A new generalization of the Natural-Residual function

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(1)國立臺灣師範大學數學系碩士班碩士論文. 指導教授：. 陳界山. 博士. A new generalization of the Natural-Residual function. 研究生：. 中華民國. 蘇揚善. 一零七年八. 月.

(2) Contents 1 Introduction 2 Fischer-Burmeister Functions 3 Natural Residual Functions 4 Main Results References. 1 5 7 11 20. 1.

(3) A new generalization of the Natural-Residual function Yang-San Su Jun 12, 2018 Abstract. NCP-functions play an important role in nonlinear complementarity problems(NCP). In this paper, we recall some definitions and properties of NCP-functions such as generalized Fisher-Burmeister function, ϕpFB (a, b) = ∥(a, b)∥p − (a + b), and the generalized Natural-Residual function, ϕpNR (a, b) = ap −(a−b)p+ . We attempt to generalize Natural-Residual function as a new NCP-function: ϕ˜pNR (a, b) = (a + b)p − |a − b|p . Keywords. NCP, Fisher-Burmeister, Natural-Residual, complementarity.. 1 Introduction Consider the quadratic program 1 min f (x ) = cT x + xT Qx, 2 s.t. Ax ≥ b,. (1). x ≥ 0, where Q ∈ Rn×n is symmetric, c ∈ Rn , A ∈ Rm×n , x ∈ Rn , and b ∈ Rm . If x is a locally optimal solution of the program (1), then there exists a vector y ∈ Rm such that the pair (x, y) satisfies the Karush-Kuhn-Tucker (KKT) conditions u = c + Qx − AT y ≥ 0, x ≥ 0, xT u = 0,. (2). v = −b + Ax ≥ 0, y ≥ 0, y T v = 0.. (3). Hence, the KKT conditions in (2)(3) corresponds to a linear complementarity problem x¯ ≥ 0, F (¯ x) ≥ 0, ⟨¯ x, F (¯ x)⟩ = 0, [ ] [ ] [ ] Q −AT c x where F (¯ x) = M x¯ + q, M = ,q= , x¯ = . A 0 −b y Now, we consider another optimal problem min f (x ) s.t. Ax = b, g(x) ≤ 0, 1.

(4) where f is nonlinear. Note that the Lagrangian is L(x, λ, µ) = f (x) + ⟨λ, Ax − b⟩ + ⟨µ, g(x)⟩, thus its KKT conditions are ∇x L(x, λ, µ) = ∇f (x) + AT λ + ∇g(x) · µ = 0, Ax = b,. (4). g(x) ≤ 0,. (5). µ ≥ 0,. (6). g(x) · µ = 0.. (7). Thus, ∇g(x) · µ = −∇f (x) − AT λ. If (∇g(x))T ∇g(x) is nonsingular, then from (7) we have µ = −[(∇g(x))T ∇g(x)]−1 · ∇g(x)T · (∇f (x) + AT λ). Hence, the KKT conditions in (4), (5), (6), and (7) are corresponding to equations Ax = b and a nonlinear complementarity problem µ ≥ 0, −g(x) ≥ 0, ⟨µ, −g(x)⟩ = 0. Therefore, the nonlinear complementarity problem 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 maps from Rn to Rn . We assume that F is continuously differentiable throughout this paper. The nonlinear complementarity problem has attracted much attention due to its various applications in operations research, economics, and engineering, see [18, 26, 27] and references therein. There have been many methods proposed for solving the nonlinear complementarity problem. Among which, one of the most popular and powerful approaches that has been studied intensively recently is to reformulate the nonlinear complementarity problem as a system of nonlinear equations [21]. For constructing a merit function, a class of functions, called NCP-functions and defined below, plays a significant role. Definition 1.1. A function ϕ : R2 → R is called an NCP-function if it satisfies ϕ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0. The nonlinear complementarity problem can be reformulated as a system of nonsmooth equations   ϕ(x1 , F1 (x))   ..  = 0. Φ(x) =  .   ϕ(xn , Fn (x)) 2.

(5) Therefore, the function Ψ : Rn → R+ defined as below is a merit function for the nonlinear complementarity problem Ψ(x) = ∥Φ(x)∥2 =. n ∑. ψ(xi , Fi (x)). i=1. where ψ : R2 → R+ is the square of ϕ. Such a function that can constitute an equivalent unconstrained minimization problem for the nonlinear complementarity problem is called a merit function. Consequently, the nonlinear complementarity problem is equivalent to an unconstrained minimization problem[13, 14]: min ∥Φ(x)∥2 .. x∈Rn. In other words, a merit function is a function whose global minimum are coincident with the solutions of the original nonlinear complementarity problem. Many NCP-functions and merit functions have been explored and proposed in many literature, see [16] for a survey. Among them, the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are two famous and effective NCP-functions. The FB function ϕFB : R2 → R is defined by √ (8) ϕFB (a, b) = a2 + b2 − (a + b), ∀(a, b) ∈ R2 and the NR function ϕNR : R2 → R is defined by ϕNR (a, b) = a − (a − b)+ = min {a, b} , ∀(a, b) ∈ R2 .. (9). Recently, the generalized Fischer-Burmeister function ϕpFB which includes the FischerBurmeister as a special case was considered in [1, 2, 3, 7, 24]. Indeed, the function ϕpFB is a natural extension of the ϕFB function, in which the 2-norm in ϕFB is replaced by general p-norm. In other words, ϕpFB : R2 → R is defined as ϕpFB (a, b) = ∥(a, b)∥p − (a + b), (10) √ where p > 1 and ∥(a, b)∥p = p |a|p + |b|p . The detailed geometric view of ϕpFB is depicted p in [24]. Corresponding to ϕpFB , there is a merit function ψFB : R2 → R+ given by p ψFB (a, b) =.

