基於費雪-博美斯特函數以及自然殘留函數建構出的非線性互補性問題函數的性質

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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. Properties of Some NCP-functions based on the Fischer-Burmeister function and the Natural-Residual function. 研究生：翁康鈞. 中華民國一零四年七月.

(2) 致. 謝. 首先誠摯的感謝指導教授陳界山博士，老師學識淵博、治學嚴謹，其悉心的教導使我得以一窺應用數學領域的深奧，不時的討論並指點我正確的方向，使我在這些年中獲益匪淺。陳老師對學問的嚴謹更是我輩學習的典範。感激口試委員柯春旭老師與張毓麟老師為我的著作提供許多寶貴的建議，使內容更加充實。碩士生涯裡，師大為我營造了良好的精神氣氛。除了眾位幫助過我的師長外，我的身邊還充滿著一群風華正茂的有志青年。感謝學長不厭其煩的指出我研究中的缺失，且總能在我迷惘時為我解惑，也感謝各位同學的幫忙，恭喜我們順利走過這些年。研究室共同生活的點滴我永遠不會忘記。論文最後衝刺期間紮實討論，言不及義的閒扯搞笑，點點滴滴我都銘感在心。來自四方的好友們的支持鼓勵體諒包容也是能讓我不斷前進的動力。光陰似箭，歲月如梭，琴棋書畫煮酒論劍的研究生涯也將隨著這篇論文的定稿而告一段落。這段時光短暫而充實，時而匆匆，時而寫意，宛如一道畫過人生旅途的璀璨流星。稍縱即逝，讓人措手不及。最後，謹以此文獻給我親愛的父母和姊姊，感謝你們從小到大的支持與栽培，我愛你們！. 翁康鈞謹致 2015 年 7 月.

(3) 論文摘要非線性互補問題函數在非線性互補性問題裡扮演很重要的角色。在本篇文章中，我們討論幾個非線性互補性問題函數，其中包含一般化的費雪－博美斯特函數以及一般化的自然殘留函數。我們企圖提出幾個關於這些函數的基本性質。關鍵字：非線性互補問題函數，費雪－博美斯特函數，. 自然殘留函數，互補性. Abstract It is well known that NCP-functions play an important role in nonlinear complementarity problem (NCP). In this paper, we study several NCP-functions including the generalized Fisher-Burmeister function, and the generalized Natural-Residual function. We attempt to give some basic properties for these NCP-functions. Keywords. NCP, Fisher-Burmeister, Natural-Residual, complementarity..

(4) Contents 1 Introduction. 1. 2 Extensions of FB function. 3. 3 Extensions of NR function. 5. 4 Some properties. 9. 5 Conclusion. 13. References. 13.

(5) Properties of some NCP-functions based on the Fischer-Burmeister function and the Natural-Residual function Kang-Jun Weng Jul 27, 2015 Abstract. It is well known that NCP-functions play an important role in nonlinear complementarity problem (NCP). In this paper, we study several NCP-functions including the generalized Fisher-Burmeister function, φpFB (a, b) = k(a, b)kp − (a + b), and the generalized Natural-Residual function, φpNR (a, b) = ap − (a − b)p+ . We attempt to give some basic properties for these NCP-functions. Keywords. NCP, Fisher-Burmeister, Natural-Residual, complementarity.. 1. Introduction. The nonlinear complementarity problem (NCP) (Harker and Pang,1990 [18];Pang 1994 [27]) is to find a point x ∈ Rn such that x ≥ 0, F (x) ≥ 0, hx, F (x)i = 0.. (1). where h·, ·i is the Euclidean inner product and F = (F1 , F1 , F1 , · · · , Fn )T maps from Rn to Rn . We assume that F is continuously differentiable throughout this paper. The NCP has attracted much attention due to its various applications in operations research, economics, and engineering (Ferris and Pang, 1997 [26]; Harker and Pang,1990 [18]; Pang, 1994 [27]). There have been many methods proposed for solving the NCP (Harker and Pang,1990 [18]; Pang, 1994 [27]). Among which, one of the most popular and powerful approaches that has been studied intensively recently is to reformulate the NCP as a system of nonlinear equations (Mangasario, 1976 [21]) or as an unconstrained minimization problem (Facchinei and Soares, 1997 [13]; Fisher, 1992 [14]; Kanzow, 1996 [19]). Such a function that can constitute an equivalent unconstrained minimization problem for the NCP is called a merit function. In other words, a merit function is a function whose global minima are coincident with the solutions of the original NCP. For constructing a merit function, the class of functions, so-called NCP-functions and defined as below, serves an important role. A function φ : R2 → R is called an NCP-function if it satisfies φ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0. 1. (2).

