whereas the natural residual function ϕNR:R2→ R is given by ϕNR(a, b) = a− (a − b)+= min{a, b}.
Recently, the generalized Fischer-Burmeister function ϕp
FB : R2 → R, which includes the Fischer-Burmeister as a special case, was considered in [1, 2, 3, 6, 17, 25]. Indeed, the function ϕp
FB is a natural extension of the popular ϕFB function, in which the 2-norm in ϕFB(a, b) is replaced by general p-norm. In other words, ϕp
FB is defined as ϕp
FB(a, b) =∥(a, b)∥p− (a + b), p > 1 (1.1) and its geometric view is depicted in [25]. The effect of perturbing p for different kinds of algorithms are investigated in [4, 5, 6, 7, 8]. To the contrast, “Is there an extension of natural residual function?” and “If yes, how does the extension of ϕNR look like?” remain open. As mentioned, there exist many NCP-functions which are variants of ϕNR, but there is no literature talking about the extension of natural residual function. The main hurdle lies on lacking continuous norm generalization like what we do for ϕp
FB. In this paper, we give an affirmative answer to the long-standing open question. In fact, the main ideas rely on “discrete generalization”, not the “continuous generalization”. More specifically, the generalized natural residual function, denoted by ϕpNR, is defined by
ϕp
NR(a, b) = ap− (a − b)p+ with p > 1 being a positive odd integer, (1.2) where (a−b)p+= [(a−b)+]pand (a−b)+= max{a−b, 0}. Here p being a positive odd integer is necessary (that is, we require that p = 2k + 1, where k = 1, 2, 3,· · · ). We will explain this in Section 2. Notice that when p = 1, ϕp
NR reduces to the natural residual function ϕNR, i.e., when k = 0, it corresponds to
ϕ1
NR(a, b) = a− (a − b)+= min{a, b} = ϕNR(a, b).
This is why we call it the “generalized natural residual function”. We point it out again that the considered extension is based on “discrete generalization”. For different values of p, it is no longer an NCP-function. A special feature of ϕp
NR is that it is twice differentiable which will be proved in Section 2. It is well known that the generalized Fischer-Burmeister ϕp
FB given as in (1.1) is not differentiable, while ∥ϕpFB(a, b)∥2 is differentiable everywhere.
This yields that ∥ϕpFB(a, b)∥2 is usually adapted when using merit function approach and ϕpFB(a, b) is employed when applying nonsmooth function approach. Compared to the non- differentiability of ϕpFB, the function ϕpNRwith p = 2k + 1 is twice continuously differentiable.
This feature enables that many methods like Newton method can be employed directly for solving NCP. This is a new discovery to the literature and is the main contribution of this paper.
2 Generalized Natural Residual Function
In this section, we show that the function ϕpNR defined as in (1.2) is an NCP-function and present its twice differentiability.
Proposition 2.1. Let ϕp
NR be defined as in (1.2). Then, ϕp
NR is an NCP-function.
Proof. First, we note that for any fixed real number ξ≥ 0 and odd integer p, the equation tp− ξp= 0 has exactly one real solution t = ξ because the function tp is strictly monotone.
Thus, we observe that
ϕpNR(a, b) = 0
⇐⇒ ap− (a − b)p+= 0
⇐⇒ a − (a − b)+= 0
⇐⇒ min{a, b} = 0
⇐⇒ a, b ≥ 0, ab = 0.
This shows that ϕpNR is an NCP-function.
Remarks: We elaborate more about the function ϕpNR.
(a) For p being an even integer, ϕpNR is not a NCP-function. A counterexample is given as below.
ϕ2NR(−2, −4) = (−2)2− (−2 + 4)2+= 0.
(b) The function ϕpNR is neither convex nor concave function. To see this, taking p = 3 and using the following argument verify the assertion.
−1 = ϕ3NR(−1, −1) > 1
2ϕ3NR(−2, −1) +1
2ϕ3NR(0,−1) =−8 2 +−1
2 =−9 2 Proposition 2.2. Let p > 1 be a positive odd integer. Then, we have
[(a− b)+]p= [(a− b)p]+, (2.1) and hence
ϕpNR(a, b) = ap− [(a − b)+]p= ap− [(a − b)p]+.
