https://doi.org/10.1007/s40314-018-0660-0

**Discovery of new complementarity functions for NCP** **and SOCCP**

**Peng-Fei Ma**^{1}**· Jein-Shan Chen**^{2}**·**
**Chien-Hao Huang**^{2}**· Chun-Hsu Ko**^{3}

Received: 25 February 2017 / Revised: 20 May 2018 / Accepted: 30 May 2018

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

**Abstract It is well known that complementarity functions play an important role in dealing**
with complementarity problems. In this paper, we propose a few new classes of com-
plementarity functions for nonlinear complementarity problems and second-order cone
complementarity problems. The constructions of such new complementarity functions are
based on discrete generalization which is a novel idea in contrast to the continuous general-
ization of Fischer–Burmeister function. Surprisingly, these new families of complementarity
functions possess continuous differentiability even though they are discrete-oriented exten-
sions. This feature enables that some methods like derivative-free algorithm can be employed
directly for solving nonlinear complementarity problems and second-order cone comple-
mentarity problems. This is a new discovery to the literature and we believe that such new
complementarity functions can also be used in many other contexts.

Communicated by Jinyun Yuan.

Peng-Fei Ma This research was supported by a grant from the National Natural Science Foundation of China(No.11626212).

Jein-Shan Chen The author’s work is supported by Ministry of Science and Technology, Taiwan.

### B

Jein-Shan Chen jschen@math.ntnu.edu.tw Peng-Fei Mamathpengfeima@126.com Chien-Hao Huang qqnick0719@ntnu.edu.tw Chun-Hsu Ko

chko@isu.edu.tw

1 Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, P.R. China

2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan 3 Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan

**Keywords NCP**· SOCCP · Natural residual · Complementarity function
**Mathematics Subject Classification 26B05**· 26B35 · 90C33 · 65K05

**1 Introduction**

In general, the complementarity problem comes from the Karush–Kuhn–Tucker (KKT) con- ditions of linear and nonlinear programming problems. For different types of optimization problems, there arise various complementarity problems, for example, linear complemen- tarity problem, nonlinear complementarity problem (NCP), semidefinite complementarity problem, second-order cone complementarity problem (SOCCP), and symmetric cone complementarity problem. To deal with complementarity problems, the so-called comple- mentarity functions play an important role therein. In this paper, we focus on two classes of complementarity functions, which are used for the NCP and SOCCP, respectively.

The first class is the NCP that has attracted much attention since 1970s because of its wide
applications in the fields of economics, engineering, and operations research, see (Cottle
et al.1992; Facchinei and Pang 2003; Harker and Pang1990) and references therein. In
*mathematical format, the NCP is to find a point x*∈ R* ^{n}* such that

*x≥ 0, F(x) ≥ 0, x, F(x) = 0,*

where*·, · is the Euclidean inner product and F = (F*1*, . . . , F**n**)** ^{T}* is a map fromR

*toR*

^{n}*. For solving NCP, the so-called NCP function*

^{n}*φ : R*

^{2}→ R defined as below

*φ(a, b) = 0 ⇐⇒ a, b ≥ 0, ab = 0*

plays a crucial role. Generally speaking, with such NCP functions, the NCP can be refor- mulated as nonsmooth equations (Mangasarian1976; Pang1990; Yamashita and Fukushima 1997) or unconstrained minimization (Facchinei and Soares1997; Fischer1992; Geiger and Kanzow1996; Jiang1996; Kanzow1996; Pang and Chan1982; Yamashita and Fukushima 1995). Then, different kinds of approaches and algorithms are designed based on the afore- mentioned reformulations and various NCP functions. During the past four decades, around thirty NCP functions are proposed, see (Galántai2012) for a survey.

The second class is the SOCCP, which can be viewed as a natural extension of NCP and
is to seek a*ζ ∈ R** ^{n}*such that

*ζ ∈K, F(ζ ) ∈K, ζ, F(ζ ) = 0,*

*where F*: R* ^{n}* → R

*is a map and*

^{n}*K*is the Cartesian product of second-order cones (SOC), also called Lorentz cones (Faraut and Korányi1994). In other words,

*K*is expressed as:

*K*=*K*^{n}^{1}× · · · ×*K*^{n}^{m}*,*
*where m, n*1*, . . . , n**m**≥ 1, n*1*+ · · · + n**m**= n, and*

*K*^{n}^{i}*:= {(x*1*, x*2*) ∈ R × R*^{n}^{i}^{−1}*| x*2* ≤ x*1*},*

with · denoting the Euclidean norm. The SOCCP has important applications in engineering problems (Kanno et al.2006) and robust Nash equilibria (Hayashi et al.2005). Another important special case of SOCCP corresponds to the KKT optimality conditions for the

*minimize c*^{T}*x*

*subject to Ax* *= b, x ∈K,*

*where A* ∈ R^{m×n}*has full row rank, b* ∈ R^{m}*and c* ∈ R* ^{n}*. Many solution methods have
been proposed for solving SOCCP, see (Chen and Pan2012) for a survey. For example,
merit function approach based on reformulating the SOCCP as an unconstrained smooth
minimization problem is studied in Chen and Tseng (2005), Chen (2006b), Pan et al. (2014).

In such approach, it is to find a smooth function*ψ : R** ^{n}*× R

*→ R+such that*

^{n}*ψ(x, y) = 0 ⇐⇒ x, y = 0, x ∈K*^{n}*, y ∈K*^{n}*.* (1)
Then, the SOCCP can be expressed as an unconstrained smooth (global) minimization prob-
lem:

*ζ ∈R*min^{n}*ψ(ζ, F(ζ )).* (2)

In fact, a function *ψ satisfying the condition in (1) (not necessarily smooth) is called a*
complementarity function for SOCCP (or complementarity function associated with*K** ^{n}*).

