DOI 10.1007/s10589-015-9781-1

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

**Xin-He Miao**^{1}**· Shengjuan Guo**^{1}**· Nuo Qi**^{1}**·**
**Jein-Shan Chen**^{2}

Received: 29 March 2015 / Published online: 20 August 2015

© Springer Science+Business Media New York 2015

**Abstract In this paper, we consider complementarity problem associated with circu-**
lar 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 func-
tions for such nonsymmetric cone complementarity problem, and propose a few merit
functions for solving such a complementarity problem. In addition, we study the con-
ditions 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

### B

Jein-Shan Chen jschen@math.ntnu.edu.tw Xin-He Miaoxinhemiao@tju.edu.cn Shengjuan Guo gshengjuan@163.com Nuo Qi

qinuo@163.com

1 Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, People’s Republic of China

2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

**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

*x, F(x) = 0,*(1) where

*·, · denotes the Euclidean inner product, F : IR*

*→ IR*

^{n}*is a continuously differentiable mapping,*

^{n}*K represents a closed convex cone, and K*

^{∗}is the dual cone of

*K given by*

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

When*K is a symmetric cone, the problem (1) is called the symmetric cone comple-*
mentarity 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}*| x*2* ≤ x*1*},*

the problem (1) reduces to the second-order cone complementarity problem [1,3–5,10, 11]. In contrast to symmetric cone programming and symmetric cone complementarity 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 problems 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*

*can be expressed as*

_{θ}*L**θ* :=

*x= (x*1*, x*2*) ∈ IR × IR*^{n}^{−1}*| x cos θ ≤ x*1

=

*x= (x*1*, x*2*) ∈ IR × IR*^{n}^{−1}*| x*2* ≤ x*1tan*θ*

*.* (2)

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 comple-
mentarity problem. 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 numeri-
cal 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 complementar-
ity 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 epi-

*graph 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 problem and the second-order cone complementarity problem, the study about non- symmetric 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 believe 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 minimiza- tion problem

*x*min∈IR^{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 comple- mentarity 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

*N R* : IR* ^{n}*→ IR, which is defined as

*N R**(x) :=* 1

2* φ*NR*(x, F(x)) *^{2}= 1

2* x − (x − F(x))*+ ^{2}*,*

where*(·)*+denotes the projection onto the symmetric cone*K. It is well known that*

*N R**(x) = 0 if and only if x is a solution to the symmetric cone complementarity*
problem. 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 prop-
erties 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 circular 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 rev-

*olution axis which is the ray generated by the canonical vector e*1

*:= (1, 0, . . . , 0)*

*∈ IR*

^{T}*. Its dual cone denoted by*

^{n}*L*

^{∗}

*is given as*

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

Note that the circular cone*L**θ* is not a self-dual cone when*θ =* ^{π}_{4}, that is,*L*^{∗}_{θ}*= L**θ*,
whenever*θ = 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}*| y*2* ≤ y*1cot*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 coneL*

_{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*

*, the explicit formula of*

_{θ}*projection of x*∈ IR

*onto*

^{n}*L*

*is obtained in [23] as below:*

_{θ}*x*_{+}=

⎧⎨

⎩

*x if x∈ L*_{θ}*,*
*0 if x* *∈ −L*^{∗}_{θ}*,*

*u otherwise,* (4)

where

*u*=

⎡

⎢⎣

*x*1*+ x*2* tan θ*
1+ tan^{2}*θ*

*x*1*+ x*2* tan θ*
1+ tan^{2}*θ* tan*θ*

*x*2

* x*2

⎤

⎥*⎦ .*

*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*− x*2* cot θ*
1+ cot^{2}*θ*

*x*1*− x*2* cot θ*
1+ cot^{2}*θ* cot*θ*

*x*2

* x*2

⎤

⎥*⎦ .*

From the expressions (4)–(5) for x_{+}*and x*_{−}, it is easy to verity that*x*_{+}*, x*_{−} = 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 = (x*1*, x*2*) ∈ IR × IR*^{n}^{−1}*| x*2* ≤ x*1*}.*