(6) 2 1

(7)

(8) p ϕFB (a, b)

(9) . 2. (11). To the contrast, what does “generalized natural-residual function” look like? In [6], Chen et al. give an answer to the long-standing open question. More specifically, the generalized natural-residual function, denoted by ϕpNR , is defined by ϕpNR (a, b) = ap − (a − b)p+ 3. (12).

(10) with p > 1 being a positive odd integer. As remarked in [6], the main idea to create it relies on “discrete generalization”, not the “continuous generalization”. Note that when p = 1, ϕpNR is reduced to the natural residual function ϕNR . Unlike the surface of ϕpFB , the surface of ϕpNR is not symmetric which may cause some difficulties in further analysis in designing solution methods. To this end, Chang et al. [12] try to symmetrize the function ϕpNR . The first-type symmetrization of ϕpNR , denoted by ϕpS−NR is proposed as  p p   a − (a − b) if a > b, ϕpS−NR (a, b) = ap = b p if a = b,   p b − (b − a)p if a < b,. (13). where p > 1 being a positive odd integer. It is shown in [12] that ϕpS−NR is an NCPfunction with symmetric surface, but it is not differentiable. Therefore, it is natural to ask whether there exists another symmetrization function that has not only symmetric surface but also is differentiable. Fortunately, Chang et al. [12] also figure out the second p , which is proposed as symmetrization of ϕpNR , denoted by ψS−NR  p p p p   a b − (a − b) b if a > b, p ψS−NR (a, b) = ap bp = a2p if a = b,   p p a b − (b − a)p ap if a < b,. (14). where p > 1 being a positive odd integer. The idea of “discrete generalization” looks simple, but it is novel and important. In fact, we also apply such idea to construct more NCP-functions. For example, we apply it to the Fischer-Burmeister function to obtain ϕpD−FB : R2 → R given by ϕpD−FB (a, b) =. (√ )p a2 + b2 − (a + b)p. (15). where p > 1 being a positive odd integer. One can see that the second symmetrization of ϕpNR is differentiable and symmetric simultaneously. Now, we have a new idea comes from ϕNR (a, b) = min {a, b} = a − (a − b)+ =. (a + b) − |a − b| . 2. Hence, we can define a hole new NCP-function ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p > 1 being a positive odd integer. In fact, ϕ˜pNR (a, b) is a differentiable NCP function, and hence a symmetric surface. 4.

(11) 2. Fischer-Burmeister Functions. In this section, we focus on Fischer-Burmeister (FB) function and its generalizations as defined in (9), (11) and (12). We first recall some basic definitions and then we discuss the differentiability. Definition 2.1. Let ϕ : R2 → R (a) ϕ is sub-additive if ϕ(w + w′ ) ≤ ϕ(w) + ϕ(w′ ) ∀ w, w′ ∈ R2 . (b) ϕ is positive homogeneous if ϕ(α · w) = α · ϕ(w) ∀ w ∈ R2 and α ≥ 0. (c) ϕ is convex if ϕ(αw+(1−α)w′ ) ≤ αϕ(w)+(1−α)ϕ(w′ ) ∀ w, w′ ∈ R2 and 0 ≤ α ≤ 1. Definition 2.2. Let ϕ : Rn → Rm is locally Lipschitz continuous (also called strictly continuous) at x ∈ Rn if there exist scalars κ > 0 and δ > 0 such that ∥ϕ(y) − ϕ(z)∥ ≤ κ∥y − z∥ for all y, z ∈ Rn with ∥y − x∥ ≤ δ and ∥z − x∥ ≤ δ. Proposition 2.1. [7, 11] Let ϕpFB (a, b) be defined as in (11), then the following hold. (a) ϕpFB is a NCP-function. (b) ϕpFB is sub-additive. (c) ϕpFB is positive homogeneous of degree p. (d) ϕpFB is convex. (e) ϕpFB is Lipschitz continuous with κ1 = √ κ2 = 1 + 2 when p ≥ 2.. √. 2 + 2(1/p−1/2) when 1 < p < 2, and with. p Proposition 2.2. [7, 11] Let ψFB = 12 |ϕpFB |2 and ϕpFB be defined as in (11). Then, the following hold. p (a) ψFB is an NCP-function. p (b) ψFB (a, b) ≥ 0 for all (a, b) ∈ R2 . p (c) ψFB is continuously differentiable everywhere. p p (d) ∇a ψFB (a, b) · ∇b ψFB (a, b) ≥ 0 for all (a, b) ∈ R2 . The equality holds if and only if ϕpFB (a, b) = 0.. 5.