Various gradient methods such as conjugate gradient methods and quasi-Newton methods
(Bertsekas1999; Fletcher1987) can be applied for solving (2). In general, for this approach
to be effective, the choice of complementarity function*ψ is also crucial.*

Back to the complementarity functions for NCP, two popular choices of NCP functions
are the well-known Fischer–Burmeister function (FB function, in short)*φ*_{FB} : R^{2} → R
defined by see (Fischer1992,1997)

*φ*_{FB}*(a, b) =*

*a*^{2}*+ b*^{2}*− (a + b),*
and the squared norm of Fischer–Burmeister function given by

*ψ*FB*(a, b) =* 1

2*φ*FB*(a, b)*^{2}*.*

In addition, the generalized Fischer–Burmeister function*φ**p*: R^{2}→ R, which includes the
Fischer–Burmeister as a special case, is considered in Chen (2006a,2007), Chen et al. (2009),
Chen and Pan (2008), Hu et al. (2009), Tsai and Chen (2014). In particular, the function*φ**p*

is a natural “continuous extension” of*φ*FB, in which the 2-norm in*φ*FB*(a, b) is replaced by*
*general p-norm. In other words,φ**p* : R^{2} → R is defined as:

*φ**p**(a, b) = (a, b) **p**− (a + b), p > 1* (3)
and its geometric view is depicted in Tsai and Chen (2014). The effect of perturbing p for dif-
ferent kinds of algorithms is investigated in Chen et al. (2010,2011), and Chen and Pan (2008)
. We point it out that the generalized Fischer–Burmeister*φ**p*given as in (3) is not differentiable,
whereas the squared norm of generalized Fischer–Burmeister function is smooth so that it is
usually adapted as a differentiable NCP function Pan et al. (2014). Moreover, all the afore-
mentioned functions including Fischer–Burmeister function, generalized Fischer–Burmeister
function and their squared norm can be extended to the setting of SOCCP via Jordan algebra.

A different type of popular NCP function is the natural residual function*φ*NR: R^{2}→ R
given by

*φ*_{NR}*(a, b) = a − (a − b)*_{+}*= min{a, b}.*

Recently, Chen et al. propose a family of generalized natural residual functions*φ*_{NR}* ^{p}* defined
by

*φ*_{NR}^{p}*(a, b) = a*^{p}*− (a − b)*_{+}^{p}*,*

*where p> 1 is a positive odd integer, (a−b)*_{+}^{p}*= [(a−b)*+]* ^{p}*, and

*(a−b)*+

*= max{a−b, 0}.*

*When p= 1, φ*_{NR}* ^{p}* reduces to the natural residual function

*φ*NR, i.e.,

*φ*

_{NR}

^{1}

*(a, b) = a − (a − b)*+

*= min{a, b} = φ*NR

*(a, b).*

As remarked in Chen et al. (2016), this extension is “discrete generalization”, not “continu-
ous generalization”. Nonetheless, it possesses twice differentiability surprisingly so that the
squared norm of*φ*_{NR}* ^{p}* is not needed. Based on this discrete generalization, two families of
NCP functions are further proposed in Chang et al. (2015) which have the feature of sym-
metric surfaces. To the contrast, it is very natural to ask whether there is a similar “discrete
extension” for Fischer–Burmeister function. We answer this question affirmatively.

In this paper, we apply the idea of “discrete generalization” to the Fischer–Burmeister
function which gives the following function (denoted by*φ*_{D−FB}* ^{p}* ):

*φ*_{D−FB}^{p}*(a, b) =*

*a*^{2}*+ b*^{2}_{p}

*− (a + b)*^{p}*,* (4)

*where p* *> 1 is a positive odd integer and (a, b) ∈ R*^{2}*. Notice that when p* *= 1, φ*_{D−FB}* ^{p}*
reduces to the Fischer–Burmeister function. In Sect.3, we will see that

*φ*

_{D−FB}

*is an NCP*

^{p}*function and is twice differentiable directly without taking its squared norm. Note that if p*is even, it is no longer an NCP function. Even though we have the feature of differentiability, we point out that the Newton method may not be applied directly because the Jacobian at a degenerate solution to NCP is singular see (Kanzow1996; Kanzow and Kleinmichel1995).

Nonetheless, this feature may enable that many methods like derivative-free algorithm can be
employed directly for solving NCP. In addition, we investigate the differentiable properties
of*φ*_{D−FB}* ^{p}* , the computable formulas for their gradients and Jacobians. In order to have more
insight for this new family of NCP function, we also depict the surfaces of

*φ*

_{D−FB}

^{p}*(a, b) with*

*various values of p.*

In Sect.4, we show that the new function*φ*_{D−FB}* ^{p}* can be further employed to the SOCCP
setting as complementarity functions and merit functions. In other words, in the terms of
Jordan algebra, we define

*φ*

_{D−FB}

*: R*

^{p}*× R*

^{n}*→ R*

^{n}*by*

^{n}*φ*_{D−FB}^{p}*(x, y) =*

*x*^{2}*+ y*^{2}

_{p}

*− (x + y)*^{p}*,* (5)

*where p* *> 1 is a positive odd integer, x ∈ R*^{n}*, y* ∈ R^{n}*, x*^{2} *= x ◦ x is the Jordan*
*product of x with itself and*√

*x with x* ∈ *K** ^{n}* being the unique vector such that √

*x*◦

√*x* *= x. We prove that each φ*_{D−FB}^{p}*(x, y) is a complementarity function associated with*
*K** ^{n}* and establish formulas for its gradient and Jacobian. These properties and formulas can
be used to design and analyze non-interior continuation methods for solving second-order
cone programs and complementarity problems. In addition, several variants of

*φ*

_{D−FB}

*are also shown to be complementarity functions for SOCCP.*

^{p}Throughout the paper, we assume*K*=*K** ^{n}*for simplicity and all the analysis can be carried
over to the case where

*K*is a product of SOC without difficulty. The following notations will

*be used. The identity matrix is denoted by I and*R

^{n}*denotes the space of n-dimensional*

*real column vectors. For any given x*∈ R

^{n}*with n*

*> 1, we write x = (x*1

*, x*2

*) where*

*x*

_{1}

*is the first entry of x and x*

_{2}is the subvector that consists of the remaining entries. For

*every differentiable function f*: R

^{n}*→ R, ∇ f (x) denotes the gradient of f at x. For*

*every differentiable mapping F*: R

*→ R*

^{n}*,*

^{m}*∇ F(x) is an n × m matrix which denotes the*

*transposed Jacobian of F at x. For nonnegative scalar functionsα and β, we write α = o(β)*to mean lim

_{β→0}*α*

= 0.

**2 Preliminaries**

In this section, we review some background materials about the Jordan algebra in Faraut and Korányi (1994), Fukushima et al. (2002). Then, we present some technical lemmas which are needed in subsequent analysis.

