to appear in Computational Optimization and Applications, 2017

### On merit functions for p-order cone complementarity problem

Xin-He Miao^{1}

Department of Mathematics Tianjin University, China

Tianjin 300072, China

Yu-Lin Chang^{2}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

Jein-Shan Chen ^{3}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

August 17, 2016

(revised on December 12, 2016)

Abstract Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained mini- mization via merit function or complementarity function. In this paper, for the comple- mentarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solv- ing p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.

1E-mail: xinhemiao@tju.edu.cn. The author’s work is supported by National Natural Science Foun- dation of China (No. 11471241).

2E-mail: ylchang@math.ntnu.edu.tw

3Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan. E-mail: jschen@math.ntnu.edu.tw

Keywords. p-order cone complementarity problem, merit function, error bound.

### 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 complementarity problem [9, 10, 11, 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, 2, 3, 4, 5, 7, 8]. In contrast to symmetric cone programming and symmetric cone complementarity problem, we are not familiar with their nonsymmetric counterparts. Referring the reader to [16, 22, 14, 19] and the bibliographies therein, we observe that there is no 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 focus on a special nonsymmetric cone K for problem (1), i.e., p-
order cone. Then, the problem (1) reduces to the p-order cone complementarity problem
(POCCP for short). Indeed , the p-order cone [17, 22] is a generalization of the second-
order cone in IR^{n}, denoted by Kp, and can be expressed as

Kp :=

x ∈ IR^{n}

x1 ≥

n

X

i=2

|xi|^{p}

!^{1}_{p}

(p > 1).

If we write x := (x_{1}, ¯x) ∈ IR × IR^{n−1}, the p-order cone K_{p} can be equivalently expressed
as

K_{p} =x = (x_{1}, ¯x) ∈ IR × IR^{n−1}| x_{1} ≥ k¯xk_{p} , (p > 1).

When p = 2, it is obvious that the p-order cone is exactly the second-order cone, which
means the p-order cone complementarity problem is actually the second-order cone com-
plementarity problem. Thus, the p-order cone complementarity problem (POCCP) can
be viewed as the generalization of the second-order cone complementarity problem. As
shown in [15, 17], K_{p} is a convex cone and its dual cone is given by

K_{p}^{∗} =

y ∈ IR^{n}

y_{1} ≥

n

X

i=2

|y_{i}|^{q}

!^{1}_{q}

or equivalently

K_{p}^{∗} =y = (y_{1}, ¯y) ∈ IR × IR^{n−1}| y_{1} ≥ k¯yk_{q} = K_{q},

where q > 1 and satisfies ^{1}_{p} +^{1}_{q} = 1. In addition, the dual cone K^{∗}_{p} is also a convex cone.

For more details regarding p-order cone and its involved optimization problems, please refer to [15, 17, 22].

During the past decade, there had active research and various methods for com- plementarity problems, which include the interior-point methods, the smoothing New- ton methods, the semismooth Newton methods, and the merit function methods, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 21] and references therein. As seen in the literature, almost all the attention was paid to symmetric cone complementarity problems, that is, nonlinear complementarity problem (NCP), positive semi-definite complementarity problem (SDCP), second-order cone complementarity problem (SOCCP). As mentioned earlier, there is no unified framework to deal with general nonsymmetric cone comple- mentarity problems. Consequently, the study about nonsymmetric cone complementarity problem is very limited. Nonetheless, we believe that that merit function approach, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function, may be appropriately viewed as a unified way to deal with nonsymmetric cone complementarity problem. Indeed, the main difficulty lies on how to construct complementarity functions or merit functions for nonsymmetric cone complementarity problem. For circular cone setting, several successful ways were shown in [16]. Inspired by the work [16], we employ the similar ways to construct merit func- tions for solving p-order cone complementarity problem. For completeness, the idea is roughly described again as below.

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. A merit function
for the p-order cone complementarity problem is a function h : IR^{n}→ [0, +∞), provided
that

h(x) = 0 ⇐⇒ x solves the POCCP (1).

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

x∈IRmin^{n}h(x)

with the optimal value zero. Until now, for solving symmetric cone complementarity
problem, a large number of merit functions have been proposed. Among them, one of the
most popular merit functions is the natural residual (NR) merit function Ψ_{NR} : IR^{n}→ IR,
which is defined as

Ψ_{NR}(x) := 1

2kφ_{NR}(x, F (x))k^{2} = 1
2

x − (x − F (x))^{K}_{+}

2,

where (·)^{K}_{+} denotes the projection onto the symmetric cone K. Then, we know that
Ψ_{NR}(x) = 0 if and only if x is a solution to the symmetric cone complementarity prob-
lem. As remarked in [16], this function Ψ_{NR} (or φ_{NR}) can also serve as merit function
(or complementarity function) for general conic complementarity problem. Hence, it is
also applicable to p-order cone complementarity problem. Under this setting, for any
x ∈ IR^{n}, we denote x+ be the projection of x onto the p-order cone Kp, and x− be the
projection of −x onto the dual cone K^{∗}_{p} of K_{p}. By properties of projection onto the closed
convex cone, it can be verified that x = x_{+}− x−. Moreover, the formula of projection of
x ∈ IR^{n} onto K_{p} is obtained in [17]. Besides the NR merit function Ψ_{NR}, are there any
other types of merit functions for POCCP? In this paper, we answer this question by
presenting other types of merit functions for the p-order 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 p-order cone complementarity problem.