Proof. For any α∈ R, we know that [α]+= 12(α +|α|). In addition, looking the coefficients of the binomial (1 + x)p, we have
∑p j=0,even
C(p, j) =
∑p j=0,odd
C(p, j) = 1 2
∑p j=0
C(p, j) = 2p
2 = 2p−1. These two facts lead to
[(a− b)+]p
= 1
2p(a− b + |a − b|)p
= 1
2p
∑p j=0
C(p, j)|a − b|j(a− b)p−j
= 1
2p
∑p j=0,even
C(p, j)|a − b|j(a− b)p−j+
∑p
j=0,odd
C(p, j)|a − b|j(a− b)p−j
= 1
2p
∑p j=0,even
C(p, j)(a− b)p+
∑p
j=0,odd
C(p, j)|a − b|(a − b)p−1
= 1
2p
(2p−1(a− b)p+ 2p−1|a − b|(a − b)p−1)
= 1
2
((a− b)p+|a − b|(a − b)p−1)
= [(a− b)p]+
where the last equality holds because p is an positive odd integer. Thus, the proof is complete.
Remark: In Proposition 2.2, note that the equality in (2.1) holds only when p is positive odd integer. When p is an even integer, [(a− b)+]p ̸= [(a − b)p]+. This also explains that requiring p being an positive odd integer is necessary in the definition of ϕpNR. Next, we provide an alternative expression for ϕpNR and show its twice differentiability. To this end, we need a technical lemma.
Lemma 2.3. Let p > 1. Then,
(a) the function f (t) =|t|p is differentiable and f′(t) = p sgn(t)|t|p−1; (b) the function f (t) = tp|t| is differentiable and f′(t) = (p + 1)tp−1|t|.
Proof. The proofs are straightforward which are omitted here.
Proposition 2.4. Let p = 2k + 1 where k = 1, 2, 3· · · . Then, we have (a) ϕp
NR(a, b) = a2k+1−12(
(a− b)2k+1+ (a− b)2k|a − b|)
; (b) ϕp
NR is continuously differentiable with
∇ϕpNR(a, b)
= p
[ ap−1− (a − b)p−2(a− b)+
(a− b)p−2(a− b)+
]
;
(c) ϕpNR is twice continuously differentiable with
∇2ϕpNR(a, b)
= p(p− 1)
[ ap−2− (a − b)p−3(a− b)+ (a− b)p−3(a− b)+
(a− b)p−3(a− b)+ −(a − b)p−3(a− b)+
] .
Proof. (a) This alternative expression follows from Proposition 2.2.
(b) From Lemma 2.3, we compute that
∂ϕpNR
∂a (a, b)
= ∂
∂a (
a2k+1−1
2((a− b)2k+1+ (a− b)2k|a − b|
)
= (2k + 1)a2k−(2k + 1)
2 (a− b)2k−(2k + 1)
2 (a− b)2k−1|a − b|
and
∂ϕp
NR
∂b (a, b)
= ∂
∂b (
a2k+1−1
2((a− b)2k+1+ (a− b)2k|a − b|
)
= (2k + 1)
2 (a− b)2k+(2k + 1)
2 (a− b)2k−1|a − b|.
Hence, we obtain
∇ϕpNR(a, b)
= 2k + 1 2
[ 2a2k− (a − b)2k− (a − b)2k−1|a − b|
(a− b)2k+ (a− b)2k−1|a − b|
]
= 2k + 1 2
[ 2a2k− 2(a − b)2k−1(a− b)+
2(a− b)2k−1(a− b)+
]
= p
[ ap−1− (a − b)p−2(a− b)+
(a− b)p−2(a− b)+
]
which proves part (b).
(c) Similarly, with Lemma 2.3 again, the Hessian matrix can be calculated as below.
