+} and x−, it is easy to verity that hx_{+}, x−i = 0 for any
x ∈ IR^{n}.

Next, we introduce the Jordan product associated with second-order cone. As men-
tioned earlier, the SOC in IR^{n} (also called Lorentz cone or ice-cream cone) is defined
by

K^{n}:= {x = (x1, x2) ∈ IR × IR^{n−1}| kx2k ≤ x1}.

It is well known that the dual cone of K^{n} is itself, and the second-order cone K^{n} belongs
to a class of symmetric cones. In addition, K^{n} is a special case of L_{θ} corresponding to
θ = ^{π}_{4}. In fact, there is a relationship between L_{θ} and K^{n}, which is described in (3). In
the SOC setting, there is so-called Jordan algebra associated with SOC. More specifically,
for any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}, in the setting of the SOC,
the Jordan product of x and y is defined as

x ◦ y :=

hx, yi
y_{1}x_{2}+ x_{1}y_{2}

.

The Jordan product “◦”, unlike scalar or matrix multiplication, is not associative. The
identity element under Jordan product is e = (1, 0, · · · , 0)^{T} ∈ IR^{n}. In this paper, we
write x^{2} to mean x ◦ x. It is known that x^{2} ∈ K^{n} for any x ∈ IR^{n}, and if x ∈ K^{n}, there
exists a unique vector denoted by x^{1}^{2} in K^{n} such that (x^{1}^{2})^{2} = x^{1}^{2} ◦ x^{1}^{2} = x. For any
x ∈ IR^{n}, we denote |x| := √

x^{2} and x^{soc}_{+} means the orthogonal projection of x onto the
second-order cone K^{n}. Then, it follows that x^{soc}_{+} = x + |x|

2 . For further details regarding the SOC and Jordan product, please refer to [1, 3, 5, 10].

Lemma 2.1. ([10, Proposition 2.1]) For any x, y ∈ IR^{n}, the following holds:

x ∈ K^{n}, y ∈ K^{n}, and hx, yi = 0 ⇐⇒ x ∈ K^{n}, y ∈ K^{n}, and x ◦ y = 0.

With the help of (3) and Lemma 2.1, we obtain the following theorem which explains the relationship between SOCCP and CCCP.

Theorem 2.1. Let A =tan θ 0

0 I

. For any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈
IR × IR^{n−1}, the following are equivalent:

(a) x ∈ Lθ, y ∈ L^{∗}_{θ} and hx, yi = 0.

(b) Ax ∈ K^{n}, A^{−1}y ∈ K^{n} and hAx, A^{−1}yi = 0.

(c) Ax ∈ K^{n}, A^{−1}y ∈ K^{n} and Ax ◦ A^{−1}y = 0.

(d) x ∈ L_{θ}, y ∈ L^{∗}_{θ} and Ax ◦ A^{−1}y = 0.

In each case, elements x and y satisfy the condition that either y_{2} is a multiple of x_{2} or
x2 is a multiple of y2.

Proof. From the relation between K^{n} and L_{θ} given as in (3), we know that
x ∈ L_{θ} ⇐⇒ Ax ∈ K^{n} and y ∈ L^{∗}_{θ} ⇐⇒ A^{−1}y ∈ K^{n}.
Moreover, under condition (a), there holds

hAx, A^{−1}yi = hA^{−1}Ax, yi = hx, yi = 0.

Hence, it follows that (a) and (b) are equivalent. The equivalence of (b) and (c) has been
shown in Lemma 2.1. In addition, based on the relation between K^{n} and L_{θ} again, the
equivalence of (c) and (d) is obvious.

Now, under condition (a), we prove that either y_{2} is a multiple of x_{2} or x_{2} is a multiple
of y_{2}. To see this, note that x ∈ L_{θ} and y ∈ L^{∗}_{θ} which gives

kx_{2}k ≤ x_{1}tan θ and ky_{2}k ≤ y_{1}cot θ.

This together with hx, yi = 0 yields

0 = hx, yi

= x_{1}y_{1}+ hx_{2}, y_{2}i

≥ x1y1− kx2kky2k

≥ x_{1}y_{1}− x_{1}y_{1}

= 0

which implies hx_{2}, y_{2}i = kx_{2}kky_{2}k. This says that either y_{2} is a multiple of x_{2} or x_{2} is a
multiple of y_{2}. Thus, the proof is complete. 2

### 3 C-functions for CCCP

In this section, we define C-functions for CCCP and the product of elements in the setting of the circular cone. Moreover, based on the product of elements, we construct some C- functions which play an important role in solving the circular cone complementarity problems by merit function methods.

Definition 3.1. Given a mapping φ : IR^{n} × IR^{n} → IR^{n}, we call φ an C-function for
CCCP if, for any (x, y) ∈ IR^{n}× IR^{n}, it satisfies

φ(x, y) = 0 ⇐⇒ x ∈ L_{θ}, y ∈ L^{∗}_{θ}, hx, yi = 0.

When θ = ^{π}_{4}, an C-function for CCCP reduces to an C-function for SOCCP, i.e.,
φ(x, y) = 0 ⇐⇒ x ∈ K^{n}, y ∈ K^{n}, hx, yi = 0.

Two popular and well-known C-functions for SOCCP are Fischer-Burmeister (FB) func- tion and natural residual (NR) function:

φ_{FB}(x, y) = (x^{2}+ y^{2})^{1/2}− (x + y),
φ_{NR}(x, y) = x − (x − y)^{soc}_{+} .