### 2 Preliminaries

In this section, we briefly review some basic concepts and background materials about the p-order cone, and define one type of product associated with p-order cone, which will be extensively used in subsequent analysis.

As mentioned, the p-order cone K_{p} is a pointed closed convex cone, and its dual cone
denoted by K_{p}^{∗} is given as

K_{p}^{∗} =

y ∈ IR^{n}

y_{1} ≥

n

X

i=2

|y_{i}|^{q}

!^{1}_{q}

or equivalently

K_{p}^{∗} =y = (y1, ¯y) ∈ IR × IR^{n−1}| y_{1} ≥ k¯yk_{q} = Kq,

where q > 1 and satisfies ^{1}_{p} +^{1}_{q} = 1. From the expression of the dual cone K^{∗}_{p}, it is easy
to know that the dual cone K^{∗}_{p} is also a closed convex cone. In addition, when p 6= q, we
have K_{p} 6= K_{q} = K_{p}^{∗}, i.e., the p-order cone K_{p} is not a self-dual cone. That is to say, the
p-order cone K_{p} is not a symmetric cone for p 6= 2.

It is well known that Jordan product plays a critical role in the study of symmetric cone programming or symmetric cone complementarity problems. However, there is no

Jordan product for the setting of the p-order cone so far. Hence, we need to find one
type of special product for the setting of the p-order cone, which is similar to the one
for the setting of symmetric cone. To this end, for any x = (x_{1}, · · · , x_{n})^{T} ∈ IR^{n} and
y = (y_{1}, · · · , y_{n})^{T} ∈ IR^{n}, we define one type of product of x and y associated with p-order
cone K_{p} as follows:

x • y = hx, yi w

where w := (w_{2}, · · · , w_{n})^{T} with w_{i} = |x_{1}|^{p}^{q}|y_{i}| − |y_{1}||x_{i}|^{p}^{q}. (2)
From the above definition (2) of product, when p = q = 2, it is not hard to see that the
product x•y is exactly the Jordan product in the setting of second-order cone. According
to the product “•” defined as in (2), we have the following equivalence.

Proposition 2.1. For any x = (x_{1}, ¯x) ∈ IR × IR^{n−1} with ¯x = (x_{2}, · · · , x_{n})^{T} ∈ IR^{n−1} and
y = (y1, ¯y) ∈ IR × IR^{n−1} with ¯y = (y2, · · · , yn)^{T} ∈ IR^{n−1}, the following statements are
equivalent:

(a) x ∈ K_{p}, y ∈ K_{p}^{∗} = K_{q} and hx, yi = 0.

(b) x ∈ K_{p}, y ∈ K^{∗}_{p} = K_{q} and x • y = 0.

In each case, x and y satisfy the condition that there is c ≥ 0 such that |x_{i}|^{p} = c|y_{i}|^{q} or

|y_{i}|^{q}= c|x_{i}|^{p} for any i = 2, · · · , n.

Proof. (b) ⇒ (a) From the definition of product x • y of x and y associated with K_{p},
the implication is obvious.

(a) ⇒ (b) When ¯x = 0 or ¯y = 0, from (a), we know x ∈ K_{p}, y ∈ K_{q} and hx, yi = 0.

Then, it is easy to see that x • y = 0. When ¯x 6= 0 and ¯y 6= 0, by x ∈ K_{p} and y ∈ K_{q}, we
have x_{1} ≥ k¯xk_{p} and y_{1} ≥ k¯yk_{q}. Hence, it follows from hx, yi = 0 that

0 = hx, yi

= x_{1}y_{1}+ h¯x, ¯yi

≥ k¯xk_{p}k¯yk_{q}− k¯xk_{p}kk¯yk_{q}

= 0.

This implies that x_{1} = k¯xk_{p}, y_{1} = k¯yk_{q} and |x_{i}|^{p} = c|y_{i}|^{q} or |y_{i}|^{q} = c|x_{i}|^{p} with some
c ≥ 0 for any i = 2, · · · , n. Next, we only consider the case |xi|^{p} = c|yi|^{q}, and the same
arguments apply for the case |yi|^{q} = c|xi|^{p}. Because |xi|^{p} = c|yi|^{q}, we have |xi|^{p}^{q} = c^{1}^{q}|yi|.

This yields that, for any i = 2, · · · , n,

|x_{1}|^{p}^{q}|y_{i}| − |y_{1}||x_{i}|^{p}^{q} = |x_{1}|^{p}^{q}|y_{i}| − |y_{1}|c^{1}^{q}|y_{i}|

=

n

X

k=2

|x_{k}|^{p}

!^{1}_{q}

|y_{i}| −

n

X

k=2

|y_{k}|^{q}

!^{1}_{q}
c^{1}^{q}|y_{i}|

= c^{1}^{q}

n

X

k=2

|y_{k}|^{q}

!^{1}_{q}

|y_{i}| − c^{1}^{q}

n

X

k=2

|y_{k}|^{q}

!^{1}_{q}

|y_{i}|

= 0,

where the second equality holds due to x1 = k¯xkp, y1 = k¯ykq. Then, it follows that x • y = 0, and the proof is complete. 2

To close this section, we introduce some other concepts that will be needed in subse-
quent analysis. 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 result is crucial for achieving the property of bounded level sets. Although the analysis technique is similar to [16, Lemma 4.1], we present the details for completeness.