*For any x= (x*1*, x*2*), y = (y*1*, y*2*) ∈ R × R*^{n−1}*, we define the Jordan product associated*
with*K** ^{n}*as:

*x◦ y := (x, y, y*1*x*2*+ x*1*y*2*).*

*The identity element under this product is e:= (1, 0, . . . , 0)** ^{T}* ∈ R

^{n}*. For any given x*=

*(x*1

*, x*2

*) ∈ R × R*

*, we define symmetric matrix*

^{n−1}*L** _{x}* :=

*x*1 *x*_{2}^{T}*x*2 *x*1*I*

which can be viewed as a linear mapping fromR* ^{n}*toR

*. It is easy to verify that*

^{n}*L*

_{x}*y= x ◦ y, ∀x ∈ R*

^{n}*.*

*Moreover, we have L*_{x}*is invertible for x**K** ^{n}* 0 and

*L*^{−1}* _{x}* = 1
det

*(x)*

⎡

⎣*x*_{1} *−x*_{2}^{T}

*−x*2 det(x)
*x*1

*I*+ 1
*x*1

*x*2*x*_{2}^{T}

⎤

*⎦ ,*

where det(x) = x_{1}^{2}*− x*2 ^{2}. We next recall from Chen and Pan (2012); Fukushima et al.

(2002) that each x *= (x*1*, x*2*) ∈ R × R** ^{n−1}*admits a spectral factorization, associated with

*K*

*, of the form*

^{n}*x* *= λ*1*u*^{(1)}*+ λ*2*u*^{(2)}*,* (6)

where*λ*1*, λ*2*and u*^{(1)}*, u*^{(2)}*are the spectral values and the associated spectral vectors of x*
given by

*λ**i* *= x*1*+ (−1)*^{i}* x*2* ,*

*u** ^{(i)}*=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 2

1, (−1)^{i}*x*2

* x*2

*if x*2= 0;

1 2

1*, (−1)*^{i}*w*2

*if x*_{2}*= 0,*

*for i* *= 1, 2, with w*2 being any vector in R* ^{n−1}* satisfying

*w*2

*= 1. If x*2 = 0, the factorization is unique.

*Given a real-valued function g* : R → R, we can define a vector-valued SOC function
*g*^{soc}: R* ^{n}*→ R

*by*

^{n}*g*^{soc}*(x) := g(λ*1*)u*^{(1)}*+ g(λ*2*)u*^{(2)}*.*

*If g is defined on a subset ofR, then g*^{soc}is defined on the corresponding subset ofR* ^{n}*. The

*definition of g*

^{soc}

*is unambiguous whether x*

_{2}

*= 0 or x*2= 0. In this paper, we will often use

*the vector-valued functions corresponding to t*

^{p}*(t ∈ R) and*√

*t(t ≥ 0), respectively, which*
are expressed as:

*x*^{p}*:= (λ*1*(x))*^{p}*u*^{(1)}*+ (λ*2*(x))*^{p}*u*^{(2)}*,* *∀x ∈ R*^{n}

√*x* :=√

*λ*1*(x)u** ^{(1)}*+√

*λ*2*(x)u*^{(2)}*,* *∀x ∈K*^{n}*.*

We will see that the above two vector-valued functions play a role, showing that*φ*_{D−FB}* ^{p}* given
as in (5) is well defined in the SOC setting for any x

*, y ∈ R*

*. Note that the other way to*

^{n}*define x*

*and√*

^{p}*x is through Jordan product. In other words, x*^{p}*represents x◦ x ◦ · · · ◦ x for*
*p-times and*√

*x* ∈*K** ^{n}* satisfies√

*x*◦√

*x= x.*

**Lemma 2.1 Suppose that p**= 2k + 1 where k = 1, 2, 3, · · · . Then, for any u, v ∈ R, we*have u*^{p}*= v*^{p}*if and only if u= v.*

*Proof The proof is straightforward and can be found in (Baggett et al.*2012, Theorem 1.12).

Here, we provide an alternative proof.

“⇐” It is trivial.

“⇒” For v = 0, since u^{p}*= v*^{p}*, we have u= v = 0. For v = 0, from f (t) = t** ^{p}*− 1 being

*a strictly monotone increasing function for any t*∈ R, we have

*u*

*v*

_{p}

− 1 = 0 if and only if
*u*

*v* *= 1, which implies u = v. Thus, the proof is complete.*

**Lemma 2.2 For p***= 2m + 1 with m = 1, 2, 3, · · · and x = (x*1*, x*2*), y = (y*1*, y*2*) ∈*
R × R^{n−1}*, suppose that x*^{p}*and y*^{p}*represent x◦ x ◦ · · · ◦ x and y ◦ y ◦ · · · ◦ y for p-times,*
*respectively. Then, x*^{p}*= y*^{p}*if and only if x= y.*

*Proof “*⇐” This direction is trivial.

“*⇒” Suppose that x*^{p}*= y** ^{p}*. By the spectral decomposition (6), we write

*x*

*= λ*1

*(x)u*

^{(1)}*x*

*+ λ*2

*(x)u*

^{(2)}*x*

*,*

*y* *= λ*1*(y)u*^{(1)}*y* *+ λ*2*(y)u*^{(2)}*y* *.*

*Then, x*^{p}*= (λ*1*(x))*^{p}*u*^{(1)}_{x}*+(λ*2*(x))*^{p}*u*^{(2)}_{x}*and y*^{p}*= (λ*1*(y))*^{p}*u*^{(1)}_{y}*+(λ*2*(y))*^{p}*u*^{(2)}_{y}*. Since x** ^{p}*=

*y*

*and eigenvalues are unique, we obtain*

^{p}*(λ*1

*(x))*

^{p}*= (λ*1

*(y))*

*and*

^{p}*(λ*2

*(x))*

^{p}*= (λ*2

*(y))*

*. By Lemma2.1, this implies*

^{p}*λ*1

*(x) = λ*1

*(y) and λ*2

*(x) = λ*2

*(y). Moreover, {u*

^{(1)}*x*

*, u*

^{(2)}*x*} and

*{u*

^{(1)}*y*

*, u*

^{(2)}*y*

*} are Jordan frames, we have u*

^{(1)}*x*

*+ u*

^{(2)}*x*

*= u*

^{(1)}*y*

*+ u*

^{(2)}*y*

*= e, where e is the identity*

*element. From x*

^{p}*= y*

^{p}*and u*

^{(1)}

_{x}*+ u*

^{(2)}*x*

*= u*

^{(1)}*y*

*+ u*

^{(2)}*y*, we get

*(λ*1*(x))*^{p}*− (λ*2*(x))*^{p}

*(u*^{(1)}*x* *− u*^{(1)}*y* *) = 0.*

If*(λ*1*(x))*^{p}*= (λ*2*(x))** ^{p}*, we have

*λ*1

*(x) = λ*2

*(x) and λ*1

*(y) = λ*2

*(y), that is, x = λ*1

*(x)e = y.*

Otherwise, if*(λ*1*(x))*^{p}*= (λ*2*(x))*^{p}*, we must have u*^{(1)}_{x}*= u*^{(1)}*y* *, which implies u*^{(2)}_{x}*= u*^{(2)}*y* .