We may ask whether we can modify the above two C-functions for SOCCP to form C-functions for CCCP. The answer is affirmative. In fact, we consider

φg_{FB}(x, y) := (Ax)^{2}+ (A^{−1}y)^{2}^{1}_{2}

− (Ax + A^{−1}y),
φg_{NR}(x, y) := Ax − [Ax − A^{−1}y]^{soc}_{+} .

Then, these two functions are C-functions for CCCP.

Proposition 3.1. Let gφ_{FB} and gφ_{NR} be defined as above where (Ax)^{2} equals (Ax) ◦ (Ax)
under Jordan product. Then, gφ_{FB} and gφ_{NR} are both C-functions for CCCP.

Proof. In view of Theorem 2.1 and Definition 3.1, it is not hard to verify that
gφ_{FB}(x, y) = 0 ⇐⇒ x ∈ L_{θ}, y ∈ L^{∗}_{θ}, hx, yi = 0,

φg_{NR}(x, y) = 0 ⇐⇒ x ∈ Lθ, y ∈ L^{∗}_{θ}, hx, yi = 0,
which says that these two functions are C-functions for CCCP. 2

We point out that if we consider directly the FB function φ_{FB}(x, y) for CCCP, un-
fortunately, it cannot be C-function for CCCP because x^{2} is not well-defined associated
with the circular cone L_{θ} for any x ∈ IR^{n}. More specifically, because x^{2} is defined under
the Jordan product in the setting of SOC, i.e.,

x^{2} := x ◦ x =

hx, yi
x_{1}y_{2}+ y_{1}x_{2}

,

it follows that x^{2} ∈ K^{n}, which implies x^{2} may not belong to L_{θ} or L^{∗}_{θ}. Furthermore, when
φ_{FB}(x, y) = 0, we have x + y = (x^{2}+ y^{2})^{1}^{2} ∈ K^{n}, which yields that x, y ∈ K^{n}. This says
that either x /∈ L_{θ} or y /∈ L^{∗}_{θ}. All the above explains that the FB function φ_{FB} cannot be
an C-function for CCCP. Nonetheless, the NR function φ_{NR} : IR^{n}× IR^{n} → IR^{n} given by

φ_{NR}(x, y) := x − (x − y)_{+} (6)

is always an C-function for CCCP. Moreover, it is also an C-function for general cone complementarity problem, see [11, Proposition 1.5.8].

Are there any other types of C-functions for CCCP and how to construct an C-
function for CCCP? As mentioned earlier, The FB function φ_{FB} cannot serve as C-
functions for CCCP because “x^{2}” is not well-defined in the setting of circular cone. This
inspires us to define a special product associated with circular cone, and find other C-
functions for CCCP.

For any x = (x1, x2) ∈ IR × IR^{n−1} and y = (y1, y2) ∈ IR × IR^{n−1}, we define one type of
product of x and y as follows:

x • y = x_{1}
x2

• y_{1}
y2

=

hx, yi

max{tan^{2}θ, 1} x1y2+ max{cot^{2}θ, 1} y1x2

. (7) From the above product and direct calculation, it is easy to verify that

hx • y, zi = hx, z • yi, ∀z ∈ IR^{n} with θ ∈
0,π

4 i

(8) and

hx • y, zi = hy, x • zi, ∀z ∈ IR^{n} with θ ∈hπ
4,π

2

. (9)

Moreover, we also obtain the following inequalities which are crucial to establishing our main results.

Lemma 3.1. For any x, y ∈ IR^{n},

(a) if θ ∈ (0,^{π}_{4}], we have hx−, x_{+}• (−y)−i ≤ 0;

(b) if θ ∈ [^{π}_{4},^{π}_{2}), we have h(−y)+, x+• (−y)−i ≤ 0.

Proof. (a) When θ ∈ (0,^{π}_{4}], let x_{+} := (s, u) ∈ IR × IR^{n−1}, x_{−} := (t, v) ∈ IR × IR^{n−1} and
(−y)− := (k, w) ∈ IR × IR^{n−1}. For the elements x_{+}, x− and (−y)−, if there exist at least
one in them is zero, it is easy to obtain

hx−, x_{+}• (−y)−i = 0.

If all the three elements are not equal to zero, from the definition of x+, x−, and (−y)−, we have k cot θ ≥ kwk, s tan θ = kuk, t cot θ = kvk and

u = αv or v = αu with α < 0.

Without loss of generality, we consider the case u = αv with α < 0 for the following analysis. In fact, using this, we know that

hx−, x_{+}• (−y)−i

= stk + thu, wi + shv, wi + khu, vi cot^{2}θ

= kukkvkk − kkukkvk cot^{2}θ − kukhv, wi tan θ + kukhv, wi cot θ

= (1 − cot^{2}θ)kkukkvk − (1 − cot^{2}θ)(kukhv, wi tan θ)

= (1 − cot^{2}θ)[kkukkvk − kukhv, wi tan θ]

≤ (1 − cot^{2}θ)[kkukkvk − kukkvkkwk tan θ]

= (1 − cot^{2}θ)kukkvk[k − kwk tan θ]

≤ 0.

Here the second equality is true due to αt = αkvk tan θ = −kuk tan θ. The last inequality
holds due to k cot θ ≥ kwk and θ ∈ (0,^{π}_{4}]. Hence, the desired result follows.

