Plane Section Curves on Surfaces of NCP Functions

: The goal of this paper is to investigate the curves intersected by a vertical plane with the surfaces based on certain NCP functions. The convexity and differentiability of these curves are studied as well. In most cases, the inﬂection points of the curves cannot be expressed exactly. Therefore, we instead estimate the interval where the curves are convex under this situation. Then, with the help of differentiability and convexity, we obtain the local minimum or maximum of the curves accordingly. The study of these curves is very useful to binary quadratic programming.


Introduction
The nonlinear complementarity problem (NCP) is finding a vector x ∈ R n such that where ·, · is the Euclidean inner product and F is a function from R n to R n . Since a few decades ago, the NCP has attracted significant attention due to its various applications in areas such as economics, engineering, and information engineering [1]. There are many methods proposed for solving the NCP. One popular approach is to reformulate the NCP as a system of nonlinear equations, whereas the other approach is to recast the NCP as an unconstrained minimization problem. Both methods rely on the so-called NCP function. A function φ : R 2 → R is said to be an NCP function if it satisfies φ(a, b) = 0 ⇐⇒ a ≥ 0, b ≥ 0 and ab = 0.
In light of the NCP function, one can define the vector-valued function Φ F (x) : R n → R n by where F(x) = (F 1 (x), · · · , F n (x)) is a mapping from R n to R n . Consequently, solving the NCP is equivalent to solving a system of equation Φ F (x) = 0. In particular, it also induces a merit function of the NCP which is given by It is clear that the global minimizer of Ψ F (x) is the solution to the NCP. During the past few decades, several NCP functions have been discovered [2][3][4][5][6][7]. A well-known NCP function is the Fischer-Burmeister function [8,9] φ FB : R 2 → R, defined as φ FB (a, b) = ||(a, b)|| − (a + b), √ a 2 + b 2 . In [10], Tseng did an extension of the Fischer-Burmeister function, in which a 2-norm is relaxed to a general p-norm. In other words, the so-called generalized FB function is defined by where ||(a, b)|| p = p |a| p + |b| p and p > 1. Similarly, it induces a merit function ψ p FB : R 2 → R + given by where p > 1.
Another popular NCP function is the natural residual function [4], φ NR : R → R given by Is there a similar extension for the natural residual NCP function? Wu, Ko and Chen answered this question in [4]. The extension is kind of discrete generalization because they defined the function φ where p > 1 and p is an odd integer. Recently, the idea of discrete generalization of natural residual function has beei applied to construct discrete Fischer-Burmeister functions. More specifically, φ p D−FB : R 2 → R is defined by where p > 1 and p is an odd integer. If p = 1, then it is exactly the classical Fischer-Burmeister function (see [4,11]). The graph of φ p NR is not symmetric. Is it possible to construct a symmetric natural residual NCP function? Chang, Yang, and Chen answered this question in [2]. Note that the function φ p NR can also be expressed as a piecewise function: where p > 1 and p is an odd integer. They use this expression of φ p NR to modify the part on a < b, and achieve symmetrization of φ p S−NR (a, b) as below: where p > 1 and p is an odd integer. Surprisingly, it is still an NCP function.
How about the merit function induced by φ p S−NR (a, b)? Observing that the merit function has squared terms, Chang, Yang, and Chen combined a p and b p together and constructed ψ p S−NR (a, b) as where p > 1 and p is an odd integer.
Recently, more and more NCP functions have been discovered. As mentioned, Wu et al. [4] proposed a discrete type of natural residual function. Regarding this dis-crete counterpart, Alcantara and Chen [1] consider a continuous type of natural residual function as below: where p > 1 is a real number and The main principle behind their work is described as follows. If f (·) is a bijection mapping and φ = φ 1 − φ 2 is a given NCP function, then is also an NCP function. Hence, it can be verified that is an NCP function by employing the bijective function f (t) = sgn(t)|t| p , see [12]. Note that when p is an positive odd integer, it reduces to the discrete type of a natural residual function, that is, φ . For further symmetrization, using the above idea in (5) and (6), one can obtain a continuous type of natural residual functions [12]: and its corresponding merit function where p > 0. Again, when p is an odd integer, we see the beloe relations, The NCP functions can also be constructed by certain invertible functions. What kind of inverse functions can be applied to construct the NCP functions? Lee, Chen, and Hu [6] figured it out in ( [6], Proposition 3.8). In particular, let f : R → R be a continuous differentiable function and g : R → R with g(0) = 1. They chose functions of f (t) and g(t) satisfying the below conditions to construct new NCP functions: is an NCP function. For example, taking f (t) = ln(t), we see that f (t) is invertible on [1, ∞) and the inverse function is f −1 (t) = e t . It is easy to see that ( f −1 (t)) = e t > 0, ∀t ∈ R. Thus, f −1 is strictly monotone increasing on R. For third condition, we take g(t) = e t , which gives g(t) > 1 on (1, ∞) and − 1 2 < g(t) < 1 on (−∞, 0). We list some more examples of f and g as below. Examples of f (t) are and examples of g(t) are In summary, nine corresponding NCP functions are generated by using the above f (t) and g(t).
In [13], Tsai et al. discussed the geometry of curves on Fischer-Burmeister function surfaces, which are intersected by the plane a + b = 2r for r ∈ R. They parametrized the curves by considering a = r + t and b = r − t and defined the vector valued function α(t) : R → R 3 and β(t) : R → R 3 as α(t) = (r + t, r − t, φ(r + t, r − t)) and β(t) = (r + t, r − t, ψ(r + t, r − t)), respectively. Tsai et al. also found the local maxima and minima and studied the convexity of curves.
In this paper, we follow a similar idea to the one in [13] to investigate the curves, which are the intersection of a vertical plane a + b = 1 and surfaces based on NCP functions. We also have to point out that the study on these curves is very useful to binary quadratic programming. See [14] for the details. We parametrize the curves by the vector functions ). Then, we explore the behavior of the curves when the value p is perturbed. In addition, we discuss the convexity and local minimum and maximum of curves. Although the inflection points cannot be exactly determined, we can still estimate the interval in which the curves are convex such as in ( [14], Proposition 2.1(b)). With the convexity or differentiability of a curve, we discuss the local minimum and maximum.