(12) p p (a, b) = 0 ⇐⇒ ∇b ψFB (a, b) = 0 ⇐⇒ ϕpFB (a, b) = 0. (e) ∇a ψFB. Proposition 2.3. [8] Let ϕpFB be defined as in (11). Then, the generalized gradient ∂ϕpFB (a, b) of ϕpFB at a point (a, b) is equal to the set of all (va , vb ) such that  ( ) sgn(a) · |a|p−1 sgn(b) · |b|p−1    − 1, −1 if (a, b) ̸= (0, 0),  ∥(a, b)∥p−1 ∥(a, b)∥p−1 p p (va , vb ) =     (ξ − 1, ζ − 1) if (a, b) = (0, 0), p. p. where (ξ, ζ) is any vector satisfying |ξ| p−1 + |ζ| p−1 ≤ 1. √ Proposition 2.4. [11] Let ϕpD−FB (a, b) = ( a2 + b2 )p − (a + b)p : R2 → R where p is a positive odd integer and p > 1, then (a) ϕpD−FB is a NCP-function. (b) ϕpD−FB is positive homogeneous of degree p. (c) ϕpD−FB is locally Lipschitz continuous, but not (globally) Lipschitz continuous. (d) ϕpD−FB is not α-H¨ older continuous for any α ∈ (0, 1]. (e) ϕpD−FB < 0 ⇔ a > 0, b > 0. (f) ∇a ϕpD−FB (a, b) · ∇b ϕpD−FB (a, b) > 0 on the first quadrant. (g) ∇a ϕpD−FB (a, b) · ∇b ϕpD−FB (a, b) = 0 provided that ϕpD−FB (a, b) = 0. Proposition 2.5. [11] Let ϕpD−FB be defined as in (12) where p being a positive odd integer. Then, the followings hold. (a) For p > 1, ϕpD−FB is continuously differentiable with [ √ ] 2 + b2 )p−2 − (a + b)p−1 a( a √ ∇ϕpD−FB (a, b) = p . b( a2 + b2 )p−2 − (a + b)p−1 [ (b) For p > 3, ϕpD−FB is twice continuously differentiable with ∇2 ϕpD−FB (0, 0) = and for (a, b) ̸= (0, 0),. . ∂ 2 ϕpD−FB.  ∂a2  ∇ ϕD−FB (a, b) =    ∂ 2 ϕp D−FB 2 p. ∂b∂a 6. ∂ 2 ϕpD−FB ∂a∂b ∂ 2 ϕpD−FB ∂b2.      . 0 0 0 0. ] ,.

(13) where ∂ 2 ϕpD−FB ∂a2 ∂ ϕD−FB 2 p. ∂a∂b ∂ ϕD−FB 2 p. ∂b2. { } √ = p [(p − 1)a2 + b2 ]( a2 + b2 )p−4 − (p − 1)(a + b)p−2 , √ ∂ 2 ϕpD−FB = p[(p − 2)ab( a2 + b2 )p−4 − (p − 1)(a + b)p−2 ] = , ∂b∂a { } √ = p [a2 + (p − 1)b2 ]( a2 + b2 )p−4 − (p − 1)(a + b)p−2 .. 3 Natural Residual Functions In this section, we would also talk about some properties and differentiability of NaturalResidual (NR) function and its generalizations as defined in (10), (13), (14) and (15). Proposition 3.1. [6] Let ϕpNR (a, b) be defined as in (13) with p > 1 being a positive odd integer. Then, the following hold. (a) ϕpNR is a NCP-function. (b) ϕpNR (a, b) > 0 ⇐⇒ a > 0, b > 0. (c) ϕpNR is positive homogeneous of degree p. (d) ϕpNR is locally Lipschitz continuous, but not (globally) Lipschitz continuous. (e) ϕpNR is not α-H¨ older continuous    >0 (f) ∇a ϕpNR (a, b) · ∇b ϕpNR (a, b) = 0   <0. for any α ∈ (0, 1]. on {(a, b) | a > b > 0 or a > b > 2a}, on {(a, b) | a ≤ b or a > b = 2a or a > b = 0}, otherwise.. (g) ∇a ϕpNR (a, b) · ∇b ϕpNR (a, b) = 0 provided that ϕpNR (a, b) = 0. Proposition 3.2. [6] Let ϕpNR be defined as in (13) with p > 1 being a positive odd integer, and let p = 2k + 1 where k ∈ N. Then, the following hold. (a) An alternative expression of ϕpNR is ϕpNR (a, b) = a2k+1 −. ) 1( (a − b)2k+1 + (a − b)2k |a − b| . 2. (b) The function ϕpNR is continuously differentiable with [ ] p−1 p−2 a − (a − b) (a − b) + ∇ϕpNR (a, b) = p . p−2 (a − b) (a − b)+ 7.

(14) (c) The function ϕpNR is twice continuously differentiable with [ ∇ ϕNR (a, b) = p(p − 1) 2 p. 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)+. ] .. Proposition 3.3. [11] Let ϕpS−NR be defined as in (14) with p > 1 being a positive odd integer. Then, the following hold. (a) ϕpS−NR (a, b) > 0 ⇐⇒ a > 0, b > 0. (b) ϕpS−NR is positive homogeneous of degree p. (c) ϕpS−NR is not Lipschitz continuous. (d) ϕpS−NR is not α-Hölder continuous for any α ∈ (0, 1]. (e) ∇a ϕpS−NR (a, b) · ∇b ϕpS−NR (a, b) > 0 on {(a, b) | a > b > 0}. ∪. {(a, b) | b > a > 0}.. (f) ∇a ϕpS−NR (a, b) · ∇b ϕpS−NR (a, b) = 0 provided that ϕpS−NR (a, b) = 0 and (a, b) ̸= (0, 0). Proposition 3.4. [12] Let ϕpS−NR be defined in (14) with p > 1 being a positive odd integer. Then, ϕpS−NR is an NCP-function and is positive only on the first quadrant Ω = {(a, b) | a > 0, b > 0}. p Proposition 3.5. [12] Let ψS−NR be defined in (15) with p > 1 being a positive odd p integer. Then, ψS−NR is an NCP-function and is positive on the set. Ω′ = {(a, b) | ab ̸= 0} ∪ {(a, b) | a < b = 0} ∪ {(a, b) | 0 = a > b}. Now, we elaborate more about the above three functions as below. (i) For p being an even integer, all of above are not NCP-functions. A counterexample is given as below. ϕ2NR (−1, −2) = (−1)2 − (−2 + 1)2+ = 0. ϕ2S−NR (−1, −2) = (−1)2 − (−1 + 2)2 = 0. 2 ψS−NR (−1, −2) = (−1)2 (−2)2 − (−1 + 2)2 (−2)2 = 0.. 8.