(6) Many NCP-functions and merit functions have been explored and proposed in many literature. Among them,the Fischer-Burmeister(FB) function and the Natural-Residual(NR) function are two effective NCP-functions. The FB function φFB : R2 → R is defined by √ φFB (a, b) = a2 + b2 − (a + b), ∀(a, b) ∈ R2 (3) and the NR function φNR : R2 → R is defined by φNR (a, b) = a − (a − b)+ = min {a, b} , ∀(a, b) ∈ R2. (4). A novel idea arises recently, which is by means of “discrete generalization” to extend the above two functions [6]. One is the “discrete-type generalized Fischer-Burmeister function” which is defined by √ p φpD−FB (a, b) = a2 + b2 − (a + b)p (5) where p > 1 is a positive odd integer and (a, b) ∈ R2 . Notice that this φpD−FB function has some difference from the so-called “generalized Fischer-Burmeister function,” which is given by φpFB (a, b) = k(a, b)kp − (a + b) p where p > 1 and k(a, b)kp = p |a|p + |b|p .. (6). What about applying the concept of “discrete generalization” to the Natural-Residual function? In [6], there studies this extension. In particular, the “generalized naturalresidual function”, denoted by φpNR , is defined by φpNR (a, b) = ap − (a − b)p+. (7). with p > 1 being a positive odd integer. Moreover, two new NCP-functions were derived in [12]. More specifically, they are the “first-type symmetrization of φpNR ”, denoted by φpS−NR and given by  p p   a − (a − b) if a > b, φpS−NR (a, b) = (8) ap = b p if a = b,   p b − (b − a)p if a < b, and the “second-type symmetrization of φpNR ”, denoted  p p p p   a b − (a − b) b p ψS−NR (a, b) = ap bp = a2p   p p a b − (b − a)p ap. by if a > b, if a = b,. (9). if a < b,. In this thesis, we introduce some properties the two extensions of Fischer-Burmeister function, φp and φpD−FB . We compare these two functions in Section 2. The other three extensions of Natural-Residual function, φpNR , φS−NR and ψS−NR , will be studied in Section 3. We attempt to propose some properties about these functions. 2.

(7) 2. Extensions of FB function. In this section,we focus on the two extensions, φpFB and φpD−FB as defined in (5) and (6). We first give some basic property about these two functions,then we discuss the differentiability about about these two functions. Proposition 2.1. (Chen and Pan,2008,[7, Prop.3.1 and Lemma 3.1]) Let φpFB : R2 → R be defined as (5). Then, we have (a) φpFB is a NCP-function, i.e., it satisfies (1); (b) φpFB is sub-additive, i.e., φp (w + w0 ) ≤ φp (w) + φp (w0 ) ∀w, w0 ∈ R2 ; (c) φpFB is positive homogeneous, i.e., φp (αw) = αφp (w) ∀w ∈ R2 and α ≥ 0; (d) φpFB is convex, i.e., φp (αw + (1 − α)w0 ) ≤ αφp (w) + (1 − α)φp (w0 ) for all w, w0 ∈ R2 and α ≥ 0. Proposition 2.2. ([11, Prop 3.1]) Let φpD−FB : R2 → R be defined as (6) where p is a positive odd integer and greater than 1. Then, we have (a) φpD−FB is a NCP-function, i.e., it satisfies (1); (b) φpD−FB is positive homogeneous, i.e., φpD−FB (αw) = αφpD−FB (w) for all w ∈ R2 and α ≥ 0. We see the function φpFB is a convex function,but φpD−FB is not.We elaborate this as follows, taking p = 3,then: 1 1 1 9 0 = φ3D−FB (1, 1) > φ3D−FB (0, 0) + φ3D−FB (2, 2) = 0 + (2 2 − 26 ) 2 2 2 Next,we look at the gradient about these two functions. Proposition 2.3. ([28, Lemma 2.2]) Given any point (a, b) ∈ R2 , each element in the generalized gradient ∂φpFB (a, b) has the representation (ξ − 1, ζ − 1) where, if (a, b) 6= (0, 0), sgn(a) · |a|p−1 sgn(b) · |b|p−1 (ξ, ζ) = , , k(a, b)kp−1 k(a, b)kp−1 p p p. p. and otherwise (ξ,ζ) is an arbitrary vector in R2 satisfying |ξ| p−1 + |ζ)| p−1 ≤ 1.. 3.