**3 New generalized Fischer–Burmeister function for NCP**

In this section, we show that the function*φ*_{D−FB}* ^{p}* defined as in (4) is an NCP function and
present its twice differentiability. At the same time, we also depict the surfaces of

*φ*

_{D−FB}

*with*

^{p}*various values of p to have more insight for this new family of NCP functions.*

**Proposition 3.1 Let**φ_{D−FB}^{p}*be defined as in (4) where p is a positive odd integer. Then,φ*_{D−FB}^{p}*is an NCP function.*

*Proof Supposeφ*_{D−FB}^{p}*(a, b) = 0 , which says*√

*a*^{2}*+ b*^{2}_{p}

*= (a + b)*^{p}*. Using p being a*
positive odd integer and applying Lemma2.1, we have

*a*^{2}*+ b*^{2}_{p}

*= (a + b)** ^{p}* ⇐⇒

*a*^{2}*+ b*^{2}*= a + b.*

It is well known that√

*a*^{2}*+ b*^{2} *= a + b is equivalent to a, b ≥ 0, ab = 0 because φ*FB is
an NCP function. This shows that*φ*_{D−FB}^{p}*(a, b) = 0 implies a, b ≥ 0, ab = 0. The converse*
direction is trivial. Thus, we prove that*φ*_{D−FB}* ^{p}* is an NCP function.

*Remark 3.1 We elaborate more about the new NCP functionφ*_{D−FB}* ^{p}* .

*(a) For p being an even integer,φ*_{D−FB}* ^{p}* is not an NCP function. A counterexample is given
as below.

*φ*_{D−FB}^{p}*(−5, 0) = (−5)*^{2}*− (−5)*^{2}*= 0.*

(b) The surface of*φ*_{D−FB}* ^{p}* is symmetric, i.e.,

*φ*

_{D−FB}

^{p}*(a, b) = φ*

_{D−FB}

^{p}*(b, a).*

(c) The function*φ*_{D−FB}^{p}*(a, b) is positive homogenous of degree p, i.e., φ*_{D−FB}^{p}*(α(a, b)) =*
*α*^{p}*φ*_{D−FB}^{p}*(a, b) for α ≥ 0.*

(d) The function*φ*_{D−FB}^{p}*is neither convex nor concave function. To see this, taking p*= 3 and
using the following argument verify the assertion.

5^{3}− 7^{3}*= φ*_{D−FB}^{3} *(3, 4) >* 1

2*φ*^{3}_{D−FB}*(0, 0) +*1

2*φ*^{3}_{D−FB}*(6, 8)*

= 1

2× 0 +1 2

10^{3}− 14^{3}

= 4
5^{3}− 7^{3}
and

0*= φ*_{D−FB}^{3} *(0, 0) <* 1

2*φ*_{D−FB}^{3} *(−2, 0) +*1

2*φ*_{D−FB}^{3} *(2, 0) =* 1

2× 16 +1

2*× 0 = 8.*

**Proposition 3.2 Let**φ_{D−FB}^{p}*be defined as in (4) where p is a positive odd integer. Then, the*
*following hold.*

*(a) For p> 1, φ*_{D−FB}^{p}*is continuously differentiable with*

*∇φ*_{D−FB}^{p}*(a, b) = p*
*a(*√

*a*^{2}*+ b*^{2}*)*^{p−2}*− (a + b)*^{p−1}*b(*√

*a*^{2}*+ b*^{2}*)*^{p}^{−2}*− (a + b)*^{p}^{−1}

*.*

*(b) For p> 3, φ*_{D−FB}^{p}*is twice continuously differentiable with*

∇^{2}*φ*_{D−FB}^{p}*(a, b) =*

⎡

⎢⎢

⎣

*∂*^{2}*φ*_{D−FB}^{p}

*∂a*^{2}

*∂*^{2}*φ*_{D−FB}^{p}

*∂a∂b*

*∂*^{2}*φ*_{D−FB}^{p}

*∂b∂a*

*∂*^{2}*φ*_{D−FB}^{p}

*∂b*^{2}

⎤

⎥⎥

*⎦ ,*

*where*

*∂*^{2}*φ*_{D−FB}^{p}

*∂a*^{2} *= p*

*(p − 1)a*^{2}*+ b*^{2}
*(*

*a*^{2}*+ b*^{2}*)*^{p}^{−4}*− (p − 1)(a + b)*^{p}^{−2}
*,*

*∂*^{2}*φ*_{D−FB}^{p}

*∂a∂b* *= p*

*(p − 2)ab(*

*a*^{2}*+ b*^{2}*)*^{p−4}*− (p − 1)(a + b)*^{p−2}

= *∂*^{2}*φ*_{D−FB}^{p}

*∂b∂a* *,*

*∂*^{2}*φ*_{D−FB}^{p}

*∂b*^{2} *= p*

*a*^{2}*+ (p − 1)b*^{2}
*(*

*a*^{2}*+ b*^{2}*)*^{p−4}*− (p − 1)(a + b)*^{p−2}*.*

*Proof The verifications of differentiability and computations of first and second derivatives*

are straightforward, we omit them.

Next, we present some variants of*φ*_{D−FB}* ^{p}* . Indeed, analogous to those functions in Sun and
Qi (1999), the variants of

*φ*

_{D−FB}

*as below can be verified being NCP functions.*

^{p}*φ*1*(a, b) = φ*_{D−FB}^{p}*(a, b) − α(a)*+*(b)*+*, α > 0.*

*φ*2*(a, b) = φ*_{D−FB}^{p}*(a, b) − α ((a)*+*(b)*+*)*^{2}*, α > 0.*

*φ*3*(a, b) = [φ*_{D−FB}^{p}*(a, b)]*^{2}*+ α ((ab)*+*)*^{4}*, α > 0.*

*φ*4*(a, b) = [φ*_{D−FB}^{p}*(a, b)]*^{2}*+ α ((ab)*+*)*^{2}*, α > 0.*

*In the above expressions, for any t∈ R, we define t*+as max{0, t}.