It is well known that the dual cone of*K** ^{n}*is itself, and the second-order cone

*K*

*belongs to a class of symmetric cones. In addition,*

^{n}*K*

*is a special case of*

^{n}*L*

*θ*corresponding to

*θ =*

^{π}_{4}. In fact, there is a relationship between

*L*

*θ*and

*K*

*, which is described in (3). In the SOC setting, there is so-called Jordan algebra associated with SOC. More*

^{n}*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 :=*

*x, y*

*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

*. In this paper, we*

^{n}*write x*

^{2}

*to mean x◦ x. It is known that x*

^{2}

*∈ K*

^{n}*for any x*∈ IR

^{n}*, and if x*

*∈ K*

*, there*

^{n}*exists a unique vector denoted by x*

^{1}

^{2}in

*K*

*such that*

^{n}*(x*

^{1}

^{2}

*)*

^{2}

*= x*

^{2}

^{1}

*◦ x*

^{1}

^{2}

*= x. For any*

*x*∈ IR

*, we denote*

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

With the help of (3) and Lemma2.1, 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* *∈ L*_{θ}*, y ∈ L*^{∗}_{θ}*andx, y = 0.*

*(b) Ax∈ K*^{n}*, A*^{−1}*y∈ K*^{n}*andAx, A*^{−1}*y = 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 x*2*is a multiple of y*2*.*

*Proof From the relation betweenK** ^{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

*Ax, A*^{−1}*y = A*^{−1}*Ax, y = x, y = 0.*

Hence, it follows that (a) and (b) are equivalent. The equivalence of (b) and (c) has
been shown in Lemma2.1. In addition, based on the relation between*K** ^{n}* and

*L*

*again, the equivalence of (c) and (d) is obvious.*

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

* x*2* ≤ x*1tan*θ and y*2* ≤ y*1cot*θ.*

This together with*x, y = 0 yields*
0*= x, y*

*= x*1*y*1*+ x*2*, y*2

*≥ x*1*y*1*− x*2* y*2

*≥ x*1*y*1*− x*1*y*1

= 0

which implies*x*2*, y*2* = x*2* y*2* . 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.

**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 con- struct 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

*→ IR*

^{n}*, we call*

^{n}*φ an C-function for*CCCP if, for any

*(x, y) ∈ IR*

*× IR*

^{n}*, it satisfies*

^{n}*φ(x, y) = 0 ⇐⇒ x ∈ L**θ**, y ∈ L*^{∗}_{θ}*, x, y = 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}*, x, y = 0.*

Two popular and well-known C-functions for SOCCP are Fischer-Burmeister (FB) function 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

*φ*FB*(x, y) :=*

*(Ax)*^{2}*+ (A*^{−1}*y)*^{2}^{1}

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

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

* Proposition 3.1 Let φ*FB

*andφ*NR

*be defined as above where(Ax)*

^{2}

*equals(Ax)◦(Ax)*

*under Jordan product. Then, φ*FB

*andφ*NR

*are both C-functions for CCCP.*

*Proof In view of Theorem*2.1and Definition3.1, it is not hard to verify that

*φ*FB*(x, y) = 0 ⇐⇒ x ∈ L*_{θ}*, y ∈ L*^{∗}_{θ}*, x, y = 0,*
*φ*NR*(x, y) = 0 ⇐⇒ x ∈ L*_{θ}*, y ∈ L*^{∗}_{θ}*, x, y = 0,*

which says that these two functions are C-functions for CCCP.