Proposition 2.2. Suppose that the POCCP has a strictly feasible point z, i.e., z ∈
int(K_{p}) and F (z) ∈ int(K^{∗}_{p}) 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}) → ∞.

Proof. Since F is monotone, for any x^{k} ∈ IR^{n}, we have
x^{k}− z, F (x^{k}) − F (z) ≥ 0,
which leads to

x^{k}, F (x^{k}) + hz, F (z)i ≥ x^{k}, F (z) + z, F (x^{k}) . (3)

From properties of projection, we write x^{k} = x^{k}_{+} − x^{k}_{−} and F (x^{k}) = (−F (x^{k}))− −
(−F (x^{k}))+. Then, it follows from (3) that

x^{k}, F (x^{k}) + hz, F (z)i

≥ x^{k}_{+}, F (z) − x^{k}_{−}, F (z) + z, (−F (x^{k}))− − z, (−F (x^{k}))_{+} . (4)
Now, we denote x^{k}_{+} :=

[x^{k}_{+}]_{1}, x^{k}_{+}^{T}^{T}

and F (z) :=

[f (z)]_{1}, f (z)^{T}T

. With these nota- tions, we look into the first term on the right-hand side of (4):

x^{k}_{+}, F (z)

= x^{k}_{+}

1[f (z)]_{1}+D

x^{k}_{+}, f (z)E

≥ x^{k}_{+}

1[f (z)]_{1}−
x^{k}_{+}

p·

f (z)

q

≥ x^{k}_{+}

1[f (z)]_{1}−x^{k}_{+}

1

f (z)

q

= x^{k}_{+}

1

[f (z)]_{1}−
f (z)

q

(5)

≥ 0.

Note that x^{k} = x^{k}_{+}−x^{k}_{−}, which gives kx^{k}_{+}k ≥ kx^{k}k−kx^{k}_{−}k. Using 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 the POCCP has a strictly feasible point z, we know [f (z)]_{1} − kf (z)kq > 0,
which together with (5) implies that

hx^{k}_{+}, F (z)i → ∞ as k → ∞. (6)

Moreover, we also observe that lim sup

k→∞

hx^{k}_{−}, F (z)i ≤ lim sup

k→∞

kx^{k}_{−}kkF (z)k < ∞,
lim sup

k→∞

hz, (−F (x^{k}))_{+}i ≤ lim sup

k→∞

kzkk(−F (x^{k}))_{+}k < ∞
and hz, (−F (x^{k}))−i ≥ 0. All of these together with (4) and (6) yield

x^{k}, F (x^{k}) → ∞.

Then, the proof is complete. 2

### 3 Merit functions for POCCP

In this section, based on the product (2) of x and y associated with p-order cone in
IR^{n} and employing the same idea in [16], we propose several classes of merit functions
for the p-order cone complementarity problem and investigate their favorable properties,
respectively.

### 3.1 The first class of merit functions

In this subsection, we focus on the natural residual (NR) function φ_{NR} : IR^{n}× IR^{n}→ IR^{n},
which is given by:

φ_{NR}(x, y) := x − (x − y)+,

where (·)_{+}denotes the projection function. We know that the NR function φ_{NR} is always
an complementarity function for general conic complementarity problem, see [16] or [8,
Proposition 1.5.8]. In light of this, it is clear that the function Ψ_{NR}(x) = ^{1}_{2}kφ_{NR}(x, F (x))k^{2}
serves a merit function for the POCCP.

Lemma 3.1. Let x, y ∈ IR^{n} and φ_{NR}(x, y) = x − (x − y)_{+}. For any closed convex cone
K, we have

kφ_{NR}(x, y)k ≥ maxkx^{K}_{−}^{∗}k, k(−y)^{K}_{+}k ,

where z_{+}^{K} denotes the projection of z onto the closed convex cone K, and z_{−}^{K}^{∗} means the
projection of −z onto its dual cone K^{∗}.

Proof. The proof is similar to [16, Lemma 3.2] because the cone therein can be replaced by any closed convex cone. Hence, we omit it here. 2

In fact, Lu and Huang considered a more general NR merit function in [12], whose format is as bellow:

Ψ_{α}(x) = 1

2kx − (x − αF (x))_{+}k^{2}, (α > 0).

They also showed the property of error bound under the strong monotonicity and the global Lipschitz continuity of F .

Theorem 3.1. [12, Theorem 3.3] Suppose that F is strongly monotone with modulus ρ > 0 and is Lipschitz continuous with constant L > 0, Then for any fixed α > 0, the following inequality holds

1 2 + αL

pΨ_{α}(x) ≤ kx − x^{∗}k ≤ 1 + αL
αρ

pΨ_{α}(x),

where x^{∗} is the unique solution of the generally closed convex cone complementarity prob-
lems.