(15) (ii) The above three functions are neither convex nor concave function. To see this, taking p = 3 and using the following argument verify the assertion. 1 1 0 7 7 1 = ϕ3NR (1, 1) < ϕ3NR (0, 1) + ϕ3NR (2, 1) = + = . 2 2 2 2 2 1 1 7 1 1 = ϕ3NR (1, 1) > ϕ3NR (1, −1) + ϕ3NR (1, 3) = − + = −3. 2 2 2 2 1 0 8 1 1 = ϕ3S−NR (1, 1) < ϕ3S−NR (0, 0) + ϕ3S−NR (2, 2) = + = 4. 2 2 2 2 1 1 0 0 1 = ϕ3S−NR (1, 1) > ϕ3S−NR (2, 0) + ϕ3S−NR (0, 2) = + = 0. 2 2 2 2 1 1 0 64 3 3 3 1 = ψS−NR (1, 1) < ψS−NR (0, 0) + ψS−NR (2, 2) = + = 32. 2 2 2 2 1 3 1 3 0 0 3 (2, 0) + ψS−NR (0, 2) = + = 0. 1 = ψS−NR (1, 1) > ψS−NR 2 2 2 2 Proposition 3.6. [12] Let ϕpS−NR be defined as in (14) with p > 1 being a positive odd integer. Then, the following hold. (a) An alternative expression of ϕpS−NR is.  p   ϕNR (a, b) if a > b, p ϕS−NR (a, b) = ap = b p if a = b,   p ϕNR (b, a) if a < b.. (b) The function ϕpS−NR is not differentiable. However, ϕpS−NR is continuously differentiable on the set Ω := {(a, b) | a ̸= b} with { p [ ap−1 − (a − b)p−1 , (a − b)p−1 ]T if a > b, ∇ϕpS−NR (a, b) = p [ (b − a)p−1 , bp−1 − (b − a)p−1 ]T if a < b. In a more compact form, ∇ϕpS−NR (a, b) =. {. p [ ϕp−1 (a, b), (a − b)p−1 ]T if a > b, NR p [ (b − a)p−1 , ϕp−1 (b, a) ]T if a < b. NR. (c) The function ϕpS−NR is twice continuously differentiable on the set Ω = {(a, b) | a ̸= b} with [ ]  p−2 p−2 p−2 a − (a − b) (a − b)    p(p − 1) if a > b,   (a − b)p−2 −(a − b)p−2   ∇2 ϕpS−NR (a, b) = ] [   p−2 p−2  −(b − a) (b − a)    if a < b.  p(p − 1) (b − a)p−2 bp−2 − (b − a)p−2 9.

(16) In a more compact form, [ ]  p−2 p−2 ϕ (a, b) (a − b)  NR   p(p − 1) if a > b,   (a − b)p−2 −(a − b)p−2   ∇2 ϕpS−NR (a, b) = [ ]    −(b − a)p−2 (b − a)p−2    if a < b.  p(p − 1) (b − a)p−2 ϕp−2 (b, a) NR. p Proposition 3.7. [11] Let ψS−NR be defined as in (15) with p > 1 being a positive odd integer. Then, the following hold. p (a) ψS−NR (a, b) ≥ 0 for all (a, b) ∈ R2 . p (b) ψS−NR is positive homogeneous of degree 2p. p is locally Lipschitz continuous, but not Lipschitz continuous. (c) ψS−NR p (d) ψS−NR is not α-Hölder continuous for any α ∈ (0, 1]. p p (e) ∇a ψS−NR (a, b) · ∇b ψS−NR (a, b) > 0 on the first quadrant R2++ . p p p (a, b) · (a, b) = 0. In particular, we have ∇a ψS−NR (a, b) = 0 ⇐⇒ ∇ψS−NR (f) ψS−NR p p ∇b ψS−NR (a, b) = 0 provided that ψS−NR (a, b) = 0.. p Proposition 3.8. [12] Let ψS−NR be defined as in (15) with p > 1 being a positive odd p integer. Then, ψS−NR is an NCP-function and is positive on the set. Ω′ = {(a, b) | ab ̸= 0} ∪ {(a, b) | a < b = 0} ∪ {(a, b) | 0 = a > b}. p Proposition 3.9. [12] Let ψS−NR be defined as in (15) with p > 1 being a positive odd integer. Then, the following hold.. (a) An alternative expression of ϕpS−NR is  p p   ϕNR (a, b)b if a > b, p ψS−NR (a, b) = ap bp = a2p if a = b,   p ϕNR (b, a)ap if a < b.. 10.

(17) p is continuously differentiable with (b) The function ψS−NR  p−1 p p−1 p p p−1 − (a − b)p bp−1 + (a − b)p−1 bp ]T if a > b,   p [ a b − (a − b) b , a b p ∇ψS−NR (a, b) = p [ ap−1 bp , ap bp−1 ]T = pa2p−1 [1 , 1 ]T if a = b,   p [ ap−1 bp − (b − a)p ap−1 + (b − a)p−1 ap , ap bp−1 − (b − a)p−1 ap ]T if a < b.. In a more compact form,  p−1 p p p−1 + (a − b)p−1 bp ]T if a > b,   p [ ϕNR (a, b)b , ϕNR (a, b)b p ∇ψS−NR (a, b) = p [ a2p−1 , a2p−1 ]T if a = b,   p p−1 p−1 p p−1 p T p [ ϕNR (b, a)a + (b − a) a , ϕNR (b, a)a ] if a < b, p (c) The function ψS−NR is twice continuously differentiable with    (p − 1)(a − b)p−2 bp  p−2 p−2 p  − (a − b) ]b   (p − 1)[a   +p[ap−1 − (a − b)p−1 ]bp−1            p    (p − 1)[ap − (a − b)p ]bp−2   p−2 p    (p − 1)(a − b) b   +2p(a − b)p−1 bp−1   p−1 p−1 p−1   +p[a − (a − b) ]b   −(p − 1)(a − b)p−2 bp        [ ]   p−2 p p−1 p−1 (p − 1)a b pa b p p ∇2 ψS−NR (a, b) =  pap−1 bp−1 (p − 1)ap bp−2            (p − 1)[bp − (b − a)p ]ap−2   (p − 1)(b − a)p−2 ap   p−1 p−1   +2p(b − a) a     +p[bp−1 − (b − a)p−1 ]ap−1  p−2 p    −(p − 1)(b − a) a   p           (p − 1)(b − a)p−2 ap    (p − 1)[bp−2 − (b − a)p−2 ]ap   +p[bp−1 − (b − a)p−1 ]ap−1.       if a > b,    . if a = b,       if a < b.    . 4 Main Results In this section, we focus on our new generalization of the Natural-Residual (NR) function ϕ˜pNR (a, b) = (a + b)p − |a − b|p and check some basic definitions and then we discuss the differentiability.. 11. (16).