(8) Proposition 2.4. ([11, Prop. 3.2]) Let φpD−FB be defined in (5). Then, the following hold. # " √ p−2 p−1 2 + b2 a − (a + b) a (a) F or p > 1, ∇φpD−FB (a, b) = p . √ p−2 b a2 + b 2 − (a + b)p−1 . ∂ 2 φpD−FB. ∂ 2 φpD−FB.  (b) ∇2 φpD−FB (a, b) = . ∂a2 ∂ φD−FB. ∂a∂b  . ∂ φD−FB . 2 p. ∂b∂a. where ∂ 2 φpD−FB ∂a2. ∂ 2 φpD−FB ∂a∂b. ∂b2. 2 p. ∂b2. p−4 √ 2 2 p−2 = p (p − 1)a + b a2 + b 2 − (p − 1)(a + b). = p (p − 2)ab. ∂ 2 φpD−FB. . √. a2. +. b2. p−4. p−2. . − (p − 1)(a + b). =. ∂ 2 φpD−FB ∂b∂a. p−4 2 √ 2 p−2 2 2 = p a + (p − 1)b a +b − (p − 1)(a + b) .. In the end of this section, we present some other variants of the above three functions which are analogous to the functions in [23]. Proposition 2.5. ([7]) Let φpFB be defined as (5). Then, the following variants of φpFB are NCP-functions. p (a, b) = φp (a, b) − α(a) (b) , α > 0. g φ + + FB 1 FB p p (a, b) = φ (a, b) − α(ab) , α > 0. g φ + FB 2 qFB p (a, b) = g [φpFB (a, b)]2 + α ((a)+ (b)+ )2 , α > 0. φ FB 3 q p g φFB 4 (a, b) = [φpFB (a, b)]2 + α [(ab)+ ]2 , α > 0.. In the above expressions, for any t ∈ R, we define (t)+ as max{0, t}.. 4.

(9) Proposition 2.6. ([6, Prop. 2.4]) Let φpD−FB be defined in (6). Then, the following variants of φpD−FB are NCP-functions. ϕ1 (a, b) = φpD−FB (a, b) + α(a)+ (b)+ , α > 0. ϕ2 (a, b) = φpD−FB (a, b) + α ((a)+ (b)+ )2 , α > 0. ϕ3 (a, b) = φpD−FB (a, b) + α ((ab)+ )4 , α > 0. ϕ4 (a, b) = φpD−FB (a, b) + α ((ab)+ )2 , α > 0. 2 ϕ5 (a, b) = φpD−FB (a, b) + α (a)+ )2 ((b)+ , α > 0. In the above expressions, for any t ∈ R, we define (t)+ as max{0, t}.. 3. Extensions of NR function. Like the before section, we talk about extensions of NR function in this section, which are φpNR , φS−NR and ψS−NR as defined in (7),(8),(9). Proposition 3.1. ([6, Prop. 2.1]) Let φpNR be defined in (7) with p > 1 being a positive odd integer. Then, φpNR is an NCP-function. Proposition 3.2. ([12, Prop. 2.1]) Let φpS−NR be defined in (8) 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 3.3. ([12, Proposition 3.1]) Let ψS−NR be defined in (9) with p > 1 being a p positive odd integer. Then, ψS−NR is an NCP-function and is positive on the set. Ω0 = {(a, b) | ab 6= 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 (−1, −2) = (−1)2 (−2)2 − (−1 + 2)2 (−2)2 = 0. ψS−NR. 5.

(10) (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 0 7 7 1 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 1 0 8 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 1 = ψS−NR (1, 1) > ψS−NR (2, 0) + ψS−NR (0, 2) = + = 0. 2 2 2 2 Next, we try to find the gradients of these three NCP-functions. Proposition 3.4. ([6, Prop. 2.2]) Let φpNR be defined as in (7) with p > 1 being a positive odd integer, and let p = 2k + 1 where k = 1, 2, 3 . . .. 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 ; (a − b)p−2 (a − b)+ (c) The function φpNR is twice continuously differentiable with " p−2 # p−3 p−3 a − (a − b) (a − b) (a − b) (a − b) + + ∇2 φpNR (a, b) = p(p − 1) . (a − b)p−3 (a − b)+ −(a − b)p−3 (a − b)+. Proposition 3.5. ([12, Prop. 2.2]) Let φpS−NR be defined in (8) 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. 6.

