Discovery of new complementarity functions for NCP and SOCCP

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

Discovery of new complementarity functions for NCP and SOCCP

Peng-Fei Ma¹ · Jein-Shan Chen² · Chien-Hao Huang² · Chun-Hsu Ko³

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

mathpengfeima@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∈ Rⁿ such that

x≥ 0, F(x) ≥ 0, x, F(x) = 0,

where·, · is the Euclidean inner product and F = (F1, . . . , Fn)^T is a map fromRⁿtoRⁿ. For solving NCP, the so-called NCP functionφ : R²→ 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 aζ ∈ Rⁿsuch that

ζ ∈K, F(ζ ) ∈K, ζ, F(ζ ) = 0,

where F: Rⁿ → Rⁿis a map andKis the Cartesian product of second-order cones (SOC), also called Lorentz cones (Faraut and Korányi1994). In other words,Kis expressed as:

K=Kⁿ¹× · · · ×K^nm, where m, n1, . . . , nm≥ 1, n1+ · · · + nm= n, and

Kⁿⁱ := {(x1, x2) ∈ R × Rⁿⁱ⁻¹| x2 ≤ x1},

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^Tx

subject to Ax = b, x ∈K,

where A ∈ R^m×nhas full row rank, b ∈ R^m and c ∈ Rⁿ. 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 functionψ : Rⁿ× Rⁿ→ R+such that

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

ζ ∈Rminⁿ 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 withKⁿ).

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 functionψ 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 : R² → R defined by see (Fischer1992,1997)

φ_FB(a, b) =

a²+ b²− (a + b), and the squared norm of Fischer–Burmeister function given by

ψFB(a, b) = 1

2φFB(a, b)².

In addition, the generalized Fischer–Burmeister functionφp: R²→ 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 functionφp

is a natural “continuous extension” ofφFB, in which the 2-norm inφFB(a, b) is replaced by general p-norm. In other words,φp : R² → R is defined as:

φp(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–Burmeisterφpgiven 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 functionφNR: R²→ R given by

φ_NR(a, b) = a − (a − b)₊= min{a, b}.

Recently, Chen et al. propose a family of generalized natural residual functionsφ_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 functionφNR, i.e., φ_NR¹ (a, b) = a − (a − b)+= min{a, b} = φNR(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 ofφ_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 byφ_D−FB^p ):

φ_D−FB^p (a, b) =

a²+ b²_p

− (a + b)^p, (4)

where p > 1 is a positive odd integer and (a, b) ∈ R². Notice that when p = 1, φ_D−FB^p reduces to the Fischer–Burmeister function. In Sect.3, we will see thatφ_D−FB^p is an NCP 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 ofφ_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 ofφ_D−FB^p (a, b) with various values of p.

In Sect.4, we show that the new functionφ_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 defineφ_D−FB^p : Rⁿ× Rⁿ → Rⁿby

φ_D−FB^p (x, y) =

x²+ y²

_p

− (x + y)^p, (5)

where p > 1 is a positive odd integer, x ∈ Rⁿ, y ∈ Rⁿ, x² = x ◦ x is the Jordan product of x with itself and√

x with x ∈ Kⁿ 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 Kⁿ 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 ofφ_D−FB^p are also shown to be complementarity functions for SOCCP.

Throughout the paper, we assumeK=Kⁿfor simplicity and all the analysis can be carried over to the case whereKis a product of SOC without difficulty. The following notations will be used. The identity matrix is denoted by I andRⁿ denotes the space of n-dimensional real column vectors. For any given x ∈ Rⁿ with n > 1, we write x = (x1, x2) where x₁is the first entry of x and x₂ is the subvector that consists of the remaining entries. For every differentiable function f : Rⁿ → R, ∇ f (x) denotes the gradient of f at x. For every differentiable mapping F: Rⁿ → R^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→0α

= 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= (x1, x2), y = (y1, y2) ∈ R × Rⁿ⁻¹, we define the Jordan product associated withKⁿas:

x◦ y := (x, y, y1x2+ x1y2).

The identity element under this product is e:= (1, 0, . . . , 0)^T ∈ Rⁿ. For any given x = (x1, x2) ∈ R × Rⁿ⁻¹, we define symmetric matrix

L_x :=

x1 x₂^T x2 x1I

which can be viewed as a linear mapping fromRⁿtoRⁿ. It is easy to verify that L_xy= x ◦ y, ∀x ∈ Rⁿ.