Preliminaries
In this section, we review some prerequisite knowledge about the convexity and differentiability of NCP functions which will be applied to investigate the curves. First, it is known that the convexity and differentiability of an NCP function cannot hold simultaneously (see [15]). The convexity of NCP functions has been thoroughly investigated in the literature. We will now quickly recall some results directly.  (1), (2) and (4) respectively. Then, the following hold.   (7), (8) and (9) respectively. Then, for p > 1, the following hold.
We can apply Proposition 1 to check the convexity of NCP functions as in (10). In particular, based on Proposition 1, the following NCP functions are nonconvex and not differentiable at (0, 0).
. Moreover, the below NCP functions are nonconvex as well.

The Differentiability of the Curves
In this section, we investigate the differentiability of the curves, which are the intersection of surfaces of NCP functions φ(a, b), (or merit functions ψ(a, b)) with the vertical plane a + b = 1. To proceed, we set a = x and b = 1 − x. Then, the curves are parameterized as From the aforementioned NCP functions in Section 2, the parametrized curves are listed as below: (19) (11), (12) and (13) respectively. Then, the following hold.
Proof. The results follow immediately from Lemma 1.
Proof. The results are immediate consequences of Lemma 2.
Proof. The results follow from Lemma 3 directly. (a) For i = 1, 2 and j = 1, 2, 3, the function Proof. (a) Based on Proposition 2(a), the function τ In addition, we know that the exponential function and (1 − x) 2 + 4 are differentiable on R. Therefore,