(11) (b) The function φpS−NR is not differentiable. However, φpS−NR is continuously differentiable on the set Ω := {(a, b) | a 6= 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−1 p [ φNR (a, b), (a − b)p−1 ]T if a > b, p−1 p [ (b − a)p−1 , φNR (b, a) ]T if a < b.. (c) The function φpS−NR is twice continuously differentiable on the set Ω = {(a, b) | a 6= b} with  " #  ap−2 − (a − b)p−2 (a − b)p−2   if a > b,   p(p − 1) (a − b)p−2 −(a − b)p−2 2 p # " ∇ φS−NR (a, b) =  −(b − a)p−2 (b − a)p−2   if a < b.   p(p − 1) (b − a)p−2 bp−2 − (b − a)p−2 In a more compact form,  "      p(p − 1) 2 p " ∇ φS−NR (a, b) =      p(p − 1). p−2 (a, b) φNR. (a − b)p−2. #. (a − b)p−2 −(a − b)p−2 # −(b − a)p−2 (b − a)p−2 (b − a)p−2. φp−2 (b, a) NR. if a > b, if a < b.. p Proposition 3.6. ([12, Prop. 3.2]) Let ψS−NR be defined as in (9) 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. p (b) The function ψS−NR is continuously differentiable with.  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. 7.

(12) 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 [ φpNR (b, a)ap−1 + (b − a)p−1 ap , φp−1 (b, a)ap ]T if a < b. NR p (c) The function ψS−NR is twice continuously differentiable with.     p−2  − (a − b)p−2 ]bp   (p − 1)[a              p        (p − 1)(a − b)p−2 bp        +p[ap−1 − (a − b)p−1 ]bp−1     "   (p − 1)ap−2 bp pap−1 bp−1 p ∇2 ψS−NR (a, b) = p  pap−1 bp−1 (p − 1)ap bp−2      (p − 1)[bp − (b − a)p ]ap−2        +2p(b − a)p−1 ap−1        −(p − 1)(b − a)p−2 ap  p             (p − 1)(b − a)p−2 ap      +p[bp−1 − (b − a)p−1 ]ap−1. (p − 1)(a − b)p−2 bp. .  +p[ap−1 − (a − b)p−1 ]bp−1     if a > b, (p − 1)[ap − (a − b)p ]bp−2    +2p(a − b)p−1 bp−1  p−2 p # −(p − 1)(a − b) b if a = b,  (p − 1)(b − a)p−2 ap   +p[bp−1 − (b − a)p−1 ]ap−1    if a < b.     p−2 p−2 p (p − 1)[b − (b − a) ]a. In the end of this section, we present some other variants of the above three functions which are analogous to the functions in [23]. Proposition 3.7. ([6, Prop. 2.4]) Let φpNR be defined in (7) with p > 1 being a positive odd integer. Then, the following variants of φpNR are NCP-functions. p g φ (a, b) = φpNR (a, b) + α(a)+ (b)+ , α > 0. NR 1 p g (a, b) = φpNR (a, b) + α ((a)+ (b)+ )2 , α > 0. φ NR 2 p g (a, b) = φp (a, b) + α ((ab)+ )4 , α > 0. φ NR 3. NR. p g φ (a, b) = φpNR (a, b) + α ((ab)+ )2 , α > 0. NR 4 2 p 2 p g φ (a, b) = φ (a, b) + α (a) ) ((b) , α > 0. + + NR 5 NR. In the above expressions, for any t ∈ R, we define (t)+ as max{0, t}.. 8.