Moreover, we have L_xis invertible for xKⁿ 0 and

L⁻¹_x = 1 det(x)

⎡

⎣x₁ −x₂^T

−x2 det(x) x1

I+ 1 x1

x2x₂^T

⎤

⎦ ,

where det(x) = x₁²− x2 ². We next recall from Chen and Pan (2012); Fukushima et al.

(2002) that each x = (x1, x2) ∈ R × Rⁿ⁻¹admits a spectral factorization, associated with Kⁿ, of the form

x = λ1u⁽¹⁾+ λ2u⁽²⁾, (6)

whereλ1, λ2and u⁽¹⁾, u⁽²⁾are the spectral values and the associated spectral vectors of x given by

λi = x1+ (−1)ⁱ x2 ,

u⁽ⁱ⁾=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 2



1, (−1)ⁱ x2

x2



if x2= 0;

1 2



1, (−1)ⁱw2



if x₂= 0,

for i = 1, 2, with w2 being any vector in Rⁿ⁻¹ satisfying w2 = 1. If x2 = 0, the factorization is unique.

Given a real-valued function g : R → R, we can define a vector-valued SOC function g^soc: Rⁿ→ Rⁿby

g^soc(x) := g(λ1)u⁽¹⁾+ g(λ2)u⁽²⁾.

If g is defined on a subset ofR, then g^socis defined on the corresponding subset ofRⁿ. The definition of g^socis unambiguous whether x₂= 0 or x2= 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 := (λ1(x))^pu⁽¹⁾+ (λ2(x))^pu⁽²⁾, ∀x ∈ Rⁿ

√x :=√

λ1(x)u⁽¹⁾+√

λ2(x)u⁽²⁾, ∀x ∈Kⁿ.

We will see that the above two vector-valued functions play a role, showing thatφ_D−FB^p given as in (5) is well defined in the SOC setting for any x, y ∈ Rⁿ. Note that the other way to define x^pand√

x is through Jordan product. In other words, x^prepresents x◦ x ◦ · · · ◦ x for p-times and√

x ∈Kⁿ 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^pif 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 haveu

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 = (x1, x2), y = (y1, y2) ∈ R × Rⁿ⁻¹, suppose that x^pand y^prepresent x◦ x ◦ · · · ◦ x and y ◦ y ◦ · · · ◦ y for p-times, respectively. Then, x^p= y^pif and only if x= y.

Proof “⇐” This direction is trivial.

“⇒” Suppose that x^p = y^p. By the spectral decomposition (6), we write x = λ1(x)u⁽¹⁾x + λ2(x)u⁽²⁾x ,

y = λ1(y)u⁽¹⁾y + λ2(y)u⁽²⁾y .

Then, x^p = (λ1(x))^pu⁽¹⁾_x +(λ2(x))^pu⁽²⁾_x and y^p = (λ1(y))^pu⁽¹⁾_y +(λ2(y))^pu⁽²⁾_y . Since x^p= y^p and eigenvalues are unique, we obtain(λ1(x))^p = (λ1(y))^pand(λ2(x))^p = (λ2(y))^p. By Lemma2.1, this impliesλ1(x) = λ1(y) and λ2(x) = λ2(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^pand u⁽¹⁾_x + u⁽²⁾x = u⁽¹⁾y + u⁽²⁾y , we get

(λ1(x))^p− (λ2(x))^p

(u⁽¹⁾x − u⁽¹⁾y ) = 0.

If(λ1(x))^p= (λ2(x))^p, we haveλ1(x) = λ2(x) and λ1(y) = λ2(y), that is, x = λ1(x)e = y.

Otherwise, if(λ1(x))^p= (λ2(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 functionφ_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 ofφ_D−FB^p with various values of p to have more insight for this new family of NCP functions.

Proposition 3.1 Letφ_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√

a²+ b²_p

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

a²+ b²_p

= (a + b)^p ⇐⇒ 

a²+ b²= a + b.