(18) (a) p = 3. (b) p = 5. (c) p = 7. (d) p = 9. Figure 1: The surfaces of ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p = 3, 5, 7, 9. The surfaces of ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p = 3, 5, 7, 9 are show in Figure 1. We can easily see that the surfaces of ϕ˜pNR (a, b) are symmetric to a=b. Theorem 4.1. Let ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p > 1 being a positive odd integer. Then, ϕ˜pNR is an NCP-function.. 12.

(19) Proof. For any a, b ∈ R, we note ϕ˜pNR (a, b) = 0 ⇐⇒ (a + b)p − |a − b|p = 0 ⇐⇒ (a + b) − |a − b| = 0 { ⇐⇒ 2 min a, b} = 0 ⇐⇒ a ≥ 0, b ≥ 0 , ab = 0. Hence, ϕ˜pNR is an NCP-function.. 2. Lemma 4.1. Let p > 1, then (a) the function f (t) = |t|p is differentiable and f ′ (t) = sgn(t)p|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.. 2. Theorem 4.2. Let ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p > 1 being a positive odd integer. The function ϕ˜pNR is continuously differentiable with [ ] p−2 p−1 p(a + b) − (a − b)|a − b| ∇ϕ˜pNR (a, b) = p . p(a + b)p−1 + (a − b)|a − b|p−2 Proof. By Lemma 4.1, one can directly calculate the partial derivative of ϕ˜pNR ∂ ϕ˜pNR = p(a + b)p−1 − sgn(a − b)p|a − b|p−1 , ∂a = p(a + b)p−1 − (a − b)|a − b|p−2 , and ∂ ϕ˜pNR = p(a + b)p−1 + sgn(a − b)p|a − b|p−1 , ∂b = p(a + b)p−1 + (a − b)|a − b|p−2 , ∂ ϕ˜p NR ∂a. ˜p. ∂ϕ and [ ∂bNR are both continuous. Hence]ϕ˜pNR is continuously differentiable p(a + b)p−1 − (a − b)|a − b|p−2 p ˜ . 2 and ∇ϕNR (a, b) = p p(a + b)p−1 + (a − b)|a − b|p−2. Note that. 13.

(20) Theorem 4.3. Let ϕ˜pNR (a, b) = (a + b)p − |a − b|p with p > 1 being a positive odd integer. The function ϕ˜pNR is twice continuously differentiable with [ ] p−2 p−2 p−2 p−2 (a + b) − |a − b| (a + b) + |a − b| . ∇2 ϕ˜pNR (a, b) = p(p − 1) (a + b)p−2 + |a − b|p−2 (a + b)p−2 − |a − b|p−2 Proof. By Theorem 4.2, we can directly calculate the second partial derivative of ϕ˜pNR : ∂ 2 ϕ˜pNR = p(p − 1)(a + b)p−2 − p(p − 1)|a − b|p−2 , ∂a∂a ∂ 2 ϕ˜pNR = p(p − 1)(a + b)p−2 − p(p − 1)|a − b|p−2 , ∂b∂b ∂ 2 ϕ˜pNR = p(p − 1)(a + b)p−2 + p(p − 1)|a − b|p−2 , ∂a∂b ∂ 2 ϕ˜pNR = p(p − 1)(a + b)p−2 + p(p − 1)|a − b|p−2 . ∂b∂a Also, every second partial derivatives of ϕ˜pNR are continuous. Then, the function ϕ˜pNR is twice continuously differentiable. 2 Theorem 4.4. The function ϕ˜pNR (a, b) = (a + b)p − |a − b|p is positive homogeneous of degree p. Proof. Let ω = (a, b) ∈ R2 and α ≥ 0 be given. Then ϕ˜pNR (αω) = (αa + αb)p − |αa − αb|p = [α(a + b)]p − |α(a − b)|p = αp (a + b)p − |α|p |a − b|p = αp [(a + b)p − |a − b|p ] = αp ϕ˜p (ω). NR. Thus, the function ϕ˜pNR is positive homogeneous of degree p with p > 1 being odd positive integer. 2 Theorem 4.5. The function ϕ˜pNR (a, b) = (a + b)p − |a − b|p with positive odd integer p is locally Lipschitz continuous, but not (globally) Lipschitz continuous. Proof. By theorem 4.2, since ϕ˜pNR (a, b) = (a + b)p − |a − b|p is continuously differentiable, which implies locally Lipschitz continuity. Consider the restriction of ϕ˜pNR (a, b) on the { } line L = (a, b)|a = b > 0 . Note that for any a > 0, ϕ˜pNR (a, b) = (2a)p , it suffices to 14.