(13) Proposition 3.8. ([12, Prop. 2.3]) Let φpS−NR be defined in (8) with p > 1 being a positive odd integer. Then the following variants of φpS−NR are NCP-functions. φe1 (a, b) = φpS−NR (a, b) + α(a)+ (b)+ , α > 0. φe2 (a, b) = φpS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. φe3 (a, b) = φp (a, b) + α ((ab)+ )4 , α > 0. S−NR. φe4 (a, b) = φpS−NR (a, b) + α ((ab)+ )2 , α > 0. 2 φe5 (a, b) = φpS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0. In the above expressions, for any t ∈ R, we define (t)+ as max{0, t}. Proposition 3.9. ([12, Prop. 3.3]) p Let ψS−NR be defined in (9) with p > 1 being a positive odd integer. Then the following p variants of ψS−NR are NCP-functions. p ψe1 (a, b) = ψS−NR (a, b) + α(a)+ (b)+ , α > 0. p ψe2 (a, b) = ψS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. ψe3 (a, b) = ψ p (a, b) + α ((ab)+ )4 , α > 0. S−NR. p (a, b) + α ((ab)+ )2 , α > 0. ψe4 (a, b) = ψS−NR 2 p ψe5 (a, b) = ψS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0.. In the above expressions, for any t ∈ R, we define (t)+ as max{0, t}.. 4. Some properties. We would like to show some properties about these NCP functions. Proposition 4.1. Let φpNR be defined in (7) with p > 1 being a positive odd integer. Then ∂φpNR ∂φpNR · ≥ 0, ∀(a, b) ∈ {(a, b)|a ≤ b} ∪ {(a, b)|a > b > 0}. ∂a ∂b Moreover. ∂φpNR ∂φpNR · = 0, ∀(a, b) ∈ {(a, b)|a ≤ b}. ∂a ∂b. Proof. According proposition 3.4, we have that ∂φpNR = p · ap−1 − (a − b)p−2 (a − b)+ ∂a 9.

(14) ∂φpNR = p · (a − b)p−2 (a − b)+ . ∂b Therefore we obtain ∂φpNR ∂φpNR · = p2 · ap−1 − (a − b)p−2 (a − b)+ (a − b)p−2 (a − b)+ . ∂a ∂b Since p2 > 0, so we could only consider the other two product. As a > b > 0, ap−1 − (a − b)p−2 (a − b)+ = ap−1 − (a − b)p−1 > 0 and (a − b)p−2 (a − b)+ = (a − b)p−1 > 0, both terms are positive because p is an odd integer. As a ≤ b, (a − b)p−2 (a − b)+ = 0. This completes the proof. 2 Proposition 4.2. Let φpS−NR be defined in (8) with p > 1 being a positive odd integer. Then ∂φpS−NR ∂a. ·. ∂φpS−NR ∂b. > 0, ∀(a, b) ∈ {(a, b)|a 6= b} ∪ {(a, b)|a > b > 0} ∪ {(a, b)|b > a > 0}. Proof. (i) For a > b, according to Proposition 3.5, we have that ∂φpS−NR ∂a. = p · ap−1 − (a − b)p−1. ∂φpS−NR ∂b. . = p · ((a − b)p−1 ).. Therefore we have ∂φpS−NR ∂a. ·. ∂φpS−NR ∂b. = p2 · ap−1 − (a − b)p−1 )((a − b)p−1 .. Observe that (a − b)p−1 > 0 since p is an odd integer, and through basic algebra, we k know ap−1 − (a − b)p−1 = ak − (a − b) ak + (a − b)k , where k ∈ N. As a > b > 0, ak − (a − b)k > 0 and ak + (a − b)k > 0. (ii) For a < b, the same arguments apply for this case.. 2. p Proposition 4.3. Let ψS−NR be defined in (9) with p > 1 being a positive odd integer. Then, we have p p (a, b) = 0 ψS−NR (a, b) = 0 ⇐⇒ ∇ψS−NR p Proof. We already know that ψS−NR is a NCP-function from proposition 3.3, which implies that p ψS−NR (a, b) = 0 ⇐⇒ a ≥ 0, b ≥ 0, ab = 0 p This is sufficient to know that ∇ψS−NR (a, b) = 0. On the other hand, suppose that p p ∇ψS−NR (a, b) = 0, then if a = b, we have a = b = 0, this gives that ψS−NR (a, b) = 0.. 10.