It is well known that√

a²+ b² = a + b is equivalent to a, b ≥ 0, ab = 0 because φFB is an NCP function. This shows thatφ_D−FB^p (a, b) = 0 implies a, b ≥ 0, ab = 0. The converse direction is trivial. Thus, we prove thatφ_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 ofφ_D−FB^p is symmetric, i.e.,φ_D−FB^p (a, b) = φ_D−FB^p (b, a).

(c) The functionφ_D−FB^p (a, b) is positive homogenous of degree p, i.e., φ_D−FB^p (α(a, b)) = α^pφ_D−FB^p (a, b) for α ≥ 0.

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

5³− 7³= φ_D−FB³ (3, 4) > 1

2φ³_D−FB(0, 0) +1

2φ³_D−FB(6, 8)

= 1

2× 0 +1 2

10³− 14³

= 4 5³− 7³ and

0= φ_D−FB³ (0, 0) < 1

2φ_D−FB³ (−2, 0) +1

2φ_D−FB³ (2, 0) = 1

2× 16 +1

2× 0 = 8.

Proposition 3.2 Letφ_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(√

a²+ b²)^p−2− (a + b)^p−1 b(√

a²+ b²)^p−2− (a + b)^p−1

.

(b) For p> 3, φ_D−FB^p is twice continuously differentiable with

∇²φ_D−FB^p (a, b) =

⎡

⎢⎢

⎣

∂²φ_D−FB^p

∂a²

∂²φ_D−FB^p

∂a∂b

∂²φ_D−FB^p

∂b∂a

∂²φ_D−FB^p

∂b²

⎤

⎥⎥

⎦ ,

where

∂²φ_D−FB^p

∂a² = p

(p − 1)a²+ b² (

a²+ b²)^p−4− (p − 1)(a + b)^p−2 ,

∂²φ_D−FB^p

∂a∂b = p

(p − 2)ab(

a²+ b²)^p−4− (p − 1)(a + b)^p−2

= ∂²φ_D−FB^p

∂b∂a ,

∂²φ_D−FB^p

∂b² = p

a²+ (p − 1)b² (

a²+ b²)^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 ofφ_D−FB^p . Indeed, analogous to those functions in Sun and Qi (1999), the variants ofφ_D−FB^p as below can be verified being NCP functions.

φ1(a, b) = φ_D−FB^p (a, b) − α(a)+(b)+, α > 0.

φ2(a, b) = φ_D−FB^p (a, b) − α ((a)+(b)+)², α > 0.

φ3(a, b) = [φ_D−FB^p (a, b)]²+ α ((ab)+)⁴, α > 0.

φ4(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 Letφ_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

⇐⇒

a²+ b² < a + b

⇐⇒

a²+ b²p

< (a + b)^p

⇐⇒ φ_D−FB^p (a, b) < 0,

which proves the desired result. 

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

Proof Applying Lemma3.1, 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 ofφ_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 ofφ_D−FB^p .

φ5(a, b) =

g(a, b)

a²+ b²+ f (a, b)2m+1^2k+1

− g(a, b)

a+ b + f (a, b)_2m+1^2k+1 . φ6(a, b) =

g(a, b)(

a²+ b²− a − b)^k

m . φ7(a, b) =

g(a, b)(

a²+ b²− a + f (a, b))_2m+1^2k+1

− [g(a, b)(b + f (a, b))]^2m+12k+1 . φ8(a, b) =

g(a, b)(

a²+ b²− a + f (a, b))_2m+1^2k+1

− [g(a, b)(b + f (a, b))]^2m+12k+1 . φ9(a, b) = e^φi(a,b)− 1 where i = 5, 6, 7, 8.

φ10(a, b) = ln(|φi(a, b)| + 1) where i = 5, 6, 7, 8.

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

Proof This is an immediate consequence of Propositions3.1,3.1,3.3. By Lemma2.1and g(a, b) = 0 for a, b ∈ R, we have

φ5(a, b) = 0

⇐⇒

g(a, b)

a²+ b²+ f (a, b)2m+1^2k+1

= g(a, b)

a+ b + f (a, b)_2m+1^2k+1

⇐⇒ 

g(a, b)

a²+ b²+ 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)

a²+ b²+ f (a, b)^2k+1

= g(a, b)

a+ b + f (a, b)_2k+1

⇐⇒ g(a, b)

a²+ b²+ f (a, b)

= g(a, b)

a+ b + f (a, b)

⇐⇒

a²+ b²+ f (a, b)

=

a+ b + f (a, b)

⇐⇒

a²+ b²= a + b.