(b) When θ ∈ [^{π}_{4},^{π}_{2}), with the same skills, we also conclude that
h(−y)+, x+• (−y)−i ≤ 0.

Then, the desired result follows. 2

Besides the inequalities in Lemma 3.1, “•” defined as in (7) plays the similar role like what “◦” does in the setting of second-order cone. This is shown as below.

Theorem 3.1. For any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}, the
following statements are equivalent:

(a) x ∈ Lθ, y ∈ L^{∗}_{θ} and hx, yi = 0.

(b) x ∈ L_{θ}, y ∈ L^{∗}_{θ} and x • y = 0.

In each case, x and y satisfy the condition that either y_{2} is a multiple of x_{2} or x_{2} is a
multiple of y_{2}.

Proof. In view of Theorem 2.1, we know that part (a) is equivalent to
x ∈ L_{θ}, y ∈ L^{∗}_{θ} and Ax ◦ A^{−1}y = 0.

To proceed the proof, we discuss the following two cases.

Case 1: For θ ∈ (0,^{π}_{4}], from the definition of the product of x and y, we have
x • y =

hx, yi

x_{1}y_{2}+ cot^{2}θ y_{1}x_{2}

which implies

Ax ◦ A^{−1}y =

hx, yi

x_{1}tan θ y_{2}+ cot θ y_{1}x_{2}

= 1 0

0 (tan θ)I

(x • y).

This together with Theorem 2.1 yields the conclusion.

Case 2: For θ ∈ [^{π}_{4},^{π}_{2}), from the definition of the product of x and y again, we have
x • y =

hx, yi

tan^{2}θ x_{1}y_{2}+ y_{1}x_{2}

which says

Ax ◦ A^{−1}y = 1 0
0 (cot θ)I

(x • y).

Then, applying Theorem 2.1 again, the desired result follows. 2

Based on the product x • y of x and y. we now introduce a class of functions φ_{p} :
IR^{n}× IR^{n}→ IR^{n}, which is called the penalized natural residual function and defined as

φ_{p}(x, y) = x − (x − y)_{+}+ p (x_{+}• (−y)−) , p > 0. (10)
Note that when p = 0, φ_{p}(x, y) reduces to φ_{NR}(x, y). In the following, we show that the
function φp is an C-function for CCCP. To achieve the conclusion, a technical lemma is
needed.

Lemma 3.2. Let φ_{p} : IR^{n}× IR^{n} → IR^{n} be defined as in (10). Then, for any x, y ∈ IR^{n},
we have

kφp(x, y)k ≥ max {kx−k, k(−y)+k} .

Proof. First, we prove that kφ_{p}(x, y)k ≥ kx−k. To see this, we observe that
kφ_{p}(x, y)k^{2}

= hx − (x − y)_{+}+ p x_{+}• (−y)−, x − (x − y)_{+}+ p x_{+}• (−y)−i

= hx_{+}− x_{−}− (x − y)_{+}+ p x_{+}• (−y)_{−}, x_{+}− x_{−}− (x − y)_{+}+ p x_{+}• (−y)_{−}i

= kx−k^{2}+ kx+− (x − y)++ p x+• (−y)−k^{2}− 2 hx−, x+− (x − y)++ p x+• (−y)−i

≥ kx−k^{2}− 2hx−, x_{+}i + 2 hx−, (x − y)_{+}i − 2 hx−, p x_{+}• (−y)−i

≥ kx−k^{2}− 2p hx−, x_{+}• (−y)−i .

Here, the last inequality is true due to x_{+}, (x − y)_{+}∈ L_{θ}, x− ∈ L^{∗}_{θ}, hx_{+}, x−i = 0 and the
relation between Lθ and L^{∗}_{θ}. When θ ∈ (0,^{π}_{4}], by Lemma 3.1(a), we have

hx−, x_{+}• (−y)−i ≤ 0.

When θ ∈ [^{π}_{4},^{π}_{2}), from equation (9), we have

hx−, x+• (−y)−i = h(−y)−, x+• x−i = 0

where the second equality holds due to x_{+}• x−= 0. In summary, from all the above, we
prove that

kφ_{p}(x, y)k^{2} ≥ kx−k^{2}.

With similar arguments, we also obtain
kφ_{p}(x, y)k^{2}

= hx − (x − y)_{+}+ p x_{+}• (−y)−, x − (x − y)_{+}+ p x_{+}• (−y)−i

= hy − (x − y)−+ p x_{+}• (−y)−, y − (x − y)−+ p x_{+}• (−y)−i

= h(−y)−− (−y)_{+}− (x − y)−+ p x_{+}• (−y)−, (−y)−− (−y)_{+}− (x − y)−

+px_{+} • (−y)_{−}i

= k(−y)+k^{2}+ k(−y)−− (x − y)−+ p x+• (−y)−k^{2}− 2h(−y)+, (−y)−− (x − y)−

+px_{+} • (−y)−i

≥ k(−y)_{+}k^{2}− 2h(−y)_{+}, (−y)−i + 2h(−y)_{+}, (x − y)−i − 2h(−y)_{+}, p x_{+}• (−y)−i

≥ k(−y)_{+}k^{2}− 2p h(−y)_{+}, x_{+}• (−y)_{−}i

≥ k(−y)+k^{2},

where the second inequality holds due to due to (−y)_{+} ∈ L_{θ}, (−y)−, (x − y)− ∈ L^{∗}_{θ},
h(−y)_{+}, (−y)−i = 0 and the relation between L_{θ} and L^{∗}_{θ}. The last inequality holds due
to equation (8) and Lemma 3.1(b). Therefore, we prove that kφ_{p}(x, y)k ≥ k(−y)_{+}k.