We point out that if we consider directly the FB function*φ*FB*(x, y) for CCCP, unfor-*
*tunately, 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 =*

*x, y*

*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*

^{∗}

*. Furthermore, when*

_{θ}*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*

^{∗}

*. All the above explains that the FB function*

_{θ}*φ*FB

cannot be an C-function for CCCP. Nonetheless, the NR function*φ*NR : IR* ^{n}*× IR

*→ IR*

^{n}*given by*

^{n}*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* *= (x*1*, x*2*) ∈ IR × IR*^{n}^{−1}*and y= (y*1*, y*2*) ∈ IR × IR*^{n}^{−1}, we define one
*type of product of x and y as follows:*

*x• y =*

*x*1

*x*2

•

*y*1

*y*2

=

*x, y*

max{tan^{2}*θ, 1} x*1*y*2+ max{cot^{2}*θ, 1} y*1*x*2

*. (7)*

From the above product and direct calculation, it is easy to verify that

*x • y, z = x, z • y, ∀z ∈ IR** ^{n}*with

*θ ∈*0,

*π*

4

(8) and

*x • y, z = y, x • z, ∀z ∈ IR** ^{n}*with

*θ ∈π*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) ifθ ∈ (0,*^{π}_{4}*], we have x*_{−}*, x*_{+}*• (−y)*_{−} ≤ 0;

*(b) ifθ ∈ [*^{π}_{4}*,*^{π}_{2}*), we have (−y)*_{+}*, x*_{+}*• (−y)*_{−}* ≤ 0.*

*Proof (a) Whenθ ∈ (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

*x*_{−}*, x*_{+}*• (−y)*_{−}* = 0.*

*If all the three elements are not equal to zero, from the definition of x*_{+}*, x*_{−}, and*(−y)*_{−},
*we have k cotθ ≥ w , s tan θ = u , t cot θ = v 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

*x*−*, x*+*• (−y)*−

*= stk + tu, w + sv, w + ku, v cot*^{2}*θ*

*= u v k − k u v cot*^{2}*θ − u v, w tan θ + u v, w cot θ*

*= (1 − cot*^{2}*θ)k u v − (1 − cot*^{2}*θ)( u v, w tan θ)*

*= (1 − cot*^{2}*θ)[k u v − u v, w tan θ]*

*≤ (1 − cot*^{2}*θ)[k u v − u v w tan θ]*

*= (1 − cot*^{2}*θ) u v [k − w tan θ]*

*≤ 0.*

Here the second equality is true due to *αt = α v tan θ = − u tan θ. The last*
*inequality holds due to k cotθ ≥ w 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*

*(−y)*_{+}*, x*_{+}*• (−y)*_{−}* ≤ 0.*

Then, the desired result follows.

Besides the inequalities in Lemma3.1, “•” 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* *∈ L*_{θ}*, y ∈ L*^{∗}_{θ}*andx, y = 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 =*

*x, y*

*x*1*y*2+ cot^{2}*θ y*1*x*2

which implies

*Ax* *◦ A*^{−1}*y*=

*x, y*

*x*1tan*θ y*2*+ cot θ y*1*x*2

=

1 0
0*(tan θ)I*

*(x • y).*

This together with Theorem2.1yields 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 =*

*x, y*

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 Theorem2.1again, the desired result follows.

*Based on the product x* *• y of x and y. we now introduce a class of functions*
*φ**p* : IR* ^{n}*× IR

*→ IR*

^{n}*, which is called the penalized natural residual function and defined as*

^{n}*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*φ**p* is an C-function for CCCP. To achieve the conclusion, a technical
lemma is needed.

**Lemma 3.2 Let**φ*p*: IR* ^{n}*×IR

*→ IR*

^{n}

^{n}*be defined as in (10). Then, for any x, y ∈ IR*

^{n}*,*

*we have*

* φ**p**(x, y) ≥ max { x*_{−}* , (−y)*_{+}* } .*

*Proof First, we prove that φ**p**(x, y) ≥ x*_{−} . To see this, we observe that

* φ**p**(x, y) *^{2}

*= x − (x − y)*+*+ p x*+*• (−y)*−*, x − (x − y)*+*+ p x*+*• (−y)*−

*= x*+*− x*−*− (x − y)*+*+ p x*+*• (−y)*−*, x*+*− x*−*− (x − y)*+*+ p x*+*• (−y)*−