(15) Without losing of generality, let a > b, with the assumption, we know ap−1 bp − (a − b)p−1 bp = 0 and ap bp−1 − (a − b)p bp−1 + (a − b)p−1 bp = 0. Therefore, ap−1 bp − (a − b)p−1 bp = 0 ⇐⇒ bp · [ap−1 − (a − b)p−1 ] = 0 p If bp = 0, then b = 0, this shows ψS−NR (a, 0) = 0. If ap−1 − (a − b)p−1 = 0, with the ap bp−1 − (a − b)p bp−1 + (a − b)p−1 bp = 0, we find that. a(a − b)p−1 bp−1 − (a − b)p bp−1 + (a − b)p−1 bp = 0 ⇔ abp−1 − (a − b)bp−1 + bp = 0 ⇔ bp−1 [a − (a − b) + b] = 0 ⇔ 2bp = 0. This completes the proof.. 2. Proposition 4.4. Let φpD−FB be defined in (5) with p > 1 being a positive odd integer. Then ∂φpD−FB ∂φpD−FB · > 0, ∂a ∂b if a > 0 and b > 0. Proof. According proposition 2.4, we have that ∂φpD−FB ∂a ∂φpD−FB ∂b. =p·a =p·b. √ √. a2. +. b2 )p−2. p−1. − (a + b). a2 + b2 )p−2 − (a + b)p−1. . . Therefore we know ∂φpD−FB ∂a. ·. ∂φpD−FB. h √ i h √ i = p a( a2 + b2 )p−2 − (a + b)p−1 · p b( a2 + b2 )p−2 − (a + b)p−1. ∂b h i √ = p2 ab(a2 + b2 )p−2 + (a + b)2p−2 − (a + b)p−1 ( a2 + b2 )p−2 · (a + b) = p2 ab(a2 + b2 )p−2 + (a + b)2p−2 − (a + b)p (c)p−2 h i √ 2 2 2 p−2 p p−2 p−2 2 2 = p ab(a + b ) + (a + b) [(a + b) −( a +b ) ]. Since , b > 0 and p is an odd number, hence it is crucial to verify that (a + b)p−2 − √ a > 0p−2 ( a2 + b 2 ) > 0 is always true. Consider f (t) = tp is a strictly increasing function when p is odd. Using the fact yields 11.

(16) a > 0, b > 0 ⇐⇒ a√ + b > 0 and ab > 0 ⇐⇒ a2 + b2 < a2 + b2 √ ⇐⇒ ( a2 + b2 )p−2√< (a2 + b2 )p−2 ⇐⇒ (a + b)p−2 − ( a2 + b2 )p−2 > 0 2. This completes the proof.. Next, we try to scribe the ”generalized gradient” about the φpS−NR . Proposition 4.5. Let φpS−NR be defined in (9) with p > 1 being a positive odd integer. Then the generalized gradient of φpS−NR as follows:  p−1 − (a − b)p−1 , (a − b)p−1 ]T if a > b,   p[a ∂φpS−NR (a, b) = {p[αap−1 , (1 − α)bp−1 ]T |α ∈ [0, 1]} if a = b,   p [ (b − a)p−1 , bp−1 − (b − a)p−1 ]T if a < b. Proof. We’ve already seen the ∂φpS−NR (a, b) as a 6= b in the reference [11]. Now consider a = b and according the definition of Clarke’s generalized gradient, φpS−NR (ai , bi ) = conv{. lim. (ai ,bi )→(a,a). ∇φpS−NR (ai , bi )|φpS−NR is dif f erentiable at (ai , bi ) ∈ R2 }. Since (ai , bi ) → (a, a), there are three cases we ought to discuss. If ai > bi for some sufficiently large i, # " # " p−1 p−1 p−1 a a − (a − b ) i i i =p . lim ∇φpS−NR (ai , bi ) = lim p p−1 (ai ,bi )→(a,a) (ai ,bi )→(a,a) 0 (ai − bi ) If ai < bi for some sufficiently large i, " lim (ai ,bi )→(a,a). If ai = bi ,then. ∇φpS−NR (ai , bi ) = lim. (ai ,bi )→(a,a). lim. p. (ai ,bi )→(a,a). (bi − ai )p−1 bip−1 − (bi − ai )p−1. #. " =p. 0 ap−1. # .. ∇φpS−NR (ai , bi ) does not exist. Therefore we know that ". φpS−NR (ai , bi ) = conv{p. ap−1 0. #. " ,p. 0 ap−1. 2. 12. #. " } = {p. αap−1 (1 − α)bp−1. # |α ∈ [0, 1]}..

(17) 5. Conclusion. In this thesis,we collect some new NCP-functions which are extensions that based on Fisher-burmeister and Natural-Residual functions.We propose some basic properties about these function. To discover more properties or apply these properties to numerical computation are two future research topics.. 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 NCPfunctions 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. [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.. 13.

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