Then, the proof is complete. 2

Remark 3.1. From the proof of Lemma 3.2, it also can be seen that
kφ_{NR}(x, y)k ≥ max{kx−k, k(−y)_{+}k}.

Theorem 3.2. Let φ_{p} : IR^{n}× IR^{n}→ IR^{n} be defined as in (10). Then, φ_{p} is an C-function
for CCCP, i.e., for any x, y ∈ IR^{n},

φ_{p}(x, y) = 0 ⇐⇒ x ∈ L_{θ}, y ∈ L^{∗}_{θ} and hx, yi = 0.

Proof. “=⇒” Suppose that φ_{p}(x, y) = 0. If either x /∈ L_{θ} or y /∈ L^{∗}_{θ}, applying Lemma
3.2 yields

kφ_{p}(x, y)k ≥ max{kx_{−}k, k(−y)_{+}k} > 0.

This contradicts with φ_{p}(x, y) = 0. Hence, there must have x ∈ L_{θ} and y ∈ L^{∗}_{θ}. Next,
we argue that hx, yi = 0. To see this, we consider the first component of φp(x, y), which
is denoted by [φ_{p}(x, y)]_{1}. In other words,

[φ_{p}(x, y)]_{1} = [x − (x − y)_{+}+ p x • y]_{1}

=

y_{1}+ p hx, yi if x − y ∈ L_{θ},
x_{1}+ p hx, yi if x − y ∈ −L^{∗}_{θ},
w + p hx, yi otherwise,

where

w = x1− x_{1}− y_{1}+ kx_{2}− y_{2}k tan θ

1 + tan^{2}θ = x_{1}tan^{2}θ + y_{1}− kx_{2}− y_{2}k tan θ
1 + tan^{2}θ .
Since x ∈ L_{θ} and y ∈ L^{∗}_{θ}, it follows that x_{1}, y_{1} ≥ 0, hx, yi ≥ 0 and

x_{1}tan^{2}θ + y_{1}− kx_{2}− y_{2}k tan θ

1 + tan^{2}θ ≥ tan θ(x_{1}tan θ − kx_{2}k + y_{1}cot θ − ky_{2}k)

1 + tan^{2}θ ≥ 0.

This together with φ_{p}(x, y) = 0 gives phx, yi = 0. Thus, we conclude that hx, yi = 0
because p > 0.

“⇐=” Suppose that x ∈ Lθ, y ∈ L^{∗}_{θ} and hx, yi = 0. Since φ_{NR} is always an C-function for
CCCP, we have x − (x − y)_{+}= 0. Using Theorem 3.1 again yields x_{+}• (−y)− = x • y = 0,
which says φ_{p}(x, y) = 0. 2

Remark 3.2. In fact, for any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1},
we define another type of product of x and y as follows:

x • y = x_{1}
x_{2}

• y_{1}
y_{2}

=

hx, yi

min{tan^{2}θ, 1} x_{1}y_{2}+ min{cot^{2}θ, 1} y_{1}x_{2}

. With the same skills, we may obtain the same results.

Motivated by the construction of φ_{p} given as in (10), we consider another function
φ_{r} : IR^{n}× IR^{n}→ IR^{n} defined by

φ_{r}(x, y) = x − (x − y)_{+}+ r (x • y)^{Ω}_{+} r > 0, (11)

where Ω := L_{θ}∩ L^{∗}_{θ} = L_{θ} if θ ∈ (0,^{π}_{4}],

L^{∗}_{θ} if θ ∈ [^{π}_{4},^{π}_{2}). We point out that the function φ_{r} defined
as in (11) is not an C-function for CCCP. The reason come from that if φ_{r}(x, y) = 0, we
have φ_{NR}(x, y) = x − (x − y)_{+} = −r (x • y)^{Ω}_{+}. Combining with the expression of φ_{p}, this
implies that

−r (x • y)^{Ω}_{+}+ p (x_{+}• (−y)−) 6= 0

due to (x • y)^{Ω}_{+} ∈ Ω = L_{θ} ∩ L^{∗}_{θ} and x_{+} • (−y)− ∈ K/ ^{n} ⊇ L_{θ} (or L^{∗}_{θ}) when θ ∈ (0,^{π}_{4}]
(or θ ∈ [^{π}_{4},^{π}_{2})). This explains that φ_{p}(x, y) 6= 0, which contradicts φ_{p}(x, y) being an
C-function for CCCP.

However, there is a merit function related to φ_{r} which possesses property of bounded
level sets. We will explore it in next section.

### 4 Merit functions for circular cone complementarity problem

In this section, based on the product (7) of x and y in IR^{n}, we propose four classes of
merit functions for the circular cone complementarity problem and investigate their im-
portant properties, respectively.