(21) p show that { f (t) }= ct , c′ > 1 is not Lipschitz continuous. Indeed, for any M > 0, choosing t = max 1, M and t = t + 1. Then. |f (t) − f (t′ )| = c(t + 1)p − ctp |t − t′ | = c[(t + 1)p − tp ] = c[(t + 1) − t][(t + 1)p−1 · 1 + · · · + 1 · tp−1 ] > cptp−1 > M. Hence, it follows that f is not Lipschitz continuous.. 2. Remark 4.1. In fact, we can use mean value theorem to say ϕ˜pNR (a, b) is not globally Lipschitz continuous for any p ∈ R. Indeed, for any M > 0, there exists a real number ξ ∈ [t, t′ ] such that |f (t) − f (t′ )| = f ′ (ξ) = cpξ p−1 ≥ cptp−1 > M. |t − t′ | Hence, ϕ˜pNR (a, b) is not globally Lipschitz continuous for any p ∈ R. Finally, we note that ϕNR and ϕpFB have the following inequality (see [3]) (2 − 21/p )|ϕNR | ≤ |ϕpFB | ≤ (2 + 21/p )|ϕNR |. Consequently, we try to find a similar inequality about ϕpD−FB and ϕ˜pNR . Let ϕpD−F B , ϕ˜pNR be defined in (15) and (16), and ratio(R) be defined as below R=. |ϕpD−FB (a, b)| . |ϕ˜p (a, b)| NR. According to the following tables ,we make the following conjecture α1 |ϕ˜pNR | ≤ |ϕpD−FB | ≤ α2 |ϕ˜pNR | where α1 =. √1 ( 2)p. and α2 = 2.. 15.

(22) a b 21.01 43.21 24.44 -42.92 -9.21 27.92 -23.20 46.63 6.29 47.70 -27.78 -41.78 11.20 35.84 -43.11 -1.38 -9.41 -46.57 44.46 36.39 -26.79 11.83 -32.35 -11.14 -26.86 11.05 -19.49 8.78 -32.12 45.54 0.46 -8.73 32.76 33.25 26.71 28.93 5.51 -35.73 -12.83 23.97 7.28 -9.61 -33.41 -2.40 9.71 3.41 21.31 6.20 37.65 22.75 6.38 9.19 11.67 -10.10 -22.93 19.55 24.70 9.90 -42.30 7.06. |ϕpD−FB | |ϕ˜pNR | R α2 153903.69 253956.17 0.61 2 126785.33 311953.65 0.41 2 18867.94 44657.36 0.42 2 128380.42 327552.67 0.39 2 46026.57 86409.14 0.53 2 462788.88 339254.64 1.36 2 51159.45 89152.88 0.57 2 168341.73 160772.39 1.05 2 282661.43 226715.52 1.25 2 338844.16 527971.72 0.64 2 28466.45 60958.74 0.47 2 122302.10 91793.25 1.33 2 28438.68 58405.57 0.49 2 10992.35 23821.50 0.46 2 170677.68 466023.83 0.37 2 1232.79 1339.99 0.92 2 185850.15 287507.82 0.65 2 111245.35 172301.92 0.65 2 74858.05 97729.85 0.77 2 18719.61 48470.72 0.39 2 1764.52 4828.82 0.37 2 83500.35 75742.54 1.10 2 1168.32 2008.19 0.58 2 9882.42 17361.07 0.57 2 135258.25 217092.28 0.62 2 2373.00 3750.31 0.63 2 3672.25 10313.52 0.36 2 27381.57 76647.08 0.36 2 22571.10 38167.16 0.59 2 122558.32 163942.554 0.75 2. α1 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35. α2 − R 1.39 1.59 1.58 1.61 1.47 0.64 1.43 0.95 0.75 1.36 1.53 0.67 1.51 1.53 1.63 1.08 1.35 1.35 1.23 1.61 1.63 0.90 1.42 1.43 1.38 1.37 1.64 1.64 1.41 1.25. R − α1 0.25 0.05 0.07 0.04 0.18 1.01 0.22 0.69 0.89 0.29 0.11 0.98 0.13 0.11 0.01 0.57 0.29 0.29 0.41 0.03 0.01 0.75 0.23 0.22 0.27 0.28 0.01 0.01 0.24 0.39. Table 1: p = 3 a b |ϕpD−FB | |ϕ˜pNR | R α2 12.48 -25.74 1.96E+7 8.20E+7 0.24 2 -19.26 44.03 2.47E+8 1.00E+9 0.25 2 -35.28 22.96 1.32E+8 6.70E+8 0.20 2 12.29 6.61 1.88E+6 2.41E+6 0.78 2 4.86 13.55 1.50E+6 2.06E+6 0.72 2 Table 2: p = 5 16. α1 0.18 0.18 0.18 0.18 0.18. α2 − R 1.76 1.75 1.80 1.22 1.28. R − α1 0.06 0.07 0.02 0.61 0.55.