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

*≥ x*− ^{2}*− 2x*−*, x*+* + 2 x*−*, (x − y)*+* − 2 x*−*, p x*+*• (−y)*−

*≥ x*− ^{2}*− 2p x*−*, x*+*• (−y)*−* .*

*Here, the last inequality is true due to x*_{+}*, (x − y)*_{+}*∈ L*_{θ}*, x*_{−} *∈ L*^{∗}_{θ}*, x*_{+}*, x*_{−} = 0
and the relation between*L** _{θ}*and

*L*

^{∗}

*. When*

_{θ}*θ ∈ (0,*

^{π}_{4}], by Lemma3.1(a), we have

*x*_{−}*, x*_{+}*• (−y)*_{−}* ≤ 0.*

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

*x*_{−}*, x*_{+}*• (−y)*_{−}* = (−y)*−*, x*_{+}*• x*_{−} = 0

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

* φ**p**(x, y) *^{2}*≥ x*− ^{2}*.*

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

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

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

*= (−y)*_{−}*− (−y)*_{+}*− (x − y)*_{−}*+ p x*_{+}*• (−y)*_{−}*, (−y)*_{−}*− (−y)*_{+}*− (x − y)*_{−}
*+px*+ *• (−y)*−

*= (−y)*_{+} ^{2}*+ (−y)*_{−}*− (x − y)*_{−}*+ p x*_{+}*• (−y)*_{−} ^{2}*− 2(−y)*_{+}*, (−y)*_{−}

*−(x − y)*−*+ px*+ *• (−y)*−

*≥ (−y)*_{+} ^{2}*−2(−y)*_{+}*, (−y)*_{−}*+2(−y)*_{+}*, (x −y)*_{−}*−2(−y)*_{+}*, p x*_{+}*• (−y)*_{−}

*≥ (−y)*_{+} ^{2}*− 2p (−y)*_{+}*, x*_{+}*• (−y)*_{−}

*≥ (−y)*+ ^{2}*,*

where the second inequality holds due to due to*(−y)*_{+} *∈ L*_{θ}*, (−y)*_{−}*, (x − y)*_{−} ∈
*L*^{∗}_{θ}*, (−y)*_{+}*, (−y)*_{−}* = 0 and the relation between L** _{θ}* and

*L*

^{∗}

*. The last inequality holds due to equation (8) and Lemma3.1(b). Therefore, we prove that*

_{θ}*φ*

*p*

*(x, y) ≥*

* (−y)*+ . Then, the proof is complete.

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

**Theorem 3.2 Let**φ*p* : IR* ^{n}*× IR

*→ IR*

^{n}

^{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}*andx, y = 0.*

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

* φ**p**(x, y) ≥ max{ x*−* , (−y)*+* } > 0.*

This contradicts with*φ**p**(x, y) = 0. Hence, there must have x ∈ L*_{θ}*and y∈ L*^{∗}* _{θ}*. Next,
we argue that

*x, y = 0. To see this, we consider the first component of φ*

*p*

*(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 x, y if x − y ∈ L**θ**,*
*x*1*+ p x, y if x − y ∈ −L*^{∗}_{θ}*,*
*w + p x, y otherwise,*
where

*w = x*1−*x*1*− y*1*+ x*2*− y*2* tan θ*

1+ tan^{2}*θ* = *x*1tan^{2}*θ + y*1*− x*2*− y*2* tan θ*

1+ tan^{2}*θ* *.*

*Since x* *∈ L*_{θ}*and y∈ L*^{∗}_{θ}*, it follows that x*1*, y*1*≥ 0, x, y ≥ 0 and*
*x*1tan^{2}*θ + y*1*− x*2*− y*2* tan θ*