From Theorem 3.1, we know that the NR merit function Ψ_{NR} (i.e., α = 1) provides
an error bound for the POCCP. Unfortunately, when considering the boundedness of the
level set for the NR function φ_{NR}, if under the same conditions used in Proposition 2.2,
we cannot guarantee the boundedness of the level set for the function φ_{NR}. For example,

as mentioned in [16], taking F (x) = 1 − 1

x and x > 0, it is easy to verify that the level set

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

is unbounded. Thus, a different condition is needed. In fact, in order to establish the
boundedness of the level set for the natural residual function φ_{NR} or the merit function
Ψ_{α}, we need the following concept.

Definition 3.1. 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 3.2. Suppose that F is strongly coercive. Then, the level set
LNR(γ) = {x ∈ IR^{n}| kφ_{NR}(x, F (x))k ≤ γ}

or

LΨα(γ) = {x ∈ IR^{n}| Ψ_{α}(x) ≤ γ}

is bounded for all γ ≥ 0.

Proof. The proof is similar to [16, Theorem 4.2]. Hence, we omit it. 2

### 3.2 The second class of merit functions

For any x ∈ IR^{n}, we denote f_{LT} the LT (standing for Luo-Tseng) merit function associated
with the p-order cone complementarity problem, whose mathematical formula is given as
follows:

f_{LT}(x) := ϕ(hx, F (x)i) + 1

2k(x)−k^{2}+ k(−F (x))_{+}k^{2}, (7)
where ϕ : IR → IR+ is an arbitrary smooth function satisfying

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

It is easy to see that ϕ(t) ≥ 0 for all t ∈ IR from the above condition. This class of functions has been considered by Tseng for the positive semidefinite complementarity problem in [21], for the second-order cone complementarity problem by Chen in [2], and for the general SCCP case by Pan and Chen in [18], respectively. For the setting of general closed convex cone complementarity problems, the LT merit function has also been studied by Lu and Huang in [12], with some favorable properties shown as below.

Property 3.1. ([12, Lemma 3.1 and Theorem 3.4]) Let f_{LT} : IR^{n} → IR be given as in
(7). 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
p-order 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}.

Property 3.2. ([12, Theorem 3.6]) Let fLT be given as in (7). Suppose that F : IR^{n} →
IR^{n} is a strongly monotone mapping and that the p-order 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}.

Although the above properties were established in [12] for general closed convex cone
setting, there is no study about bounded level set for f_{LT} therein. Hence, we hereby
present the condition which ensures the boundedness of level sets for the LT merit func-
tion f_{LT} to solve the p-order cone complementarity problem.

Theorem 3.3. Suppose that the p-order cone complementarity problem has a strictly feasible point and that F is monotone. Then, the level set

LfLT(γ) := {x ∈ IR^{n}| f_{LT}(x) ≤ γ}

is bounded for all γ ≥ 0.

Proof. We prove this result by contradiction. Suppose there exists an unbounded
sequence {x^{k}} ⊆LfLT(γ) for some γ ≥ 0. Then, the sequences {x^{k}_{−}} and {(−F (x^{k}))_{+}}
must be bounded. If not, from the expression (7) of f_{LT} and the property ϕ(t) ≥ 0 for
all t ∈ IR, it follows that

fLT(x^{k}) ≥ 1

2kx^{k}_{−}k^{2}+ k(−F (x^{k}))+k^{2} → ∞,
which contradicts {x^{k}} ⊆LfLT(γ). Hence, we have

lim sup

k→∞

kx^{k}_{−}k < ∞ and lim sup

k→∞

k(−F (x^{k}))_{+}k < ∞.

Then, applying Proposition 2.2 yields

hx^{k}, F (x^{k})i → ∞.

Using the properties of the function ϕ again, we have ϕ(hx^{k}, F (x^{k})i) → ∞, which leads
to f_{LT}(x^{k}) → ∞. It contradicts {x^{k}} ⊆LfLT(γ). Thus, the level setLfLT(γ) is bounded
for all γ ≥ 0 and the proof is complete. 2

### 3.3 The third class of merit functions

Motivated by the construction way of the merit function f_{LT}, we make a slight modifi-
cation on the LT merit function f_{LT} associated with the p-order cone complementarity
problem, which leads to the third class of merit functions. More specifically, we first look
into the set Ω := K_{p}∩ K^{∗}_{p}. Indeed, the set Ω is characterized as follows:

Ω := K_{p}∩ K^{∗}_{p} = K_{p} for 1 ≤ p ≤ 2,
K_{p}^{∗} = K_{q} for p ≥ 2,

where q satisfies the condition q ≥ 1 and ^{1}_{p} + ^{1}_{q} = 1. Moreover, it is easy to check that
Ω is also a closed convex cone. In light of this closed convex cone 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} , (8)
where (x • y)^{Ω}_{+} denotes the projection of x • y onto Ω. As shown in the following theo-
rem, we see that the function df_{LT} is also a type of merit functions for the p-order cone
complementarity problem.

Theorem 3.4. Let the function df_{LT} be given as in (8). Then, for all x ∈ IR^{n}, we have
fd_{LT}(x) = 0 ⇐⇒ x ∈ K_{p}, F (x) ∈ K_{p}^{∗} and hx, F (x)i = 0,

where K^{∗}_{p} denotes the dual cone of K_{p}, i.e., K_{p}^{∗} = K_{q} with p, q ≥ 1 and ^{1}_{p} + ^{1}_{q} = 1.