**Lemma 3.1 Let**φ_{D−FB}^{p}*be defined as in (4) where p is a positive odd integer. Then, the value*
*ofφ*_{D−FB}^{p}*(a, b) is negative only in the first quadrant, i.e., φ*_{D−FB}^{p}*(a, b) < 0 if and only if a > 0,*
*b> 0.*

*Proof We know that f(t) = t*^{p}*is a strictly increasing function when p is odd. Using this*
fact yields

*a> 0, b > 0*

*⇐⇒ a + b > 0 and ab > 0*

⇐⇒

*a*^{2}*+ b*^{2} *< a + b*

⇐⇒

*a*^{2}*+ b*^{2}*p*

*< (a + b)*^{p}

*⇐⇒ φ*_{D−FB}^{p}*(a, b) < 0,*

which proves the desired result.

**Proposition 3.3 All the above functions**φ*i**for i∈ {1, 2, 3, 4} are NCP functions.*

*Proof Applying Lemma*3.1, the arguments are similar to those in [Chen et al. (2016), Propo-

sition 2.4], which are omitted here.

In fact, in light of Lemma2.1, we can construct more variants of*φ*_{D−FB}* ^{p}* , which are also new

*NCP function. More specifically, consider that k and m are positive integers, f*: R×R → R,

*and g*

*: R × R → R with g(a, b) = 0 for all a, b ∈ R, the following functions are new*variants of

*φ*

_{D−FB}

*.*

^{p}*φ*5*(a, b) =*

*g(a, b)*

*a*^{2}*+ b*^{2}*+ f (a, b)**2m+1*^{2k+1}

−
*g(a, b)*

*a+ b + f (a, b)*_{2m+1}^{2k+1}*.*
*φ*6*(a, b) =*

*g(a, b)(*

*a*^{2}*+ b*^{2}*− a − b)*^{k}

*m* *.*
*φ*7*(a, b) =*

*g(a, b)(*

*a*^{2}*+ b*^{2}*− a + f (a, b))*_{2m+1}^{2k+1}

*− [g(a, b)(b + f (a, b))]*^{2m+1}^{2k+1}*.*
*φ*8*(a, b) =*

*g(a, b)(*

*a*^{2}*+ b*^{2}*− a + f (a, b))*_{2m+1}^{2k+1}

*− [g(a, b)(b + f (a, b))]*^{2m+1}^{2k+1}*.*
*φ*9*(a, b) = e*^{φ}^{i}^{(a,b)}*− 1 where i = 5, 6, 7, 8.*

*φ*10*(a, b) = ln(|φ**i**(a, b)| + 1) where i = 5, 6, 7, 8.*

**Proposition 3.4 All the above functions**φ*i**for i∈ {5, 6, 7, 8, 9, 10} are NCP functions.*

*Proof This is an immediate consequence of Propositions*3.1,3.1,3.3. By Lemma2.1and
*g(a, b) = 0 for a, b ∈ R, we have*

*φ*5*(a, b) = 0*

⇐⇒

*g(a, b)*

*a*^{2}*+ b*^{2}*+ f (a, b)**2m+1*^{2k+1}

=
*g(a, b)*

*a+ b + f (a, b)*_{2m+1}^{2k+1}

⇐⇒

*g(a, b)*

*a*^{2}*+ b*^{2}*+ f (a, b)*_{2m+1}^{2k+1}*2m*+1

=
*g(a, b)*

*a+ b + f (a, b)*_{2m+1}^{2k+1}_{2m+1}

⇐⇒

*g(a, b)*

*a*^{2}*+ b*^{2}*+ f (a, b)** ^{2k}*+1

=
*g(a, b)*

*a+ b + f (a, b)*_{2k+1}

*⇐⇒ g(a, b)*

*a*^{2}*+ b*^{2}*+ f (a, b)*

*= g(a, b)*

*a+ b + f (a, b)*

⇐⇒

*a*^{2}*+ b*^{2}*+ f (a, b)*

=

*a+ b + f (a, b)*

⇐⇒

*a*^{2}*+ b*^{2}*= a + b.*

The other functions*φ**i**for i∈ {6, 7, 8, 9, 10} are similar to φ*5.

According to the above results, we immediately obtain the following theorem.

* Theorem 3.1 Suppose thatφ(a, b) = ϕ*1

*(a, b) − ϕ*2

*(a, b) is an NCP function on R × R*

*and k and m are positive integers. Then,*

*φ(a, b)*^{k}

*m* *and*

*ϕ*1*(a, b)*_{2m+1}^{2k+1}

*− [ϕ*2*(a, b)]*^{2m+1}^{2k+1}*are NCP functions.*

*Proof Using k and m being positive integers and applying Lemma* 2.1, we have

*φ(a, b)*^{k}

*m* = 0

⇐⇒

*φ(a, b)*^{k}

*m*

*m*

= 0

⇐⇒

*φ(a, b)*_{k}

= 0

*⇐⇒ φ(a, b) = 0.*

Similarly, we have

*ϕ*1*(a, b)*_{2m+1}^{2k+1}

*− [ϕ*2*(a, b)]*^{2m+1}* ^{2k+1}* = 0

⇐⇒

*ϕ*1*(a, b)*_{2m+1}^{2k+1}

*= [ϕ*2*(a, b)]*^{2m+1}^{2k+1}

⇐⇒

*ϕ*1*(a, b)*_{2m+1}^{2k+1}*2m*+1

=

*[ϕ*2*(a, b)]*^{2m+1}^{2k+1}*2m*+1

⇐⇒

*ϕ*1*(a, b)]** ^{2k+1}*=

*ϕ*2*(a, b)]*^{2k+1}

*⇐⇒ ϕ*1*(a, b) = ϕ*2*(a, b)*

*⇐⇒ φ(a, b) = 0.*

The above arguments together with the assumption of*φ(a, b) being an NCP function yield*

the desired result.

−10

−5 0

5

10 −10 −5 0 5 10

−10 0 10 20 30 40

b−axis a−axis

z−axis

**Fig. 1 The surface of z***= φ*_{D−FB}*(a, b) and (a, b) ∈ [−10, 10] × [−10, 10]*

*Remark 3.2 We elaborate more about Theorem*3.1.