1+ tan^{2}*θ* ≥ tan*θ(x*1tan*θ − x*2* + y*1cot*θ − y*2* )*

1+ tan^{2}*θ* *≥ 0.*

This together with*φ**p**(x, y) = 0 gives px, y = 0. Thus, we conclude that x, y = 0*
*because p> 0.*

“⇐” Suppose that x ∈ L_{θ}*, y ∈ L*^{∗}* _{θ}* and

*x, y = 0. Since φ*NR is always an C-

*function for CCCP, we have x*

*− (x − y)*

_{+}= 0. Using Theorem 3.1again yields

*x*

_{+}

*• (−y)*

_{−}

*= x • y = 0, which says φ*

*p*

*(x, y) = 0.*

*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

=

*x, y*

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

*→ IR*

^{n}*defined by*

^{n}*φ**r**(x, y) = x − (x − y)*+*+ r (x • y)*^{}_{+} *r* *> 0,* (11)

where* := 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)*

^{}_{+}. Combining with the expression of

*φ*

*p*, this implies that

*−r (x • y)*^{}_{+}*+ p (x*+*• (−y)*_{−}*) = 0*

due to*(x • y)*^{}_{+}*∈ = L*_{θ}*∩ L*^{∗}_{θ}*and x*_{+}*• (−y)*_{−} */∈ K*^{n}*⊇ L** _{θ}*(or

*L*

^{∗}

*) when*

_{θ}*θ ∈ (0,*

^{π}_{4}] (or

*θ ∈ [*

^{π}_{4}

*,*

^{π}_{2}

*)). This explains that φ*

*p*

*(x, y) = 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
important properties, respectively.

*First, we recall that a function F* : IR* ^{n}* → IR

^{n}*is said to be monotone if, for any*

*x, y ∈ IR*

*, there holds*

^{n}*x − y, F(x) − F(y) ≥ 0;*

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

*x − y, F(x) − F(y) ≥ ρ x − y *^{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}*}

*satisfying*

*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

*, we know*

^{n}

*x*^{k}*− ¯x, F*
*x*^{k}

*− F( ¯x)*

*≥ 0,*

which says

*x*^{k}*, F*

*x*^{k}

*+ ¯x, F( ¯x) ≥*

*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}

*+ ¯x, F( ¯x)*

≥

*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

·*[ f( ¯x)]*2

≥
*x*_{+}^{k}

1*[ f( ¯x)]*1−
*x*_{+}^{k}

1tan*θ**[ f( ¯x)]*2

=
*x*_{+}^{k}

1 *[ f( ¯x)]*1*− tan θ**[ f( ¯x)]*2*!.* (14)
*Note that x*^{k}*= x*_{+}^{k}*− x*_{−}* ^{k}*, it gives

*x*

_{+}

^{k}*≥ x*

^{k}*− x*

_{−}

*. From the assumptions on*

^{k}*{x*

^{k}*}, i.e., x*

*→ ∞, and lim sup*

^{k}*k*→∞

*x*

_{−}

^{k}*< ∞, we see that x*

_{+}

*→ ∞, and hence*

^{k}*[x*

_{+}

*]1*

^{k}*→ ∞. Because CCCP has a strictly feasible point ¯x, we have*

*[ f( ¯x)]*1

*− tan θ [ f ( ¯x)]*2

*> 0, which together with (*14) implies that

*x*_{+}^{k}*, F( ¯x)*

→ ∞ *(k → ∞).* (15)

On the other hand, we observe that lim sup

*k*→∞ *x*_{−}^{k}*, F( ¯x) ≤ lim sup*

*k*→∞ * x*_{−}^{k}* F( ¯x) < ∞*
lim sup

*k*→∞ * ¯x,*

*−F*
*x*^{k}

+ ≤ lim sup

*k*→∞ * ¯x *

*−F*
*x*^{k}

+* < ∞*
and* ¯x,*

*−F*
*x*^{k}

− ≥ 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 that

*x*_{+}^{k}*,*

*−F*
*x*^{k}

−

→ ∞. Suppose not, that is, lim*k*→∞"