Proof. From the definition of the function df_{LT} given in (8), we have
fdLT(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) ∈ −K_{p} or x • F (x) ∈ −K^{∗}_{p} , x ∈ K_{p} and F (x) ∈ K^{∗}_{p},

⇐⇒ −x • F (x) ∈ K_{p} or − x • F (x) ∈ K^{∗}_{p} , x ∈ K_{p} and F (x) ∈ K_{p}^{∗},

⇐⇒ x ∈ K_{p}, F (x) ∈ K^{∗}_{p} and hx, F (x)i = 0,

where the last equivalence holds due to the properties of K_{p} and K^{∗}_{p}. Thus, the proof is
complete. 2

In the following, we investigate the error bound property and the boundedness prop- erty of level sets of the merit function dfLT for the p-order cone complementarity problem.

In order to achieve these results, we need a novel lemma as below.

Lemma 3.2. For any x, y ∈ IR^{n}, we have
hx, yi ≤

(x • y)^{Ω}_{+}
,
where the product x • y is defined as in (2).

Proof. Given any x, y ∈ IR^{n}, let x = (x_{1}, · · · , x_{n})^{T} and y = (y_{1}, · · · , y_{n})^{T}. Recall from
the product (2), we know

x • y = hx, yi w

where w :=

|x_{1}|^{p}^{q}|y_{i}| − |y_{1}||x_{i}|^{p}^{q}n
i=2.
To proceed the arguments, we consider the following three cases.

Case 1 When x • y ∈ Ω, we have (x • y)^{Ω}_{+} = x • y. Then, it is easy to verify that
(x • y)^{Ω}_{+}

≥ hx, yi.

Case 2 When x • y ∈ −Ω^{∗}, where Ω^{∗} denotes the dual cone of Ω, we have (x • y)^{Ω}_{+}= 0
and hx, yi ≤ 0. This clearly implies that

(x • y)^{Ω}_{+}

≥ hx, yi.

Case 3 When x • y /∈ Ω ∪ (−Ω^{∗}), let (x • y)^{Ω}_{+} := (v_{1}, ¯v^{T})^{T}. If hx, yi ≤ 0, then the result
is obvious. Thus, we only need to look into the case of hx, yi > 0. If Ω = K_{p}, by
the property of projection onto the p-order cone, we have

(x • y)^{Ω}_{+}− x • y = v_{1}− hx, yi

¯ v − w

∈ Ω^{∗} = K^{∗}_{p} = K_{q}.

From the definition of dual order cone K^{∗}_{p} again, it follows that v_{1}− hx, yi ≥ 0, i.e.,
v_{1} ≥ hx, yi. Hence, this yields that

(x • y)^{Ω}_{+}

≥ |v_{1}| ≥ v_{1} ≥ hx, yi.

With similar arguments, for the case of Ω = K_{p}^{∗} = K_{q}, we also obtain that
hx, yi ≤

(x • y)^{Ω}_{+}
.
From all the above cases, we have shown that hx, yi ≤

(x • y)^{Ω}_{+}

. Thus, the proof is complete. 2

Theorem 3.5. Let the function df_{LT} be given as in (8). Suppose that F : IR^{n} → IR^{n} is
strongly monotone mapping and that x^{∗} is a solution to the p-order cone complementarity
problem. Then, there exists a scalar τ > 0 such that

τ kx − x^{∗}k^{2} ≤
2 +√

2 h

fd_{LT}(x)i^{1}_{2}
.

Proof. Since the function F is strongly monotone and x^{∗} is a solution to the p-order
cone complementarity problem, there exists a scalar ρ > 0 such that, for any x ∈ IR^{n},

ρ kx − x^{∗}k^{2} ≤ hF (x) − F (x^{∗}), x − x^{∗}i

= hF (x), xi + hF (x^{∗}), −xi + h−F (x), x^{∗}i

= hF (x), xi + hF (x^{∗}), x−− x_{+}i + h(−F (x))_{+}− (−F (x))−, x^{∗}i

≤ hF (x), xi + hF (x^{∗}), x−i + h(−F (x))_{+}, x^{∗}i

≤

(x • F (x))^{Ω}_{+}

+ kx_{−}kkF (x^{∗})k + kx^{∗}k k(−F (x))_{+}k

≤ max {1, kF (x^{∗})k, kx^{∗}k}

(x • F (x))^{Ω}_{+}

+ kx−k + k(−F (x))_{+}k ,
where the second inequality holds due to the properties of Kp and K^{∗}_{p}, and the third
inequality follows from Lemma 3.2. Then, setting τ := ρ

max {1, kF (x^{∗})k, kx^{∗}k} yields
τ kx − x^{∗}k^{2} ≤

(x • F (x))^{Ω}_{+}

+ kx_{−}k + k(−F (x))_{+}k.

Moreover, we observe that
(x • F (x))^{Ω}_{+}

=√ 2 1

2k(x • F (x))^{Ω}_{+}k^{2}

^{1}_{2}

≤√ 2h

fd_{LT}(x)i^{1}_{2}
,
and

kx−k + k(−F (x))+k ≤√

2 kx−k^{2}+ k(−F (x))+k^{2}^{1}_{2}

≤ 2h

fdLT(x)
i^{1}_{2}

. Putting all the above together gives

τ kx − x^{∗}k^{2} ≤ (2 +√
2)h

fd_{LT}(x)i^{1}_{2}
,
which is the desired result. 2

Next, we study the boundedness of level sets of merit function df_{LT}.