(a) Based on the existing well-known NCP functions, we can construct new NCP functions in light of Theorem3.1. This is a novel way to construct new NCP functions.

*(b) When k is a positive integer,*

*φ(a, b)**k*

is an NCP function. This means that perturbing
*the parameter k gives new NCP functions. In addition, ifφ(a, b) is an NCP function,*
*for any positive integer m,*

*φ(a, b)*^{k}

*m* is also an NCP function. Thus, we can determine
suitable and nice NCP functions among these functions according to their numerical
performance.

To close this section, we depict the surfaces of*φ*_{D−FB}^{p}*with different values of p so that we*
may have deeper insight for this new family of NCP functions. Figure1shows the surface if
*φ*_{D−FB}*(a, b) from which we see that it is convex. Figure*2presents the surface of*φ*_{D−FB}^{3} *(a, b)*
in which we see that it is neither convex nor concave as mentioned in Remark3.1(c). In
addition, the value of*φ*_{D−FB}^{p}*(a, b) is negative only when a > 0 and b > 0 as mentioned in*
Lemma3.1. The surfaces of*φ*_{D−FB}^{p}*with various values of p are shown in Fig.*3.

**4 Extending****φ**_{D−FB}^{p}**and****φ****NR**^{p}**to SOCCP**

In this section, we extend the new function*φ*_{D−FB}* ^{p}* and

*φ*

_{NR}

*to SOC setting. More specifically, we show that the function*

^{p}*φ*

_{D−FB}

*and*

^{p}*φ*

_{NR}

*are complementarity functions associated with*

^{p}*K*

*. In addition, we present the computing formulas for its Jacobian.*

^{n}**Proposition 4.1 Let**φ_{D−FB}^{p}*be defined by (5). Then,φ*_{D−FB}^{p}*is a complementarity function*
*associated withK*^{n}*, i.e., it satisfies*

*φ*_{D−FB}^{p}*(x, y) = 0 ⇐⇒ x ∈K*^{n}*, y ∈K*^{n}*, x, y = 0.*

−10

−5 0

5

10 −10 −5 0 5 10

−1

−0.5 0 0.5 1 1.5

x 10^{4}

b−axis a−axis

z−axis

**Fig. 2 The surface of z***= φ*_{D}^{3}_{−FB}*(a, b) and (a, b) ∈ [−10, 10] × [−10, 10]*

−5 0

−5 0 5 5

−1000

−500 0 500 1000 1500

a−axis b−axis

z−axis

**(a)***z = φ*^{3}_{D−FB}*(a, b)*

−5 0 5

−5

0

5

−1

−0.5 0 0.5 1 1.5

x 10^{5}

a−axis b−axis

z−axis

**(b)***z = φ*^{5}_{D−FB}*(a, b)*

−5

0

−5 5

0 5

−1

−0.5 0 0.5 1 1.5 x 107

a−axis b−axis

z−axis

**(c)** *z = φ*^{7}_{D−FB}*(a, b)*

−5

0

−5 5

0 5

−1

−0.5
0
0.5
1
1.5
x 10^{9}

a−axis b−axis

z−axis

**(d)***z = φ*^{9}_{D−FB}*(a, b)*
**Fig. 3 The surface of z***= φ*D* ^{p}*−FB

*(a, b) with different values of p.*

*Proof Sinceφ*_{D−FB}^{p}*(x, y) = 0 , we have*

*x*^{2}*+ y*^{2}*p*

*= (x + y)*^{p}*. Using p being a positive*
odd integer and applying Lemma2.2yield

*x*^{2}*+ y*^{2}

_{p}

*= (x + y)** ^{p}* ⇐⇒

*x*^{2}*+ y*^{2}*= x + y.*

It is known that*φ*FB*(x, y) :=*

*x*^{2}*+ y*^{2}*− (x + y) is a complementarity function associated*
with*K** ^{n}*. This indicates that

*φ*

_{D−FB}

*is a complementarity function associated with*

^{p}*K*

*.*

^{n}With similar technique, we can prove that*φ*_{NR}* ^{p}* can be extended as a complementarity
function for SOCCP.

**Proposition 4.2 The function**φ_{NR}* ^{p}* : R

*× R*

^{n}*→ R*

^{n}

^{n}*defined by*

*φ*_{NR}^{p}*(x, y) = x*^{p}*− [(x − y)*_{+}]* ^{p}* (7)

*is a complementarity function associated withK*

^{n}*, where p> 1 is a positive odd integer and*

*(·)*

_{+}

*means the projection ontoK*

^{n}*.*

*Proof From Lemma*2.2, we see that*φ*_{NR}^{p}*(x, y) = 0 if and only if x = (x − y)*+. On the
other hand, it is known that*φ*NR*(x, y) = x − (x − y)*+is a complementarity function for
*SOCCP, which implies x− (x − y)*+ *= 0 if and only if x ∈K*^{n}*, y* ∈*K** ^{n}*, and

*x, y = 0.*

Hence,*φ*_{NR}* ^{p}* is a complementarity function associated with

*K*

*.*

^{n}To compute the Jacobian of*φ*_{D−FB}* ^{p}* , we need to introduce some notations for convenience.

*For any x= (x*1*, x*2*) ∈ R × R*^{n−1}*and y= (y*1*, y*2*) ∈ R × R** ^{n−1}*, we define

*w(x, y) := x*^{2}*+ y*^{2}*= (w*1*(x, y), w*2*(x, y)) ∈ R × R*^{n}^{−1} and *v(x, y) := x + y.*

Then, it is clear that*w(x, y) ∈K** ^{n}* and

*λ*

*i*

*(w) ≥ 0, i = 1, 2.*

**Proposition 4.3 Let**φ_{D−FB}^{p}*be defined as in (5) and g*^{soc}*(x) = (*√

*|x|)*^{p}*, h*^{soc}*(x) = x*^{p}*are the*
*vector-valued functions corresponding to g(t) = |t|*^{2}^{p}*and h(t) = t*^{p}*for t∈ R, respectively.*