The Convexity of the Curves
In Section 2, we discussed the convexity of NCP functions. It naturally leads to the convexity of the curves. Although we cannot find the inflection points one by one, we focus on estimating the interval where the curves are convex. In addition, with different p, the geometric structure of the curves will be changed. The following lemma will be employed to check the convexity.
Proof. These are very basic materials which are also well known, see [16].  (11) and (13), respectively. Then, the following hold. See Figure 1.
Proof. (a) First, as indicated in (11) is the section of a plane with the surface of the function φ p FB (a, b), which is convex on R 2 according to Lemma 1(a). τ It is clear that h(x) is nondecreasing and convex; moreover, g(x) is positive and convex. Then, according to Lemma 4( (1, ∞). In addition, due to symmetry, σ From this, we know that a ± = 1 2 1 ± 2 ( 1 3 ) − 1 are two inflection points of the function σ p FB (x). Hence, the function σ p FB (x) is convex on the intervals (−∞, a − ) and (a + , ∞). For a general p > 1, we have difficulty in determining their infection points. However, let us study their behavior when p goes to ∞ on the interval (0, 1). When 1/2 < x < 1, we have |x| > |1 − x|. Hence, the function σ p FB (x) approaches 1 2 (x − 1) 2 as p goes to ∞. Similarly, provided 0 < x < 1/2, the function σ p FB (x) approaches 1 2 x 2 as p goes to ∞. Note also that σ p FB ( 1 2 ) approaches 1 8 as p goes to ∞. (ii) We also examine the behavior of the second derivative of the function σ p FB (x) at the point 0.55 which is near 1 2 . We present the numerical results in Figure 3. Observe that their inflection points a p ± approaches 1/2, and also that (σ p FB ) (0.55) approaches 1 as p goes to ∞.
According to Remark 1 and Figure 3, we make a conjecture here.

Proof. As indicated in
where p is an odd integer and p > 1. Since σ p S−NR (x) is symmetric about x = 1 2 , we divide it into two cases: Cases (i): Suppose x > 1, the first and second derivative of this function are Note that σ Because To summarize, we have shown In this case, it is clear that σ   The following proposition is simple but tedious. We list it here for the readers' convenience. Proposition 11. Let τ f i ,g j where i = 1, 2 and j = 1, 2, 3 be defined from (20)-(28). Then, the following hold.
(a) The function τ f i ,g j (·) for i = 1, 2 and j = 1, 2 is convex on R.

Proof. (a) As stated in
is convex on R according to Proposition 6(b), it suffices to show that h(x) is convex. Taking the first and second derivatives of this function give In order to verify that h (x) > 0, we divide it into three cases: This shows that h (x) is always positive, which indicates that h(x) is convex on R. Because g(x) and h(x) are convex on R, according to Lemma 4(a), the function τ f 1 ,g 1 (·) is convex on R.
. We need to verify that g(x) is convex. Taking the second derivative of g(x) gives We want to show that that g (x) > 0. The main principle of this is to check whether the minimum of the second derivative is positive. Taking the third derivative gives The critical numbers of g (x) are x ≈ 1 2 , −1.946503, and 2.946503. Moreover, g ( 1 2 ) ≈ 2.2568, and g (−1.946503) = g (2.946503) ≈ 1.945045. The intervals where it is increasing are (−1.946503, 1 2 ) and (2.946503, ∞), and the intervals where it is decreasing are (−∞, −1.946503) and ( 1 2 , 2.946503). Therefore, the local minimum is 1.945045, and the local maximum is 2.2568. Furthermore, we also find lim x→±∞ g (x) = 2. This shows that the global minimum of g (x) is positive, hence g (x) > 0 on the entire R. This implies that g(x) is convex on R. As h(x) and g(x) are convex on R according to Lemma 4(a), τ f 1 ,g 2 (·) is convex on R.
ln(e |x| + e |1−x| − 1) and g(x) := −e (1−x) x − e x (1 − x). As g(x) is convex on R based on the proof of the case for τ f 1,g1 , the convexity of h(x) is all that remains to determined. Note that h(x) is not differentiable at x = 0 and x = 1, and we need to discuss three cases: Cases (i): Suppose 0 < x < 1. Taking the first derivative and second derivative of h(x) give Since the denominator of h (x) is positive, we need to check whether the numerator is positive. The numerator is (e For 0 < x < 1, we have 1 < e x < e and 1 < e (1−x) < e, which indicates that the numerator is positive. Therefore, we conclude h (x) > 0, and hence τ f 3 ,g 1 (·) is convex on the interval (0, 1). Cases (ii): Suppose x > 1, taking the second derivative of h(x) gives We want to show that h (x) > 0 for x > 1. Taking the third derivative of h(x) yields For the first term of h (x), since e x + e x−1 > e + 1, the denominator is positive, and hence the first term is positive. For the second term of h (x), we have As 2e > 3 and e x−1 > e 1−x when x > 1, it is also positive. Therefore, we obtain h (x) > 0. This shows that h (x) is increasing. Note also that h (1) Cases (iii): Suppose x < 0. As τ f 3 ,g 1 (x) is symmetric about the point x = 1 2 according to case (ii), the function τ f 3 ,g 1 (·) is convex on interval (−∞, 0). As indicated in (27), Let h(x) := ln(e |x| + e |1−x| − 1) and g(x) := −( is convex on R according to the proof of the case for τ f 1 ,g 2 and h(x) is convex on the intervals (−∞, 0), (0, 1) and (1, ∞) according to previous arguments. Therefore, τ f 3 ,g 2 is convex on the intervals (−∞, 0), (0, 1), and (1, ∞).
As shown in (28), we know Similarly, we use the second derivative to find the inflection points. The inflection points are x ≈ −1.904132 and 2.904132. Because ln(e |x| + e |1−x| − 1) is not differentiable at the points 0 and 1, we can only assure that the interval where the curve is convex is (0, 1). 2 Recall that a function is called subdifferentiable at x if there exists at least one subgradient at x. Although τ f 3 ,g 1 (x) is not differentiable at the points 0 and 1, with the help of Proposition 11(b), we can still show that it is subdifferentiable thereat.