Theorem 3.6. Let the merit function df_{LT} be given as in (8). Suppose that the p-order
cone complementarity problem has a strictly feasible point and that F is monotone. Then,
the level set

LfdLT(γ) =

x ∈ IR^{n}

fd_{LT}(x) ≤ γ

is bounded for all γ ≥ 0.

Proof. Like the proof of Theorem 3.3, we prove this result by contradiction. Suppose
there exists an unbounded sequence {x^{k}} ⊆LfdLT(γ) for some γ ≥ 0. We claim that the
sequences {x^{k}_{−}} and {(−F (x^{k}))_{+}} are bounded. If not, by the expression (8) of df_{LT}, we
obtain

fd_{LT}(x^{k}) ≥ 1
2

kx^{k}_{−}k^{2}+ k(−F (x^{k}))_{+}k^{2}

→ ∞,
which contradicts {x^{k}} ⊆LfdLT(γ). Therefore, it follows that

lim sup

k→∞

kx^{k}_{−}k < ∞ and lim sup

k→∞

k(−F (x^{k}))+k < ∞.

Then, applying Proposition 2.2 yields hx^{k}, F (x^{k})i → ∞. This together with Lemma 3.2
implies

(x^{k}• F (x^{k}))^{Ω}_{+}

≥x^{k}, F (x^{k}) → ∞,

which leads to df_{LT}(x^{k}) → ∞. This clearly contradicts {x^{k}} ⊆L_{f}_{d}_{LT}(γ). Hence, the level
setLfdLT(γ) is bounded and the proof is complete. 2

Remark 3.1. In fact, if the term x^{k}• F (x^{k})Ω

+ in the expression of df_{LT} is replaced by
x^{k}• F (x^{k}), all Theorem 3.4, Lemma 3.2, Theorem 3.5, and Theorem 3.6 still hold.

### 3.4 The fourth class of merit function

In this subsection, in light of the product x • y and the NR merit function Ψ_{NR}, we
consider another merit function as below:

f_{r}(x) := 1

2kφ_{NR}(x, F (x))k^{2}+1
2

(x • F (x))^{Ω}_{+}

2, (9)

where (x • y)^{Ω}_{+}denotes the projection of x • y onto Ω. As seen below, we verify that f_{r}(x)
is also a merit function for the p-order cone complementarity problem.

Theorem 3.7. Let the function f_{r} be given as in (9). Then, for all x ∈ IR^{n}, we have
f_{r}(x) = 0 ⇐⇒ x ∈ K_{p}, F (x) ∈ K_{p}^{∗} and hx, F (x)i = 0,

where K^{∗}_{p} denotes the dual cone of Kp, i.e., K_{p}^{∗} = Kq.

Proof. In view of the definition of f_{r} given as in (9), we have
f_{r}(x) = 0 ⇐⇒

(x • F (x))^{Ω}_{+}

2 = 0 and Ψ_{NR}(x) = 1

2kφ_{NR}(x, F (x))k^{2} = 0,

⇐⇒ x ∈ K_{p}, F (x) ∈ K_{p}^{∗} and hx, F (x)i = 0,

where the second equivalence holds because Ψ_{NR} is a merit function for p-order cone
complementarity problem. Thus, the proof is complete. 2

From Theorem 3.7, we see that if the squared exponent in the expression of f_{r} is
deleted, i.e.,

fe_{r}(x) := kφ_{NR}(x, F (x))k +

(x • F (x))^{Ω}_{+}

, (10)

then ef_{r} is also a merit function for the POCCP. In fact, for these two merit functions f_{r}
and ef_{r}, there has no big differences between them in addition to the nature that f_{r} is
better than ef_{r}. Next, we will establish the error bound properties for f_{r} and ef_{r}.

Theorem 3.8. Let f_{r} and ef_{r} be given as in (9) and (10), respectively. Suppose that
F : IR^{n} → IR^{n} is strongly monotone mapping and that x^{∗} is a solution to the p-order
cone complementarity problem. Then, there exists a scalar τ > 0 such that

τ kx − x^{∗}k^{2} ≤ 3√

2f_{r}(x)^{1}_{2}

and τ kx − x^{∗}k^{2} ≤ 2 ef_{r}(x).

Proof. From Lemma 3.1, we know

kφ_{NR}(x, F (x))k ≥ max{kx−k, k(−F (x))_{+}k}.

Then, following similar arguments as in Theorem 3.5, we have
τ kx − x^{∗}k^{2} ≤

(x • F (x))^{Ω}_{+}

+ kx−k + k(−F (x))_{+}k

≤ √

2(f_{r}(x))^{1}^{2} + 2kφ_{NR}(x, F (x))k

= √

2(f_{r}(x))^{1}^{2} + 2√
2 1

2kφ_{NR}(x, F (x))k^{2}

^{1}_{2}

≤ √

2(f_{r}(x))^{1}^{2} + 2√

2(f_{r}(x))^{1}^{2}

= 3√ 2

fr(x)

^{1}_{2}

and

τ kx − x^{∗}k^{2} ≤

(x • F (x))^{Ω}_{+}

+ kx−k + k(−F (x))_{+}k

≤

(x • F (x))^{Ω}_{+}

+ 2 kφ_{NR}(x, F (x))k

≤ 2 ef_{r}(x),

where τ := ρ

max {1, kF (x^{∗})k, kx^{∗}k}. Thus, the proof is complete. 2

The following theorem presents the boundedness of the level sets of the functions ef_{r}
and f_{r}.