(23) a b |ϕpD−FB | 42.12 -31.99 4.14E+8 -36.82 47.44 7.80E+8 47.63 35.60 3.25E+9 -7.22 -18.26 1.36E+7 24.43 31.73 4.55E+8 -42.96 41.53 7.61E+8 32.06 12.64 1.30E+8 6.86 -27.26 2.11E+7 4.98 40.10 7.84E+7 0.85 -13.45 760619.89 27.81 35.93 8.58E+8 -36.51 -13.03 3.86E+8 -14.92 -21.03 7.15E+7 27.87 -30.32 1.19E+8 41.14 -33.73 4.26E+8 9.80 -12.38 980799.73 -8.58 -21.15 2.94E+7 23.31 28.23 2.98E+8 15.11 -40.31 1.58E+8 31.41 -4.43 1.78E+7 -8.98 48.17 1.90E+8 17.26 -40.79 1.78E+8 39.91 -17.86 1.55E+8 7.34 -29.46 3.11E+7 -17.84 42.59 2.01E+8 -48.05 5.10 4.09E+8. |ϕ˜pNR | 2.24E+9 4.25E+9 4.00E+9 1.09E+7 5.58E+8 4.30E+9 1.76E+8 4.98E+7 1.33E+8 913935.23 1.05E+9 3.06E+8 6.01E+7 6.67E+8 2.35E+9 5365252.50 2.36E+7 3.64E+8 5.33E+8 4.48E+7 5.17E+8 6.67E+8 6.38E+8 7.28E+7 7.96E+8 5.70E+8. R α2 0.19 2 0.18 2 0.81 2 1.25 2 0.82 2 0.18 2 0.74 2 0.42 2 0.59 2 0.83 2 0.82 2 1.26 2 1.19 2 0.18 2 0.18 2 0.18 2 1.25 2 0.82 2 0.30 2 0.40 2 0.37 2 0.26 2 0.24 2 0.43 2 0.25 2 0.72 2. α1 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18. α2 − R 1.81 1.82 1.19 0.75 1.18 1.82 1.26 1.58 1.41 1.17 1.18 0.74 0.81 1.82 1.82 1.82 0.75 1.18 1.70 1.60 1.63 1.73 1.76 1.57 1.75 1.28. R − α1 0.01 0.01 0.64 1.08 0.64 0.0001 0.56 0.25 0.41 0.66 0.64 1.09 1.01 0.0007 0.004 0.006 1.07 0.64 0.12 0.22 0.19 0.09 0.07 0.25 0.08 0.54. α1 0.088 0.088 0.088 0.088 0.088. α2 − R 0.84 1.80 1.91 1.90 0.85. R − α1 1.07 0.11 0.0028 0.01 1.06. Table 3: p = 5. a b |ϕpD−FB | |ϕ˜pNR | R -5.55 -13.90 1.22E+9 1.06E+9 1.16 9.27 -28.15 2.10E+10 1.04E+11 0.20 -20.96 17.32 1.10E+10 1.20E+11 0.09 -31.65 21.97 1.26E+11 1.27E+12 0.098 -47.21 -19.93 7.08E+12 6.16E+12 1.15 Table 4: p = 7. 17. α2 2 2 2 2 2.

(24) a -26.60 -14.29 -45.24 27.62 39.03 -19.28 10.96 35.26 -18.46 -20.53 -45.47 -27.48 33.46 -26.27 48.48 -0.30 41.34 32.74 -33.71 23.48 34.27 2.64 -15.40 15.56 -31.57 -40.66. b 30.34 -18.94 2.26 3.84 -34.92 -15.27 -6.23 -13.19 19.11 23.86 -36.76 -43.20 44.03 35.12 -39.69 -39.08 37.66 -28.02 -38.00 -16.65 -21.98 -21.76 45.11 -31.09 16.33 -32.70. |ϕpD−FB | 1.74E+11 4.90E+10 6.62E+11 1.74E+10 1.08E+12 6.43E+10 5.07E+10 1.05E+11 9.34E+10 3.06E+10 2.78E+13 9.73E+12 1.52E+13 3.12E+11 3.79E+12 2.86E+11 1.75E+13 2.76E+11 1.06E+13 1.64E+10 1.85E+11 3.37E+10 5.39E+11 6.16E+10 7.18E+10 1.25E+13. |ϕ˜pNR | R α2 1.94E+12 0.089 2 4.48E+10 1.09 2 8.16E+11 0.81 2 2.62E+10 0.66 2 1.21E+13 0.09 2 5.88E+10 1.09 2 4.45E+10 0.11 2 6.24E+11 0.17 2 1.06E+11 0.09 2 3.39E+11 0.09 2 2.54E+13 1.09 2 8.81E+12 1.10 2 1.68E+13 0.91 2 3.28E+12 0.09 2 4.14E+13 0.09 2 2.79E+11 1.03 2 1.92E+13 0.91 2 3.06E+12 0.09 2 9.75E+12 1.09 2 1.68E+11 0.097 2 1.78E+12 0.104 2 6.07E+10 0.55 2 2.95E+12 0.18 2 4.81E+11 0.19 2 5.79E+11 0.124 2 1.14E+13 1.09 2. α1 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088. α2 − R 1.91 0.91 1.19 1.34 1.91 0.91 1.89 1.83 1.91 1.91 0.91 0.895 1.09 1.90 1.90 0.97 1.09 1.91 0.91 1.902 1.895 1.45 1.82 1.87 1.876 0.907. R − α1 0.0013 1.006 0.72 0.58 0.0009 1.004 0.03 0.08 9.39 0.0017 1.0035 1.02 0.82 0.0066 0.0031 0.94 0.82 0.0018 1.001 0.0092 0.015 0.47 0.094 0.039 0.036 1.0036. Table 5: p = 7. a b |ϕpD−FB | |ϕ˜pNR | 33.46 -10.67 7.97E+13 6.34E+14 -45.47 33.58 5.89E+15 1.205E+17 20.58 -47.72 2.77E+15 3.22E+16 -4.72 -30.99 1.24E+14 1.003E+14 -22.28 41.52 1.145E+15 1.752E+16. R α2 0.13 2 0.049 2 0.09 2 1.23 2 0.065 2. Table 6: p = 9. 18. α1 0.044 0.044 0.044 0.044 0.044. α2 − R 1.87 1.951 1.91 0.77 1.935. R − α1 0.08 0.0046 0.04 1.19 0.021.