*Then,φ*_{D−FB}^{p}*is continuously differentiable at any(x, y) ∈ R** ^{n}*× R

^{n}*. Moreover, we have*

∇*x**φ*_{D−FB}^{p}*(x, y) = 2L**x**∇g*^{soc}*(w) − ∇h*^{soc}*(v),*

∇*y**φ*_{D−FB}^{p}*(x, y) = 2L**y**∇g*^{soc}*(w) − ∇h*^{soc}*(v),*

*wherew := w(x, y) = x*^{2}*+ y*^{2}*,v := v(x, y) = x + y, t → sign(t) is the sign function,*
*and*

*∇g*^{soc}*(w) =*

⎧⎪

⎨

⎪⎩
*p*

2*|w*1|^{p}^{2}^{−1}*· sign(w*1*)I* if *w*2= 0;

*b*1*(w)* *c*1*(w) ¯w*_{2}^{T}

*c*1*(w) ¯w*2 *a*1*(w)I + (b*1*(w) − a*1*(w)) ¯w*2*¯w*_{2}^{T}

if *w*2= 0;

*¯w*2 = *w*2

* w*2 *,*

*a*_{1}*(w) =* *|λ*2*(w)|*^{p}^{2} *− |λ*1*(w)|*^{p}^{2}
*λ*2*(w) − λ*1*(w)* *,*
*b*_{1}*(w) =* *p*

4

*|λ*2*(w)|*^{p}^{2}^{−1}*+ |λ*1*(w)|*^{p}^{2}^{−1}
*,*
*c* *(w) =* *p*

*|λ* *(w)|*^{p}^{2}^{−1}*− |λ* *(w)|*^{p}^{2}^{−1}
*,*

*and*

*∇h*^{soc}*(v) =*

⎧⎨

⎩

*pv*_{1}^{p−1}*I* if *v*2= 0;

*b*_{2}*(v) c*2*(v)¯v*_{2}^{T}

*c*2*(v)¯v*2 *a*2*(v)I + (b*2*(v) − a*2*(v)) ¯v*2*¯v*_{2}^{T}

if *v*2= 0; (8)

*¯v*2 = *v*2

* v*2 *,* (9)

*a*_{2}*(v) =* *(λ*2*(v))*^{p}*− (λ*1*(v))*^{p}

*λ*2*(v) − λ*1*(v)* *,* (10)

*b*_{2}*(v) =* *p*
2

*(λ*2*(v))*^{p}^{−1}*+ (λ*1*(v))*^{p}^{−1}

*,* (11)

*c*_{2}*(v) =* *p*
2

*(λ*2*(v))*^{p−1}*− (λ*1*(v))*^{p−1}

*,* (12)

*Proof From the definition ofφ*_{D−FB}* ^{p}* , it is clear to see that for any

*(x, y) ∈ R*

*× R*

^{n}*,*

^{n}*φ*

_{D−FB}

^{p}*(x, y) =*

*x*^{2}*+ y*^{2}

_{p}

*− (x + y)*^{p}

=

*|x*^{2}*+ y*^{2}|

_{p}

*− (x + y)*^{p}

=

*|λ*1*(w)|*^{p}^{2}*u*^{(1)}*(w) + |λ*2*(w)|*^{p}^{2}*u*^{(2)}*(w)*

−

*(λ*1*(v))*^{p}*u*^{(1)}*(v) + (λ*2*(v))*^{p}*u*^{(2)}*(v)*

*= g*^{soc}*(w) − h*^{soc}*(v).* (13)

*For p≥ 3, since both |t|*^{p}^{2}*and t** ^{p}*are continuously differentiable onR, by [Chen et al. (2004),
Proposition 5] and [Fukushima et al. (2002), Proposition 5.2], we know that the function g

^{soc}

*and h*

^{soc}are continuously differentiable onR

*. Moreover, it is clear that*

^{n}*w(x, y) = x*

^{2}

*+ y*

^{2}is continuously differentiable onR

*× R*

^{n}*, then we conclude that*

^{n}*φ*

_{D−FB}

*is continuously differ- entiable. Moreover, from the formula in [Chen et al. (2004), Proposition 4] and [Fukushima et al. (2002), Proposition 5.2], we have*

^{p}*∇g*^{soc}*(w) =*

⎧⎪

⎪⎨

⎪⎪

⎩
*p*

2*|w*1|^{p}^{2}^{−1}*· sign(w*1*)I* if *w*2= 0;

*b*1*(w)* *c*1*(w) ¯w*_{2}^{T}

*c*_{1}*(w) ¯w*2 *a*_{1}*(w)I + (b*1*(w) − a*1*(w)) ¯w*2*¯w*_{2}^{T}

if *w*2= 0;

*∇h*^{soc}*(v) =*

⎧⎪

⎨

⎪⎩

*pv*_{1}^{p−1}*I* if *v*2= 0;

*b*_{2}*(v) c*2*(v)¯v*^{T}_{2}

*c*2*(v)¯v*2 *a*2*(v)I + (b*2*(v) − a*2*(v)) ¯v*2*¯v*_{2}^{T}

if *v*2= 0;

where

*¯w*2 =_{ w}^{w}^{2}_{2}_{ }*,* *¯v*2= _{ v}^{v}^{2}_{2}_{ }
*a*_{1}*(w) =* ^{|λ}^{2}^{(w)|}

*p*

2*−|λ*1*(w)|*^{p}^{2}

*λ*2*(w)−λ*1*(w)* *,* *a*_{2}*(v) =* ^{(λ}^{2}^{(v))}_{λ}_{2}_{(v)−λ}^{p}^{−(λ}_{1}^{1}_{(v)}^{(v))}^{p}*,*
*b*1*(w) =* _{4}^{p}

*|λ*2*(w)|*^{2}^{p}^{−1}*+ |λ*1*(w)|*^{p}^{2}^{−1}

*, b*2*(v) =* _{2}^{p}

*(λ*2*(v))*^{p−1}*+ (λ*1*(v))*^{p−1}*,*
*c*_{1}*(w) =* _{4}^{p}

*|λ*2*(w)|*^{2}^{p}^{−1}*− |λ*1*(w)|*^{p}^{2}^{−1}

*, c*2*(v) =* _{2}^{p}

*(λ*2*(v))*^{p−1}*− (λ*1*(v))*^{p−1}*.*

*By taking differentiation on both sides about x and y for (13), respectively, and applying the*
chain rule for differentiation, it follows that

∇*x**φ*_{D−FB}^{p}*(x, y) = 2L**x**∇g*^{soc}*(w) − ∇h*^{soc}*(v),*

∇*y**φ*_{D−FB}^{p}*(x, y) = 2L**y**∇g*^{soc}*(w) − ∇h*^{soc}*(v).*

Hence, we complete the proof.