*x*_{+}^{k}*,*

*−F*
*x*^{k}

−

*< ∞. Then, we obtain*

*x*_{+}^{k}*,*

*−F*
*x*^{k}

−

* x*_{+}* ^{k}* =

# *x*^{k}_{+}
* x*^{k}_{+} *,*

*−F*
*x*^{k}

−

$

*→ 0.*

This means that there exists *¯x ∈ IR** ^{n}*such that

*x*

_{+}

^{k}* x*_{+}* ^{k}* →

*¯x*+

* ¯x*+ and

*¯x*_{+}

* ¯x*_{+} *, (−F( ¯x))*_{−}

*= 0.* (16)

*Denote z*:= *¯x*_{+}

* ¯x*+ and apply Theorem3.1, 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, Eq. (16) implies that

*α < 0,*

*which says that z*2and

*(−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))*_{+}

2and

*(−F( ¯x))*_{−}

2*are in the opposite direction, to each other. These conclude that z*2and

*(−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

* ¯x*+*, (−F( ¯x))*+* ≥ 0.*

Similarly, by the the relation between *¯x*_{+}and *¯x*_{−}, we know*[ ¯x*_{−}]2and*[(−F( ¯x))*_{−}]2

are in the same direction. Then, combining with *¯x*−*, (−F( ¯x))*−*∈ L*^{∗}* _{θ}*, it leads to

* ¯x*−*, (−F( ¯x))*_{−}* ≥ 0.*

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

* ¯x, F( ¯x)= ¯x*+*, (−F( ¯x))*−*− ¯x*+*, (−F( ¯x))*+*− ¯x*−*, (−F( ¯x))*−*+ ¯x*−*, (−F( ¯x))*+*.*

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 ¯x, F( ¯x) < ∞, which contradict*

* ¯x, F( ¯x) = lim*

*k*→∞

*x*^{k}*, F*

*x*^{k}

*= ∞.*

Thus, we prove that

*x*_{+}^{k}*,*

*−F*
*x*^{k}

−

→ ∞.

**4.1 The first class of merit functions**

*For any x* ∈ IR* ^{n}*, from the analysis of the Sect.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}*|

*φ*

*p*

*(x, F(x)) ≤ α}*

*is bounded for allα ≥ 0.*

*Proof We prove this result by contradiction. Suppose there exists an unbounded*
sequence*{x*^{k}*} ⊂ L**p**(α) for some α ≥ 0. If x*_{−}* ^{k}* → ∞ or

*−F*
*x*^{k}

+ → ∞, by
Lemma3.2, we have* φ**p**(x*^{k}*, F*

*x*^{k}

*) → ∞, which contradicts φ**p**(x*^{k}*, F*
*x*^{k}*α. On the other hand, if* *) ≤*

lim sup

*k*→∞ * x*_{−}^{k}* < ∞ and lim sup*

*k*→∞

*−F*
*x*^{k}

+

* < ∞,*
it follows from Lemma4.1that

*x*_{+}^{k}*,*

*−F*
*x*^{k}

−

→ ∞. From the proof of Lemma
4.1, there exists a constant*κ*0such 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

*− [x*_{+}* ^{k}*]2

*tan θ− [*

*−F*
*x*^{k}

−]2* tan θ*
1+tan^{2}*θ*

−^{2}^{κ}_{1+tan}^{0}* ^{(1+tan θ)}*2

*θ*

*,*

*if x*

^{k}*− F*

*x*

^{k}*/∈ L**θ**∪ −L*^{∗}_{θ}*,*

which means lim inf

*φ*NR*(x*^{k}*, f*
*x*^{k}

*)*

1*> −∞. Hence, it follows that*

*φ**p*

*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* φ**p**(x*^{k}*, F*
*x*^{k}

*) → ∞ which contradicts φ**p**(x*^{k}*, F*
*x*^{k}

*α. Then, the proof is complete.* *) ≤*