Theorem 3.9. Let f_{r} and ef_{r} be given as in (9) and (10), respectively. Suppose that
that the p-order cone complementarity problem has a strictly feasible point and that F is
monotone. Then, the level sets

Lfr(γ) = {x ∈ IR^{n}| fr(x) ≤ γ}

and

Lfer(γ) = n

x ∈ IR^{n}

efr(x) ≤ γ o are both bounded for all γ ≥ 0.

Proof. Here we only show the boundedness of the level sets of the function ef_{r} for all
γ ≥ 0 because the same arguments can be easily applied to the case of f_{r}.

As the proof in Theorem 3.3 and 3.6, we prove this result by contradiction. Suppose
there exists an unbounded sequence {x^{k}} ⊂ Lfer(γ) for some γ ≥ 0. If kx^{k}_{−}k → ∞ or
(−F (x^{k}))_{+}

→ ∞, by Lemma 3.1, we know
fe_{r}(x^{k}) ≥

φ_{NR}(x^{k}, F (x^{k}))

→ ∞,
which contradicts x^{k}∈Lfer(γ). Hence, we have

lim sup

k→∞

x^{k}_{−}

< ∞ and lim sup

k→∞

(−F (x^{k}))+

< ∞.

Then, applying Proposition 2.2 yields hx^{k}, F (x^{k})i → ∞. This together with Lemma 3.2
gives

x^{k}, F (x^{k}) ≤

(x^{k}• F (x^{k}))^{Ω}_{+}

→ ∞,

which leads to ef_{r}(x^{k}) → ∞. This is a contradiction because ef_{r}(x^{k}) ≤ γ. Thus, the proof
is complete. 2

Remark 3.2. As Remark 3.1, if the term x^{k}• F (x^{k})Ω

+ in the expressions of f_{r} and ef_{r}
is replaced by x^{k}• F (x^{k}), all Theorem 3.7, Theorem 3.8, and Theorem 3.9 still hold.

### 3.5 The fifth class of merit functions

In this subsection, we introduce the implicit Lagrangian merit associated with the POC-
COP. For any x ∈ IR^{n} and α > 0, the implicit Lagrangian merit function is given by

Mα(x) := hx, F (x)i + 1

2αk(x − αF (x))+k^{2}− kxk^{2}+ k(αx − F (x))−k^{2}− kF (x)k^{2} .
(11)

This class of functions was first introduced by Mangasarian and Solodov [13] for solving nonlinear complementarity problems, and was extended by Kong, Tuncel and Xiu [11]

to the setting of symmetric cone complementarity problems. Moreover, for the setting
of general closed convex cone complementarity problems in Hilbert space, Lu and Huang
[12] further investigated this merit function. Accordingly, the corresponding results in
[12] can be applied to the the setting of POCCP. For completeness, as below, the error
bound property of the merit function M_{α} for the POCCP is also presented.

Property 3.3. ([12, Theorem 3.9]) Let M_{α} be given as in (11). Suppose that F : IR^{n}→
IR^{n} is a strongly monotone mapping with modulus ρ > 0 and is Lipschitz continuous with
L > 0. Assume that the p-order cone complementarity problem has a solution x^{∗}. Then,
for any fixed α > 0, the following inequality holds

1

(α − 1)(2 + L)^{2} M_{α}(x) ≤ kx − x^{∗}k ≤ α(1 + L)^{2}

(α − 1)ρ^{2} M_{α}(x).

In the following theorem, we present the boundedness property of the level sets on
the merit function M_{α} for solving the p-order cone complementarity problem.

Theorem 3.10. Suppose that the p-order cone complementarity problem has a strictly feasible point and that F is monotone. Then, the level set

LMα(γ) := {x ∈ IR^{n}| M_{α}(x) ≤ γ}

is bounded for all γ ≥ 0.

Proof. First, we note that
M_{α}(x) = hx, F (x)i + 1

2αk(x − αF (x))_{+}k^{2}− kxk^{2}+ k(αx − F (x))−k^{2}− kF (x)k^{2}

= hx, F (x)i + 1

αhx − αF (x), (x − αF (x))_{+}i − 1

2αk(x − αF (x))_{+}k^{2}+ kxk^{2}
+

F (x), 1

α[αx − F (x) + (αx − F (x))−] − x

+ 1

2αkαx − F (x) + (αx − F (x))−− αxk^{2}

= − hF (x), (x − αF (x))_{+}− xi − 1

2αk(x − αF (x))_{+}− xk^{2}
+

F (x), 1

α(αx − F (x))_{+}− x

+ α

2 1

α(αx − F (x))_{+}− x

2

≥ 1 α

−

αF (x),1

α(αx − F (x))_{+}− x

− 1 2k1

α(αx − F (x))_{+}− xk^{2}

+α 1

αF (x), 1

α(αx − F (x))_{+}− x

+ α

2k1

α(αx − F (x))_{+}− xk^{2}

= α^{2}− 1
2α

1

α(αx − F (x))_{+}− x

2

= α^{2}− 1
2α

(x − 1

αF (x))_{+}− x

2

.