Proposition 12. (a)
The function τ f 3 ,g 1 (·) is subdifferentiable at the points 0 and 1 and the subdifferential is described by Moreover, The function τ f 3 ,g 2 (·) is subdifferentiable at the points 0 and 1 and the subdifferential is described by Moreover, τ f 3 ,g 2 (·) is convex on R.

The Local Minimum and Maximum of the Curves
After discussing the convexity and differentiability, we now work on finding the local minimum or maximum value of the curves. In addition, we shall investigate the convergent behavior of local minimum or maximum values when p becomes very large.   Proof. From (14), we know that τ p NR (x) = x p − (2x − 1) p + where p > 1 and p is an odd integer. Computing the first derivative of this function gives To proceed, we discuss two cases: 1 2 ), which indicates that it does not have local minimum or maximum value.
is the only root of p[x p−1 − 2(2x − 1) p−1 ] = 0 for p > 1. Moreover, we have that τ p NR (x) is decreasing (increasing) on the right (left) hand side of the point a. Hence, a is a local maximizer and the local maximum value is . Furthermore, the local maximum value  (15) and (16), respectively. Then, for the odd integer p, the following hold. See Figure 9. . Its local maximum value converges to 1 4 . Furthermore, it has a local minimum at x = 1 2 , which converges to 0. (b) The function σ p S−NR (·) has a local maximum at x = 1 2 and its maximum value converges to 0. In addition, it has a local minimum at x = 0 and x = 1.
Case (i): Suppose x ≥ 2 3 , the first derivative is Based on this, it is verified that a = 1 is a critical point. Because σ p S−NR (x) is nonnegative and σ p S−NR (1) = 0, we can conclude that 1 is a local minimum point and the value is 0.
Case (ii): Suppose x ≤ 1 3 . Based on symmetry, the local minimum point is a = 0 and the value is 0.
Case (iii): Suppose 1 3 < x < 2 3 , we know that σ p S−NR ( 1 2 ) = 0 and σ p S−NR (x) is decreasing (increasing) on the right (left) side of the point a = 1 2 . Hence, we obtain that a = 1 2 is a local maximizer and the maximum value is ( 1 2 ) 2p for p ≥ 3. It clearly converges to 0 when p → ∞. The local minimum for other τ f i ,g i (·) is simple.

Proof.
Because each τ f i ,g i (x) is nearly convex according to x = 1 2 and τ f i ,g i (x) has a critical number at x = 1 2 , the local minimum at x = 1 2 is confirmed and can be calculated easily. We only present the values here.
√ e This completes the proof. 2

Summary
To summarize, when comparing all the curves based on NCP functions, almost all of them are neither convex nor concave. Only the curve based on the Fischer-Burmeister function is convex due the fact that its corresponding NCP function is also convex. Nonetheless, we observe that some curves are convex whereas their corresponding NCP functions are not. For instance, the curve based on the discrete type of the Fischer-Burmeister function. This indicates that the convexity of the curves depends on the choice of vertical plane. In addition, when p is perturbed, the interval of convexity will be shrunk or stretched. For the local minimum or maximum, when p becomes very large, most of the minima and maxima converge. and the minima or maxima vary by the perturbation of p.