First, we recall that a function F : IR^{n} → IR^{n} is said to be monotone if, for any
x, y ∈ IR^{n}, there holds

hx − y, F (x) − F (y)i ≥ 0;

and strictly monotone if, for any x 6= y, the above inequality holds strictly; and strongly
monotone with modulus ρ > 0 if, for any x, y ∈ IR^{n}, the following inequality holds

hx − y, F (x) − F (y)i ≥ ρkx − yk^{2}.

The following technical lemma is crucial for achieving the property of bounded level sets.

Lemma 4.1. Suppose that CCCP has a strictly feasible point ¯x, i.e., ¯x ∈ int(L_{θ}) and
F (¯x) ∈ int(L^{∗}_{θ}) and that F is a monotone function. Then, for any sequence {x^{k}} satis-
fying

x^{k}

→ ∞, lim sup

k→∞

x^{k}_{−}

< ∞ and lim sup

k→∞

(−F (x^{k}))_{+}
< ∞,
we have

x^{k}, F (x^{k}) → ∞ and x^{k}_{+}, (−F (x^{k}))− → ∞.

Proof. Since F is monotone, for all x^{k} ∈ IR^{n}, we know
x^{k}− ¯x, F (x^{k}) − F (¯x) ≥ 0,
which says

x^{k}, F (x^{k}) + h¯x, F (¯x)i ≥x^{k}, F (¯x) + ¯x, F (x^{k}) . (12)
Using x^{k} = x^{k}_{+}− x^{k}_{−} and F (x^{k}) = (−F (x^{k}))−− (−F (x^{k}))_{+}, it follows from (12) that

x^{k}, F (x^{k}) + h¯x, F (¯x)i

≥ x^{k}_{+}, F (¯x) − x^{k}_{−}, F (¯x) + ¯x, (−F (x^{k}))− − ¯x, (−F (x^{k}))_{+} . (13)
We look into the first term in the right-hand side of (13).

x^{k}_{+}, F (¯x)

= x^{k}_{+}

1[f (¯x)]_{1}+x^{k}_{+}

2, [f (¯x)]_{2}

≥ x^{k}_{+}

1[f (¯x)]_{1}−
x^{k}_{+}

2

· k[f (¯x)]_{2}k

≥ x^{k}_{+}

1[f (¯x)]_{1}−x^{k}_{+}

1tan θ k[f (¯x)]_{2}k

= x^{k}_{+}

1{[f (¯x)]_{1}− tan θ k[f (¯x)]_{2}k} . (14)

Note that x^{k}= x^{k}_{+}−x^{k}_{−}, it gives kx^{k}_{+}k ≥ kx^{k}k−kx^{k}_{−}k. From the assumptions on {x^{k}}, i.e.,
kx^{k}k → ∞, and lim sup_{k→∞}kx^{k}_{−}k < ∞, we see that kx^{k}_{+}k → ∞, and hence [x^{k}_{+}]1 → ∞.

Because CCCP has a strictly feasible point ¯x, we have [f (¯x)]_{1}− tan θk[f (¯x)]_{2}k > 0, which
together with (14) implies that

hx^{k}_{+}, F (¯x)i → ∞ (k → ∞). (15)

On the other hand, we observe that lim sup

k→∞

hx^{k}_{−}, F (¯x)i ≤ lim sup

k→∞

kx^{k}_{−}kkF (¯x)k < ∞
lim sup

k→∞

h¯x, (−F (x^{k}))+i ≤ lim sup

k→∞

k¯xkk(−F (x^{k}))+k < ∞
and h¯x, (−F (x^{k}))−i ≥ 0. All of these together with (13) and (15) yield

x^{k}, F (x^{k}) → ∞,
which is the first part of the desired result.

Next, we prove thatx^{k}_{+}, (−F (x^{k}))_{−} → ∞. Suppose not, that is, limk→∞x^{k}_{+}, (−F (x^{k}))_{−} <

∞. Then, we obtain

x^{k}_{+}, (−F (x^{k}))_{−}
kx^{k}_{+}k =

x^{k}_{+}

kx^{k}_{+}k, (−F (x^{k}))−

→ 0.

This means that there exists ¯x ∈ IR^{n} such that x^{k}_{+}

kx^{k}_{+}k → x¯_{+}
k¯x_{+}k and

x¯_{+}

k¯x_{+}k, (−F (¯x))−

= 0. (16)

Denote z := x¯_{+}

k¯x_{+}k and apply Theorem 3.1, there exists α ∈ IR such that
[(−F (¯x))−]_{2} = αz_{2} or αz_{2} = [(−F (¯x))−]_{2}.

It is obvious that z ∈ L_{θ} and (−F (¯x))− ∈ L^{∗}_{θ}. Hence, equation (16) implies that α < 0,
which says that z_{2} and [(−F (¯x))−]_{2} are in opposite direction to each other. From the
expression of (−F (¯x))_{+}and (−F (¯x))_{−}again, it follows that [(−F (¯x))_{+}]_{2} and [(−F (¯x))_{−}]_{2}
are in the opposite direction, to each other. These conclude that z_{2} and [(−F (¯x))_{+}]_{2} are
in the same direction, which means [¯x_{+}]_{2} and [(−F (¯x))_{+}]_{2} are also in the same direction.

Now, combining with the fact that ¯x_{+}, (−F (¯x))_{+}∈ L_{θ}, we have
h¯x+, (−F (¯x))+i ≥ 0.