From Theorem 3.2, we know that the level set LΨα(γ) of the general NR merit function Ψα is bounded for all γ ≥ 0. With this, it is easy to see that the level set LMα(γ) is bounded for all γ ≥ 0. Then, the proof is complete. 2

### 4 Conclusion and future direction

Although the p-order cone complementarity problem belongs to nonsymmetric cone com- plementarity problem, for which there is no unified framework, we believe that the merit function approach may be an appropriate method that can be extended from symmetric cone complementarity problem to nonsymmetric cone complementarity problem. The key point to do such extension is constructing merit functions. Hence, in this paper, we present how to construct merit functions (by defining a new product) for the p-order cone complementarity problem. In addition, we have also shown under what conditions these merit functions have properties of error bounds and bounded level sets. These results provide a theoretical basis for designing the merit function method for solving the special nonsymmetric cone complementarity problem, i.e., p-order cone complementarity prob- lem. We leave this topic as our future direction. At last, we point out that the main idea is employed from [16] (which is for circular cone setting) and most analysis techniques look similar to those used in [16]. Nonetheless, the product is novel which contributes to the literature by providing a new way to deal with such complementarity problem.

### References

[1] J.-S. Chen, A new merit function and its related properties for the second-order cone complementarity problem, Pacific Journal of Optimization, vol. 2, pp. 167–179, 2006.

[2] J.-S. Chen, Two classes of merit functions for the second-order cone complemen- tarity problem, Mathematical Methods of Operations Research, vol. 64, pp. 495–519, 2006.

[3] J.-S. Chen, Conditions for error bounds and bounded level sets of some merit func- tions for the second-order cone complementarity problem, Journal of Optimization Theory and Applications, vol. 135, pp. 459–473, 2007.

[4] X.D. Chen, D. Sun and J. Sun, Complementarity functions and numerical exper- iments on some smoothing Newton methods for second-order cone complementarity problems, Computational Optimization and Applications, vol. 25, pp. 39–56, 2003.

[5] J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of second-order cone complementarity problem, Mathematical Programming, vol. 104, pp. 293–327, 2005.

[6] J.-S. Chen and S.-H. Pan, A survey on SOC complementarity functions and so- lution methods for SOCPs and SOCCPs, Pacific Journal of Optimization, vol. 8, pp.

33–74, 2012.

[7] M. Fukushima, Z.-Q. Luo, and P. Tseng, Smoothing functions for second- order- cone complimentarity problems, SIAM Journal on Optimization, vol. 12, pp. 436–460, 2002.

[8] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I, New York, Springer, 2003.

[9] D.R. Han, On the coerciveness of some merit functions for complimentarity problems over symmetric cones, Journal of Mathematical Analysis and Applications, vol. 336, pp. 727–737, 2007.

[10] L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for sym- metric cone complementarity problems, SIAM Journal on Optimization, vol. 19, pp.

1028–1047, 2008.

[11] L. Kong, T. Tuncel and N. Xiu, Vector-valued implicit Lagrangian for sym- metric cone complementarity problems, Asia-Pacific Journal of Operational Research, vol. 26, pp. 199–233, 2009.

[12] N. Lu and Z.-H. Huang, Three classes of merit functions for the complementarity problem over a closed convex cone, Optimization, vol. 62, pp. 545–560, 2013.

[13] O.L. Mangasarian and M.V. Solodov, Nonlinear complementarity as uncon- strained and constrained minimization, Mathematical programming, vol. 62, pp. 277–

297, 1993.

[14] Y. Matsukawa and A. Yoshise, A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems, Japan Journal of Industrial and Applied Mathematics, vol. 29, pp. 499–517, 2012.

[15] X.-H. Miao and J.-S. Chen, Characterizations of solution sets of cone-constrained convex programming problems, Optimization Letters, vol. 9, pp. 1433-1445, 2015.

[16] X.-H. Miao, S.-J. Guo, N. Qi, and J.-S. Chen, Constructions of comple- mentarity functions and merit functions for circular cone complementarity problem, Computational Optimization and Applications, vol. 63, pp. 495-522, 2016.

[17] X.H. Miao, N. Qi, and J.-S. Chen, Projection formula and one type of spectral factorization associated with p-order cone, to appear in Journal of Nonlinear and Convex Analysis, 2016.

[18] S.-H. Pan and J.-S. Chen, A one-parametric class of merit functions for the sym- metric cone complementarity problem, Journal of Mathematical Analysis and Appli- cations, vol. 355, pp. 195–215, 2009.

[19] A. Skajaa and Y. Ye, A homogeneous interior-point algorithm for nonsymmetric convex conic optimization, Mathematical Programming, vol. 150, pp. 391-422, 2015.

[20] D. Sun and J. Sun, L¨owner’s operator and spectral functions in Euclidean Jordan algebras, Mathematics of Operations Research, vol. 33, pp. 421–445, 2008.

[21] P. Tseng, Merit functions for semi-definite complementarity problems, Mathemat- ical Programming, vol. 83, pp. 159–185, 1998.

[22] G.-L. Xue and Y.-Y. Ye, An efficient algorithm for minimizing a sum of p-norm, SIAM Journal on Optimization, vol. 10, pp. 551–579, 2000.