∇2ϕp
NR(a, b)
= k(2k + 1)
[ 2a2k−1− (a − b)2k−1− (a − b)2k−2|a − b| (a− b)2k−1+ (a− b)2k−2|a − b|
(a− b)2k−1+ (a− b)2k−2|a − b| −(a − b)2k−1− (a − b)2k−2|a − b|
]
= p(p− 1)
[ ap−2− (a − b)p−3(a− b)+ (a− b)p−3(a− b)+
(a− b)p−3(a− b)+ −(a − b)p−3(a− b)+
]
Finally, we present some other variants of ϕpNR. Indeed, analogous to those functions in [24], the variants of ϕpNR as below can be verified being NCP-functions.
φ1(a, b) = ϕpNR(a, b) + α(a)+(b)+, α > 0.
φ2(a, b) = ϕp
NR(a, b) + α ((a)+(b)+)2, α > 0.
φ3(a, b) = ( ϕp
NR(a, b))2
+ α ((ab)+)4, α > 0.
φ4(a, b) = ( ϕp
NR(a, b))2
+ α ((ab)+)2, α > 0.
Lemma 2.5. The value of ϕpNR(a, b) is positive only in the first quadrant, i.e., ϕpNR(a, b) > 0 if and only if a > 0, b > 0.
Proof. We know that f (t) = tp is a strictly increasing function since p is odd. Using this fact yields
a > 0, b > 0
⇐⇒ a + b > |a − b|
⇐⇒ a > a− b + |a − b|
2
⇐⇒ a > (a − b)+
⇐⇒ ap> (a− b)p+
⇐⇒ ϕpNR(a, b) > 0, which is the desired result.
Proposition 2.6. All the above functions φi, i∈ {1, 2, 3, 4} are NCP-functions.
Proof. We will only show that φ1is an NCP-function and the same argument can be applied to the other cases. Let Ω :={(a, b) | a > 0, b > 0} and suppose φ1(a, b) = 0. If (a, b) ∈ Ω, then ϕpNR(a, b) > 0 by Lemma 2.5; and hence, φ1(a, b) > 0. This is a contradiction.
Therefore, there must have (a, b) ∈ Ωc which says (a)+(b)+ = 0. This further implies ϕpNR(a, b) = 0 which is equivalent to a, b≥ 0, ab = 0. Then, one direction is proved. The converse direction is straightforward.
3 Geometric View of ϕ
pNRIn this section, we depict the surfaces of ϕpNR with various values of p so that we may have more insight for this new family of NCP-functions. Figure 1 is the surface if ϕNR(a, b) from which we see that it is concave and increasing along the direction (t, t) in the first quadrant. Figure 2 presents the surface of ϕp
NR(a, b) in which we see that it is neither convex nor concave. In addition, the value of ϕp
NR(a, b) is positive only when a > 0 and b > 0 as mentioned in Lemma 2.5. The surfaces of ϕp
NR with various values of p is shown in Figure 3.
Figure 1: The surface of z = ϕpNR(a, b) with p = 1 and (a, b)∈ [−10, 10] × [−10, 10]
Figure 2: The surface of z = ϕp
NR(a, b) with p = 3 and (a, b)∈ [−10, 10] × [−10, 10]
Figure 3: The surface of z = ϕpNR(a, b) with different values of p
To sum up, we propose a new family of new NCP-functions in this short paper. This answers a long-standing open question: what is the generalization of natural residual NCP- function? With this new discovery, many directions can be explored in the future, including numerical comparisons between ϕp
FB and ϕp
NR involved in various algorithms, studying the effect when perturbing the parameter p, applying this new family of NCP-functions to suitable optimization problems, and extending it as complementarity function associated with second-order cone and symmetric cone.
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Manuscript received 3 December 2014 revised 12 December 2014 accepted for publication 15 December 2014
Jein-Shan Chen
Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan
E-mail address: jschen@math.ntnu.edu.tw
Chun-Hsu Ko
Department of Electrical Engineering I-Shou University
Kaohsiung 840, Taiwan
E-mail address: chko@isu.edu.tw
Xiao-Ren Wu
Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan
E-mail address: cantor0968@gmail.com