With Lemma2.2and Proposition4.1, we can construct more complementarity functions
for SOCCP which are variants of*φ*_{D−FB}^{p}*(x, y). More specifically, consider that k and m*
*are positive integers and f*^{soc}*(x, y) : R** ^{n}* × R

*→ R*

^{n}*is the vector-valued function cor-*

^{n}*responding to a given real-valued function f , the following functions are new variants of*

*φ*

_{D−FB}

^{p}*(x, y).*

*φ*1*(x, y) =*

*x*^{2}*+ y*^{2}*+ f*^{soc}*(x, y)*
_{2m+1}^{2k+1}

−

*x+ y + f*^{soc}*(x, y)*_{2m+1}^{2k+1}*.*

*φ*2*(x, y) =*

*x*^{2}*+ y*^{2}*− x − y*
^{k}

*m**.*

*φ*3*(x, y) =*

*x*^{2}*+ y*^{2}*− x + f*^{soc}*(x, y)*
_{2m+1}^{2k+1}

−

*y+ f*^{soc}*(x, y)*_{2m+1}^{2k+1}*.*

*φ*4*(x, y) =*

*x*^{2}*+ y*^{2}*− y + f*^{soc}*(x, y)*
_{2m+1}^{2k+1}

−

*x+ f*^{soc}*(x, y)*_{2m+1}^{2k+1}*.*

**Proposition 4.4 All the above functions **φ*i**for i∈ {1, 2, 3, 4} are complementarity functions*
*associated withK*^{n}*.*

*Proof The results follow from applying Lemma*2.2and Proposition4.1.

In general, for complementarity functions associated with*K** ^{n}*, we have the following
parallel result to Theorem3.1.

* Theorem 4.1 Suppose thatφ(x, y) = ϕ*1

*(x, y) − ϕ*2

*(x, y) is a complementarity function*

*associated withK*

^{n}*on*R

*× R*

^{n}

^{n}*, and k, m are positive integers. Then,*

*φ(x, y)*^{k}

*m* *and*

*ϕ*1*(x, y)*_{2m+1}^{2k+1}

*− [ϕ*2*(x, y)]*^{2m+1}^{2k+1}*are complementarity functions associated withK*^{n}*.*
*Proof According to k and m are positive integers and using Lemma* 2.2, we have

*φ(x, y)*^{k}

*m* = 0

⇐⇒

*φ(x, y)*^{k}

*m*

*m*

= 0

⇐⇒

*φ(x, y)*_{k}

= 0

*⇐⇒ φ(x, y) = 0.*

Similarly, we have

*ϕ*1*(x, y)*_{2m+1}^{2k+1}

*− [ϕ*2*(x, y)]*^{2m+1}* ^{2k+1}* = 0

⇐⇒

*ϕ* *(x, y)*_{2m+1}^{2k+1}

*= [ϕ* *(x, y)]*^{2k+1}

⇐⇒

*ϕ*1*(x, y)*_{2m+1}^{2k+1}_{2m+1}

=

*[ϕ*2*(x, y)]*^{2m+1}^{2k+1}_{2m+1}

⇐⇒

*ϕ*1*(x, y)]*^{2k}^{+1}=

*ϕ*2*(x, y)]*^{2k}^{+1}

*⇐⇒ ϕ*1*(x, y) = ϕ*2*(x, y)*

*⇐⇒ φ(x, y) = 0.*

From the above arguments and the assumption, the proof is complete.

*Remark 4.1 We elaborate more about Theorem*4.1.

(a) Based on the existing complementarity functions, we can construct new complementarity
functions associated with*K** ^{n}* in light of Theorem4.1.

*(b) When k is a positive odd integer,φ(x, y)** ^{k}* is a complementarity function associated
with

*K*

^{n}*. This means that perturbing the odd integer parameter k, we obtain the new*complementarity functions associated with

*K*

*. In addition, if*

^{n}*φ(x, y) is a complemen-*

*tarity function, then for any positive integer m,*

*φ(x, y)*^{k}

*m* is also a complementarity
function. We can determine nice complementarity functions associated with*K** ^{n}*among
these functions by their numerical performance.

Finally, we establish formula for Jacobian of*φ*_{NR}* ^{p}* and the smoothness of

*φ*

_{NR}

*. To this aim, we need the following technical lemma.*

^{p}* Lemma 4.1 Let p> 1. Then, the real-valued function f (t) = (t*+

*)*

^{p}*is continuously differ-*

*entiable with f*

^{}

*(t) = p(t*

_{+}

*)*

^{p−1}*where t*

_{+}

*= max{0, t}.*

*Proof By the definition of t*_{+}, we have
*f(t) = (t*+*)** ^{p}* =

*t*^{p}*if t≥ 0,*
0 *if t< 0,*
which implies

*f*^{}*(t) =*

*pt*^{p−1}*if t≥ 0,*
0 *if t< 0.*

*Then, it is easy to see that f*^{}*(t) = p(t*+*)*^{p}^{−1}*is continuous for p> 1.*

**Proposition 4.5 Let**φ_{NR}^{p}*be defined as in (7) and h*^{soc}*(x) = x*^{p}*, l*^{soc}*(x) = (x*+*)*^{p}*be the*
*vector-valued functions corresponding to the real-valued functions h(t) = t*^{p}*and l(t) =*
*(t*+*)*^{p}*, respectively. Then,φ*_{NR}^{p}*is continuously differentiable at any(x, y) ∈ R** ^{n}*× R

^{n}*, and*

*its Jacobian is given by*

∇*x**φ*_{NR}^{p}*(x, y) = ∇h*^{soc}*(x) − ∇l*^{soc}*(x − y),*

∇*y**φ*_{NR}^{p}*(x, y) = ∇l*^{soc}*(x − y),*
*where∇h*^{soc}*satisfies (8)–(12) and*

*∇l*^{soc}*(u) =*

⎧⎨

⎩

*p((u*1*)*_{+}*)*^{p}^{−1}*I* *if u*_{2}= 0;

*b*3*(u) c*3*(u) ¯u*^{T}_{2}

*c*_{3}*(u) ¯u*2*a*_{3}*(u)I + (b*3*(u) − a*3*(u)) ¯u*2*¯u*^{T}_{2}

*if u*_{2}= 0;

*¯u*2 = *u*2

* u*2 *,*