The other functionsφifor i∈ {6, 7, 8, 9, 10} are similar to φ5. 

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

Theorem 3.1 Suppose thatφ(a, b) = ϕ1(a, b) − ϕ2(a, b) is an NCP function on R × R and k and m are positive integers. Then,

φ(a, b)^k

m and

ϕ1(a, b)_2m+1^2k+1

− [ϕ2(a, b)]^2m+12k+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

ϕ1(a, b)_2m+1^2k+1

− [ϕ2(a, b)]^2m+12k+1 = 0

⇐⇒

ϕ1(a, b)_2m+1^2k+1

= [ϕ2(a, b)]^2m+12k+1

⇐⇒

ϕ1(a, b)_2m+1^2k+12m+1

=

[ϕ2(a, b)]^2m+12k+12m+1

⇐⇒

ϕ1(a, b)]^2k+1=

ϕ2(a, b)]^2k+1

⇐⇒ ϕ1(a, b) = ϕ2(a, b)

⇐⇒ φ(a, b) = 0.

The above arguments together with the assumption ofφ(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 Theorem3.1.

(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 ofφ_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. Figure2presents the surface ofφ_D−FB³ (a, b) in which we see that it is neither convex nor concave as mentioned in Remark3.1(c). In addition, the value ofφ_D−FB^p (a, b) is negative only when a > 0 and b > 0 as mentioned in Lemma3.1. The surfaces ofφ_D−FB^p with various values of p are shown in Fig.3.

4 Extendingφ_D−FB^p andφNR^p to SOCCP

In this section, we extend the new functionφ_D−FB^p andφ_NR^p to SOC setting. More specifically, we show that the functionφ_D−FB^p andφ_NR^p are complementarity functions associated withKⁿ. In addition, we present the computing formulas for its Jacobian.

Proposition 4.1 Letφ_D−FB^p be defined by (5). Then,φ_D−FB^p is a complementarity function associated withKⁿ, i.e., it satisfies

φ_D−FB^p (x, y) = 0 ⇐⇒ x ∈Kⁿ, y ∈Kⁿ, x, y = 0.

−10

−5 0

5

10 −10 −5 0 5 10

−1

−0.5 0 0.5 1 1.5

x 10⁴

b−axis a−axis

z−axis

Fig. 2 The surface of z= φ_D³_−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 = φ³_D−FB(a, b)

−5 0 5

−5

0

5

−1

−0.5 0 0.5 1 1.5

x 10⁵

a−axis b−axis

z−axis

(b)z = φ⁵_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 = φ⁷_D−FB(a, b)

−5

0

−5 5

0 5

−1

−0.5 0 0.5 1 1.5 x 10⁹

a−axis b−axis

z−axis

(d)z = φ⁹_D−FB(a, b) Fig. 3 The surface of z= φD^p−FB(a, b) with different values of p.

Proof Sinceφ_D−FB^p (x, y) = 0 , we have

x²+ y²p

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



x²+ y²

_p

= (x + y)^p ⇐⇒ 

x²+ y²= x + y.

It is known thatφFB(x, y) :=

x²+ y²− (x + y) is a complementarity function associated withKⁿ. This indicates thatφ_D−FB^p is a complementarity function associated withKⁿ. 

With similar technique, we can prove thatφ_NR^p can be extended as a complementarity function for SOCCP.

Proposition 4.2 The functionφ_NR^p : Rⁿ× Rⁿ→ Rⁿdefined by

φ_NR^p (x, y) = x^p− [(x − y)₊]^p (7) is a complementarity function associated withKⁿ, where p> 1 is a positive odd integer and (·)₊means the projection ontoKⁿ.

Proof From Lemma2.2, we see thatφ_NR^p (x, y) = 0 if and only if x = (x − y)+. On the other hand, it is known thatφNR(x, y) = x − (x − y)+is a complementarity function for SOCCP, which implies x− (x − y)+ = 0 if and only if x ∈Kⁿ, y ∈Kⁿ, andx, y = 0.