(25) a b |ϕpD−FB | -32.40 23.22 2.54E+14 -12.22 34.70 1.22E+14 -38.61 -37.52 8.97E+16 -12.37 35.24 1.39E+14 -14.91 -33.30 1.52E+15 -34.45 -42.95 1.04E+17 45.54 -37.27 8.46E+15 -0.87 25.66 1.31E+12 -0.76 -16.78 2.63E+11 -28.85 -48.60 1.06E+17 26.23 -11.65 1.32E+13 21.07 44.384 2.04E+16 -17.76 7.72 3.85E+11 -44.16 -5.18 2.41E+15 -48.33 13.55 2.096E+15 -17.69 41.23 7.35E+14 11.20 -48.29 1.94E+15 20.30 -8.22 1.15E+12 18.54 -26.29 3.69E+13 -5.26 7.35 4.04E+8 49.99 -35.79 1.26E+16 20.92 -42.77 1.26E+15 -31.41 0.11 5.86E+13 -32.03 -8.45 3.40E+14 -9.69 25.81 9.12E+12 22.66 36.00 7.76E+15. |ϕ˜pNR | R α2 5.09E+15 0.049 2 1.10E+15 0.11 2 8.59E+16 1.044 2 1.25E+15 0.111 2 1.41E+15 1.08 2 9.97E+16 1.046 2 1.83E+17 0.046 2 2.97E+12 0.44 2 2.27E+11 1.163 2 1.003E+17 1.059 2 1.61E+14 0.08 2 2.20E+16 0.92 2 4.54E+12 0.08 2 1.94E+15 1.24 2 1.338E+16 0.157 2 8.56E+15 0.086 2 9.47E+15 0.205 2 1.25E+13 0.092 2 7.31E+14 0.05 2 8.09E+9 0.049 2 2.51E+17 0.049 2 1.73E+16 0.073 2 5.96E+13 0.983 2 2.94E+14 1.15 2 8.93E+13 0.102 2 8.21E+15 0.945 2 Table 7: p = 9. 19. α1 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044. α2 − R 1.950 1.889 0.955 1.888 0.92 0.953 1.954 1.56 0.837 0.941 1.92 1.08 1.92 0.76 1.843 1.914 1.795 1.907 1.95 1.95 1.950 1.927 1.016 0.84 1.898 1.055. R − α1 0.0056 0.067 1.00004 0.067 1.037 1.0024 0.002 0.397 1.12 1.014 0.037 0.88 0.04 1.20 0.11 0.042 0.161 0.048 0.006 0.005 0.005 0.028 0.939 1.11 0.058 0.9003.

(26) References [1] J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, Journal of Global Optimization, 36(2006), 565-580. [2] J.-S. Chen, On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacific Journal of Operational Research, 24(2007), 401-420. [3] J.-S. Chen, H.-T. Gao and S. Pan, A R-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function, Journal of Computational and Applied Mathematics, 232(2009), 455-471. [4] J.-S. Chen, Z.-H. Huang, and C.-Y. She, A new class of penalized NCP-functions and its properties, Computational Optimization and Applications, 50(2011), 49-73. [5] J.-S. Chen, C.-H. Ko, and S.-H. Pan, A neural network based on generalized Fischer-Burmeister function for nonlinear complementarity problems, Information Sciences, 180(2010), 697-711. [6] J.-S. Chen, C.-H. Ko, and X.-R. Wu, What is the generalization of natural residual function for NCP, to appear in Pacific Journal of Optimization, January, 2016, 19-27. [7] J.-S. Chen and S. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, 40(2008), 389-404. [8] J.-S. Chen, S.-H. Pan, and T.-C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs, Nonlinear Analysis: Theory, Methods and Applications, 72(2010), 3739-3758. [9] J.-S. Chen, S.-H. Pan, and C.-Y. Yang, Numerical comparison of two effective methods for mixed complementarity problems, Journal of Computational and Applied Mathematics, 234(2010), 667-683. [10] J.-S. Chen, D. F. Sun, and J. Sun, The SC 1 property of the squared norm of the SOC Fischer-Burmeister function, Operations Research Letters, vol. 36(2008), 385-392. [11] C.-H. Huang, K.-J. Weng, J.-S. Chen, H.-W. Chu , and M.-Y. Li, On four discrete-type families of NCP-functions,to appear in Journal of Nonlinear and Convex Analysis, 2018. [12] Y.-L. Chang, J.-S. Chen, C.-Y. Yang, Symmetrization of generalized natural residual function for NCP, Operations Research Letters, 43(2015), 354-358. 20.

(27) [13] F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal on Optimization, 7(1997), 225-247. [14] A. Fischer, A special Newton-type optimization methods, Optimization, 24(1992), 269-284. [15] A. Fischer, Solution of the monotone complementarity problem with locally Lipschitzian functions, Mathematical Programming, 76(1997), 513-532. [16] A. Galántai, Properties and construction of NCP functions, Computational Optimization and Applications, 52(2012), 805-824. [17] C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Computational Optimization and Applications, 5(1996), 155-173. [18] P. T. Harker and J.-S. Pang, Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications, Mathematical Programming, 48(1990), 161-220. [19] C. Kanzow, Nonlinear complementarity as unconstrained optimization, Journal of Optimization Theory and Applications, 88(1996), 139-155. [20] C. Kanzow, N. Yamashita, and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications, 94(1997), 115-135. [21] O. L. Mangasarian, Equivalence of the Complementarity Problem to a System of Nonlinear Equations, SIAM Journal on Applied Mathematics, 31(1976), 89-92. [22] J.-S. Pang, Newton’s Method for B-differentiable Equations, Mathematics of Operations Research, 15(1990), 311-341. [23] D. Sun and L. Qi, On NCP-functions, Computational Optimization and Applications, 13(1999), 201-220. [24] H.-Y. Tsai and J.-S. Chen, Geometric views of the generalized FischerBurmeister function and its induced merit function, Applied Mathematics and Computation, 237(2014), 31-59. [25] N. Yamashita and M. Fukushima, On stationary points of the implict Lagrangian for nonlinear complementarity problems, Journal of Optimization Theory and Applications, 84(1995), 653-663. [26] M. C. Ferris and J-S Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39(1997), 669-713. [27] J-S Pang, Complementarity problems. Handbook of Global Optimization, R Horst and P Pardalos (eds.), MA: Kluwer Academic Publishers, 271-338. 21.

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