Similarly, by the the relation between ¯x_{+} and ¯x−, we know [¯x−]_{2} and [(−F (¯x))−]_{2} are in
the same direction. Then, combining with ¯x−, (−F (¯x))−∈ L^{∗}_{θ}, it leads to

h¯x−, (−F (¯x))−i ≥ 0.

Moreover, writing out the expression for h¯x, F (¯x)i, we see that

h¯x, F (¯x)i = h¯x_{+}, (−F (¯x))_{−}i − h¯x_{+}, (−F (¯x))_{+}i − h¯x_{−}, (−F (¯x))_{−}i + h¯x_{−}, (−F (¯x))_{+}i.

Note that the second and third terms of the right-hand side are nonpositive and the fourth
is bounded from above. Hence, from the assumptions lim_{k→∞}x^{k}_{+}, (−F (x^{k}))− < ∞, we
conclude that h¯x, F (¯x)i < ∞, which contradict

h¯x, F (¯x)i = lim

k→∞x^{k}, F (x^{k}) = ∞.

Thus, we prove that x^{k}_{+}, (−F (x^{k}))− → ∞. 2

### 4.1 The first class of merit functions

For any x ∈ IR^{n}, from the analysis of the section 3, we know that the function φ_{p} and φ_{NR}
are complementarity function for CCCP. In this subsection, we focus on the property of
bounded level sets of merit functions based on φ_{NR} and φ_{p} with the product of elements,
which is a property to guarantee that the existence of accumulation points of sequence
generated by some descent algorithms.

Theorem 4.1. Let φ_{p} be defined as in (10). Suppose that CCCP has a strictly feasible
point and that F is monotone. Then, the level set

L_{p}(α) = {x ∈ IR^{n}| kφ_{p}(x, F (x))k ≤ α}

is bounded for all α ≥ 0.

Proof. We prove this result by contradiction. Suppose there exists an unbounded
sequence {x^{k}} ⊂ L_{p}(α) for some α ≥ 0. If kx^{k}_{−}k → ∞ or k(−F (x^{k}))_{+}k → ∞, by Lemma
3.2, we have kφ_{p}(x^{k}, F (x^{k}))k → ∞, which contradicts kφ_{p}(x^{k}, F (x^{k}))k ≤ α. On the other
hand, if

lim sup

k→∞

kx^{k}_{−}k < ∞ and lim sup

k→∞

(−F (x^{k}))+

< ∞,

it follows from Lemma 4.1 that x^{k}_{+}, (−F (x^{k}))− → ∞. From the proof of Lemma 4.1,
there exists a constant κ_{0} such that

φ_{NR}(x^{k}, f (x^{k}))

1

≥

[x^{k}_{+}]_{1}− κ_{0} if x^{k}− F (x^{k}) ∈ −L^{∗}_{θ},

(−F (x^{k}))−

1− κ_{0} if x^{k}− F (x^{k}) ∈ L_{θ},

[x^{k}_{+}]1tan^{2}θ+[^{(−F (x}^{k}^{))}^{−}]_{1}^{−k[x}^{k}+]2k tan θ−k[(−F (x^{k}))−]2k tan θ
1+tan^{2}θ

−^{2κ}_{1+tan}^{0}^{(1+tan θ)}2θ , if x^{k}− F (x^{k}) /∈ L_{θ}∪ −L^{∗}_{θ},

which means lim infφ_{NR}(x^{k}, f (x^{k}))

1 > −∞. Hence, it follows that

φp(x^{k}, f (x^{k}))

1 = φ_{NR}(x^{k}, f (x^{k}))

1+(x^{k}_{+}• (−F (x^{k}))_{−}

1

= φ_{NR}(x^{k}, f (x^{k}))

1+x^{k}_{+}, (−F (x^{k}))−

→ ∞, where the limit comes from

x^{k}_{+}, (−F (x^{k}))− → ∞ and lim inf φ_{NR}(x^{k}, f (x^{k}))

1 > −∞.

Thus, we obtain that kφ_{p}(x^{k}, F (x^{k}))k → ∞ which contradicts kφ_{p}(x^{k}, F (x^{k}))k ≤ α.

Then, the proof is complete. 2

Note that, under the conditions of Lemma 4.1 or Theorem 4.1, we cannot guarantee
the boundedness of the level set on the NR function φ_{NR}. For example, let F (x) = 1 −1
and x > 0, it is easy to verify that the level set x

L_{NR}(2) = {x ∈ IR^{n}| kφ_{NR}(x, F (x))k ≤ 2}

is unbounded. In fact, In order to establish the boundedness of the level set on the
natural residual function φ_{NR}, we need the following concept.

Definition 4.1. A mapping F : IR^{n} → IR^{n} is said to be strongly coercive if

kxk→∞lim

hF (x), x − yi kx − yk = ∞.

holds for all y ∈ IR^{n}.

Theorem 4.2. Suppose that F is strongly coercive. Then, the level set
L_{NR}(α) = {x ∈ IR^{n}| kφ_{NR}(x, F (x))k ≤ α}

is bounded for all α ≥ 0.

Proof. Again, we prove this result by contradiction. Suppose there exists an unbounded
sequence {x^{k}} ⊂ L_{NR}(α) for some α ≥ 0, i.e.,

x^{k}

→ ∞. Note that the sequence

φ_{NR}(x^{k}, F (x^{k})) = x^{k}− (x^{k}− F (x^{k}))_{+} is bounded. It follows from the unboundedness
of the sequence {x^{k}} that the sequence {(x^{k}− F (x^{k}))+} is also unbounded. Then, for
any y ∈ L_{θ}, there exist N ∈ N and β > 0 such that

(x^{k}− F (x^{k}))_{+}− y

> β, ∀k > N.