Hence,φ_NR^p is a complementarity function associated withKⁿ. 

To compute the Jacobian ofφ_D−FB^p , we need to introduce some notations for convenience.

For any x= (x1, x2) ∈ R × Rⁿ⁻¹and y= (y1, y2) ∈ R × Rⁿ⁻¹, we define

w(x, y) := x²+ y²= (w1(x, y), w2(x, y)) ∈ R × Rⁿ⁻¹ and v(x, y) := x + y.

Then, it is clear thatw(x, y) ∈Kⁿ andλi(w) ≥ 0, i = 1, 2.

Proposition 4.3 Letφ_D−FB^p be defined as in (5) and g^soc(x) = (√

|x|)^p, h^soc(x) = x^pare the vector-valued functions corresponding to g(t) = |t|^2p and h(t) = t^pfor t∈ R, respectively.

Then,φ_D−FB^p is continuously differentiable at any(x, y) ∈ Rⁿ× Rⁿ. Moreover, we have

∇xφ_D−FB^p (x, y) = 2Lx∇g^soc(w) − ∇h^soc(v),

∇yφ_D−FB^p (x, y) = 2Ly∇g^soc(w) − ∇h^soc(v),

wherew := w(x, y) = x²+ y²,v := v(x, y) = x + y, t → sign(t) is the sign function, and

∇g^soc(w) =

⎧⎪

⎨

⎪⎩ p

2|w1|^p2−1· sign(w1)I if w2= 0;

b1(w) c1(w) ¯w₂^T

c1(w) ¯w2 a1(w)I + (b1(w) − a1(w)) ¯w2¯w₂^T

if w2= 0;

¯w2 = w2

w2 ,

a₁(w) = |λ2(w)|^p2 − |λ1(w)|^p2 λ2(w) − λ1(w) , b₁(w) = p

4

|λ2(w)|^p2−1+ |λ1(w)|^p2−1 , c (w) = p

|λ (w)|^p2−1− |λ (w)|^p2−1 ,

and

∇h^soc(v) =

⎧⎨

⎩

pv₁^p−1I if v2= 0;

b₂(v) c2(v)¯v₂^T

c2(v)¯v2 a2(v)I + (b2(v) − a2(v)) ¯v2¯v₂^T

if v2= 0; (8)

¯v2 = v2

v2 , (9)

a₂(v) = (λ2(v))^p− (λ1(v))^p

λ2(v) − λ1(v) , (10)

b₂(v) = p 2

(λ2(v))^p−1+ (λ1(v))^p−1

, (11)

c₂(v) = p 2

(λ2(v))^p−1− (λ1(v))^p−1

, (12)

Proof From the definition ofφ_D−FB^p , it is clear to see that for any(x, y) ∈ Rⁿ× Rⁿ, φ_D−FB^p (x, y) =

x²+ y²

_p

− (x + y)^p

=

|x²+ y²|

_p

− (x + y)^p

=

|λ1(w)|^p2u⁽¹⁾(w) + |λ2(w)|^p2u⁽²⁾(w)

−

(λ1(v))^pu⁽¹⁾(v) + (λ2(v))^pu⁽²⁾(v)

= g^soc(w) − h^soc(v). (13)

For p≥ 3, since both |t|^p2and t^pare 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^socare continuously differentiable onRⁿ. Moreover, it is clear thatw(x, y) = x²+ y²is continuously differentiable onRⁿ× Rⁿ, then we conclude thatφ_D−FB^p 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

∇g^soc(w) =

⎧⎪

⎪⎨

⎪⎪

⎩ p

2|w1|^p2−1· sign(w1)I if w2= 0;

b1(w) c1(w) ¯w₂^T

c₁(w) ¯w2 a₁(w)I + (b1(w) − a1(w)) ¯w2¯w₂^T

if w2= 0;