From the property of projection mapping, we have

x^{k}− F (x^{k}) − (x^{k}− F (x^{k}))_{+}, y − (x^{k}− F (x^{k}))_{+} ≤ 0 (17)
for each k > N . On the other hand,

x^{k}− F (x^{k}) − (x^{k}− F (x^{k}))_{+}, y − (x^{k}− F (x^{k}))_{+}

= x^{k}− (x^{k}− F (x^{k}))_{+}, y − (x^{k}− F (x^{k}))_{+} + F (x^{k}), (x^{k}− F (x^{k}))_{+}− y

≥ −

x^{k}− (x^{k}− F (x^{k}))_{+}
·

y − (x^{k}− F (x^{k}))_{+}

+F (x^{k}), (x^{k}− F (x^{k}))_{+}− y

≥

y − (x^{k}− F (x^{k}))+

hF (x^{k}), (x^{k}− F (x^{k}))_{+}− yi
ky − (x^{k}− F (x^{k}))_{+}k − α

.
Plugging in y^{k} := x^{k}− (x^{k}− F (x^{k}))_{+}− y, we obtain

lim

k→∞

F (x^{k}), (x^{k}− F (x^{k}))_{+}− y

ky − (x^{k}− F (x^{k}))+k = lim

k→∞

F (x^{k}), x^{k}− y^{k}
kx^{k}− y^{k}k = ∞,

where the last equality holds due to the strong coercivity of F and [22, Theorem 2.1].

This implies that

k→∞lim x^{k}− F (x^{k}) − (x^{k}− F (x^{k}))+, y − (x^{k}− F (x^{k}))+ = ∞,
which contradicts (17). Therefore, the level set

L_{NR}(α) = {x ∈ IR^{n}| kφ_{NR}(x, F (x))k ≤ α}

is bounded for all α ≥ 0. 2

### 4.2 The second class of merit functions

For any x ∈ IR^{n}, LT (standing for Luo-Tseng) merit function for the circular cone
complementarity problem is given as follows:

f_{LT}(x) := ϕ(hx, F (x)i) + 1

2k(x)−k^{2}+ k(−F (x))_{+}k^{2} , (18)
where ϕ : IR → IR_{+} is an arbitrary smooth function satisfying

ϕ(t) = 0, ∀t ≤ 0 and ϕ^{0}(t) > 0, ∀t > 0.

Notice that we have ϕ(t) ≥ 0 for all t ∈ IR from the above condition. Indeed, this class of functions has been considered for the SDCP case (positive semidefinite complementar- ity problem) by Tseng in [21], for the SOCCP case (second-order cone complementarity problem) by Chen in [2] and for the general SCCP case by Pan and Chen in [18], respec- tively. For the case of generally closed convex cone complementarity problems, the LT merit function has been studied by Lu and Huang in [15]. In view of the results in [15], it is easy to obtain the following results directly for the circular cone complementarity problem.

Proposition 4.1. Let f_{LT} : IR^{n} → IR be given as in (18). Then, the following results
hold.

(a) For all x ∈ IR^{n}, we have f_{LT}(x) ≥ 0; and f_{LT}(x) = 0 if and only if x solves the
circular cone complementarity problem.

(b) If F (·) is differentiable, then so is f_{LT}(·). Moreover,

∇f_{LT}(x) = ∇ϕ(hx, F (x)i)[F (x) + x∇F (x)] − x−− ∇F (x)(−F (x))_{+}
for all x ∈ IR^{n}.

Proof. See Lemma 3.1 and Theorem 3.4 in [15]. 2

Proposition 4.2. Let f_{LT} be given as in (18). Suppose that F : IR^{n}→ IR^{n} is a strongly
monotone mapping and that the circular cone complementarity problem has a solution
x^{∗}. Then, there exists a constant τ > 0 such that

τ kx − x^{∗}k^{2} ≤ max{0, hx, F (x)i} + kx−k + k(−F (x))+k, ∀x ∈ IR^{n}.
Moreover,

τ kx − x^{∗}k^{2} ≤ ϕ^{−1}(f_{LT}(x)) + 2[f_{LT}(x)]^{1}^{2}, ∀x ∈ IR^{n}.
Proof. See Theorem 3.6 in [15]. 2

In the following theorem, we present the condition which ensures the boundedness
of the level sets for LT merit function f_{LT} to solve the circular cone complementarity
problem.

Theorem 4.3. Suppose that the circular cone complementarity problem has a strictly feasible point and that F is monotone. Then, the level set

L_{f}_{LT}(α) := {x ∈ IR^{n}| f_{LT}(x) ≤ α}

is bounded for all α ≥ 0.

Proof. We prove this result by contradiction. Suppose there exists an unbounded
sequence {x^{k}} ⊆ L_{f}_{LT}(α) for some α ≥ 0. We may assert that the sequences {x^{k}_{−}} and
{(−F (x^{k}))_{+}} are bounded. If not, from the expression (18) of LT merit function f_{LT} and
the property ϕ(t) ≥ 0 for all t ∈ IR, it follows that

fLT(x^{k}) ≥ 1

2[kx^{k}_{−}k^{2}+ k(−F (x^{k}))+k^{2}] → ∞,

which contradicts {x^{k}} ⊆ L_{f}_{LT}(α), i.e., f_{LT}(x^{k}) ≤ α. Therefore, we have
lim sup

k→∞

kx^{k}_{−}k < ∞ and lim sup

k→∞

k(−F (x^{k}))_{+}k < ∞.

Then, by Lemma 4.1, we get that

hx^{k}, F (x^{k})i → ∞.

By the properties of the function ϕ again, we obtain that ϕ(hx^{k}, F (x^{k})i) → ∞, which
implies f_{LT}(x^{k}) → ∞. This contradicts {x^{k}} ⊆ L_{f}_{LT}(α). Hence, the level set L_{f}_{LT}(α) is
bounded for all α ≥ 0. 2

### 4.3 The third class of merit functions

To achieve the third class of merit functions, we make a slight modification of LT merit
function f_{LT} for the circular cone complementarity problem. More specifically, we con-
sider the set Ω as follows:

Ω := L_{θ}∩ L^{∗}_{θ} = L_{θ} for 0 < θ ≤ ^{π}_{4},
L^{∗}_{θ} for ^{π}_{4} < θ < ^{π}_{2}.

Indeed, Ω is also a closed convex cone. In light of this Ω, another function is considered:

fd_{LT}(x) := 1

2k(x • F (x))^{Ω}_{+}k^{2}+ 1

2kx−k^{2}+ k(−F (x))_{+}k^{2} , (19)
where (x • y)^{Ω}_{+}denotes the projection of x • y onto Ω. Then, together with the expressions
(7) of x • y, we can verify that the function df_{LT} is also a type of merit function for the
circular cone complementarity problem, which will be shown in following theorem.

Theorem 4.4. Let the function dfLT be given by (19). Then, for all x ∈ IR^{n}, we have
fd_{LT}(x) = 0 ⇐⇒ x ∈ L_{θ}, F (x) ∈ L^{∗}_{θ} and hx, F (x)i = 0,

where L^{∗}_{θ} denotes the dual cone of L_{θ}, i.e., L^{∗}_{θ} = L^{π}

2−θ.

Proof. By the definition of the function dfLT given by (19), we have
fd_{LT}(x) = 0 ⇔ k(x • F (x))^{Ω}_{+}k = 0, kx−k = 0 and k(−F (x))_{+}k = 0,

⇔ (x • F (x))^{Ω}_{+} = 0, x−= 0 and (−F (x))_{+} = 0,

⇔ x • F (x) ∈ −L_{θ} or x • F (x) ∈ −L^{∗}_{θ}, x ∈ L_{θ}, and F (x) ∈ L^{∗}_{θ},

⇔ x ∈ Lθ, F (x) ∈ L^{∗}_{θ} and hx, F (x)i = 0,

where the last equivalence holds due to the properties of the cone −L_{θ} or −L^{∗}_{θ}. Thus,
the proof is complete. 2

From Theorem 4.4, we know that the function df_{LT} is a merit function for the circular
cone complementarity problem. As below, according to the type of dot product (7), we
establish the differentiability of df_{LT}.

Theorem 4.5. Let df_{LT} : IR^{n} → IR be given by (19). Suppose that the type of dot product
(7) is employed. If F (·) is differentiable, then so is df_{LT}(·). Moreover, for all x ∈ IR^{n}, we
have

∇df_{LT}(x) = (L_{y}+ ∇F (x)L_{x}) · (x • F (x))^{Ω}_{+}− x−− ∇F (x)(−F (x))_{+},
where

L_{x} =

y1 y_{2}^{T}

max{tan^{2}θ, 1}y_{2} max{cot^{2}θ, 1}y_{1}I

and

L_{y} =

x_{1} x^{T}_{2}

max{tan^{2}θ, 1}x_{2} max{cot^{2}θ, 1}x_{1}I

with I being the identity matrix.

Proof. From the proof of Lemma 3.1(b) in [15], we have

∇ 1

2k(z)^{Ω}_{+}k^{2}

= (z)^{Ω}_{+}, ∀z ∈ IR^{n}.
Then, by the chain rule again, it follows that

∇ 1

2k(x • F (x))^{Ω}_{+}k^{2}

= ∇x(x • F (x)) · (x • F (x))^{Ω}_{+}

= [L_{y}+ ∇F (x)L_{x}] · (x • F (x))^{Ω}_{+},
where

L_{x} =

y_{1} y_{2}^{T}

max{tan^{2}θ, 1}y_{2} max{cot^{2}θ, 1}y_{1}I

and

L_{y} =

x_{1} x^{T}_{2}

max{tan^{2}θ, 1}x_{2} max{cot^{2}θ, 1}x_{1}I

with I being the identity matrix. Thus, we obtain that

∇df_{LT}(x) = (L_{y}+ ∇F (x)L_{x}) · (x • F (x))^{Ω}_{+}− (x)−− ∇F (x)(−F (x))_{+}
for all x ∈ IR^{n}. 2

In order to establish error bound property of the merit function df_{LT} for the circular
cone complementarity problem, we need a technical lemma as below.

Lemma 4.2. Let x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}. Then, we have
hx, yi ≤ max 1 + tan^{2}θ

√2 ,1 + cot^{2}θ

√2

(x • y)^{Ω}_{+}
where • is defined as in (7).