∇h^soc(v) =

⎧⎪

⎨

⎪⎩

pv₁^p−1I if v2= 0;

b₂(v) c2(v)¯v^T₂

c2(v)¯v2 a2(v)I + (b2(v) − a2(v)) ¯v2¯v₂^T

if v2= 0;

where

¯w2 = _w^w2₂ , ¯v2= _v^v2₂ a₁(w) = ^|λ2(w)|

p

2−|λ1(w)|^p2

λ2(w)−λ1(w) , a₂(v) = ^(λ2(v))_λ2(v)−λ^p−(λ₁¹_(v)^(v))p, b1(w) = ₄^p

|λ2(w)|^2p−1+ |λ1(w)|^p2−1

, b2(v) = ₂^p

(λ2(v))^p−1+ (λ1(v))^p−1 , c₁(w) = ₄^p

|λ2(w)|^2p−1− |λ1(w)|^p2−1

, c2(v) = ₂^p

(λ2(v))^p−1− (λ1(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

∇xφ_D−FB^p (x, y) = 2Lx∇g^soc(w) − ∇h^soc(v),

∇yφ_D−FB^p (x, y) = 2Ly∇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 ofφ_D−FB^p (x, y). More specifically, consider that k and m are positive integers and f^soc(x, y) : Rⁿ × Rⁿ → Rⁿ is the vector-valued function cor- responding to a given real-valued function f , the following functions are new variants of φ_D−FB^p (x, y).

φ1(x, y) = 

x²+ y²+ f^soc(x, y) _2m+1^2k+1

−

x+ y + f^soc(x, y)_2m+1^2k+1 .

φ2(x, y) = 

x²+ y²− x − y ^k

m.

φ3(x, y) = 

x²+ y²− x + f^soc(x, y) _2m+1^2k+1

−

y+ f^soc(x, y)_2m+1^2k+1 .

φ4(x, y) = 

x²+ y²− 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 φifor i∈ {1, 2, 3, 4} are complementarity functions associated withKⁿ.

Proof The results follow from applying Lemma2.2and Proposition4.1. 

In general, for complementarity functions associated withKⁿ, we have the following parallel result to Theorem3.1.

Theorem 4.1 Suppose thatφ(x, y) = ϕ1(x, y) − ϕ2(x, y) is a complementarity function associated withKⁿ on Rⁿ × Rⁿ, and k, m are positive integers. Then, 

φ(x, y)^k

m and

ϕ1(x, y)_2m+1^2k+1

− [ϕ2(x, y)]^2m+12k+1 are complementarity functions associated withKⁿ. 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

ϕ1(x, y)_2m+1^2k+1

− [ϕ2(x, y)]^2m+12k+1 = 0

⇐⇒

ϕ (x, y)_2m+1^2k+1

= [ϕ (x, y)]^2k+1

⇐⇒

ϕ1(x, y)_2m+1^2k+1_2m+1

=

[ϕ2(x, y)]^2m+12k+1_2m+1

⇐⇒

ϕ1(x, y)]^2k+1=

ϕ2(x, y)]^2k+1

⇐⇒ ϕ1(x, y) = ϕ2(x, y)

⇐⇒ φ(x, y) = 0.

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

Remark 4.1 We elaborate more about Theorem4.1.

(a) Based on the existing complementarity functions, we can construct new complementarity functions associated withKⁿ in light of Theorem4.1.

(b) When k is a positive odd integer,φ(x, y)^k is a complementarity function associated withKⁿ. This means that perturbing the odd integer parameter k, we obtain the new complementarity functions associated withKⁿ. In addition, ifφ(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 withKⁿamong these functions by their numerical performance.

Finally, we establish formula for Jacobian ofφ_NR^p and the smoothness ofφ_NR^p . To this aim, we need the following technical lemma.

Lemma 4.1 Let p> 1. Then, the real-valued function f (t) = (t+)^pis continuously differ- entiable with f^(t) = p(t₊)^p−1where 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−1is continuous for p> 1. 

Proposition 4.5 Letφ_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) ∈ Rⁿ× Rⁿ, and its Jacobian is given by

∇xφ_NR^p (x, y) = ∇h^soc(x) − ∇l^soc(x − y),

∇yφ_NR^p (x, y) = ∇l^soc(x − y), where∇h^socsatisfies (8)–(12) and

∇l^soc(u) =

⎧⎨

⎩

p((u1)₊)^p−1I if u₂= 0;

b3(u) c3(u) ¯u^T₂

c₃(u) ¯u2a₃(u)I + (b3(u) − a3(u)) ¯u2¯u^T₂

if u₂= 0;

¯u2 = u2

u2 ,