運用廣義FB函數的平滑牛頓法解混合型互補問題

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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 smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs. 研究生：林子靖. 中華民國九十八年六月.

(2) 致. 謝. 在這份論文完成的同時，十分感謝所有陪我一路走過這兩年研究所生涯的許多人。首先，要感謝我的指導老師─陳界山教授，在專業的領域上，給予我適時地引領與啟發；在論文的撰寫上，給予我不斷地指導與幫助。研究所尋覓與探索的過程中，最重要的莫過於有指導老師的從旁協助，引領著而不迷失於研究的瓶頸與窠臼，給予我許多寶貴的經驗傳承，在與指導老師不斷的討論中，點醒了我所忽略的細節與關鍵，同時為我指引出未來明確的方向。念研究所的這段日子中，無論是生活上、課業上，都受到老師許多的照顧與鼓勵，儘管人生中總是充滿著一關一關的挑戰與關卡，但老師總在許多面向給予我建議與支持，引領著我突破難關，不斷地向前邁進、逐步地實現自我的目標。在此，十分感謝老師這些年來的含辛茹苦與諄諄教誨，再次獻上最誠摯的謝意。此外，也要感謝口試委員─柯春旭教授與朱亮儒教授，感謝兩位教授在百忙之中抽空給予指導，有教授寶貴的指正與建議，才能使得本論文的內容更加精實與完備。另外，感謝同窗─淳格與晉宇在課業上的協助，有你們的陪伴與相互砥礪，研究努力的過程，不只有汗水與辛苦，還有共同攜手扶持的誠摯友誼與互相幫助的溫馨。最後，要感謝我的家人，給予我最強力的支持；並且是我最強力的後盾，無論遇到任何困難，總是對我深具信心，有爸媽在背後支持我，我才能無憂無慮的撰寫論文。最後，由衷地感謝所有協助過我的老師、同學、家人與朋友們，感謝你們的幫助，這份論文才能順利地完成，謝謝大家。林子靖敬致.

(3) 目. 次. 1 Introduction……………………………………………1 2 Preliminary…………………………………………….3 3 The smoothing function and its properties…………….6 4 Conclusions…………………………………………..15 5 References……………………………………………15.

(4) A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs June 5, 2009. Abstract. We present a smooth approximation for the generalized Fischer-Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > 1), and establish some favorable properties for it, for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations. Key Words. Mixed complementarity problem, the generalized FB function, smoothing approximation.. 1. Introduction. The mixed complementarity problem (MCP) arises in many applications including the fields of economics, engineering, and operations research [11, 17, 18, 21] and has attracted much attention in last decade [1, 2, 16, 23, 24, 25]. A collection of nonlinear mixed complementarity problems called MCPLIB can be found in [13] and two excellent books [14, 15] are good sources for seeking theoretical backgrounds and numerical methods. Let li ∈ IR ∪ {−∞} and ui ∈ IR ∪ {+∞} be given lower and upper bounds with li < ui for i = 1, 2, . . . , n. Define l = (l1 , l2 , . . . , ln )T and u = (u1 , u2 , . . . , un )T . Given a mapping F : [l, u] → IRn with F = (F1 , F2 , . . . , Fn )T . The MCP is to find a vector x∗ ∈ [l, u] such that each component x∗i satisfies exactly one of the following implications: x∗i. x∗i = li =⇒ Fi (x∗ ) ≥ 0, ∈ (li , ui ) =⇒ Fi (x∗ ) = 0, x∗i = ui =⇒ Fi (x∗ ) ≤ 0.. (1). It is not hard to see that, when li = −∞ and ui = +∞ for all i = 1, 2, . . . , n, the MCP (1) is equivalent to solving the nonlinear system of equations F (x) = 0;. (2). whereas when li = 0 and ui = +∞ for all i = 1, 2, . . . , n, it reduces to the nonlinear complementarity problems (NCP) which is to find a point x ∈ IRn such that x ≥ 0,. F (x) ≥ 0, 1. hx, F (x)i = 0.. (3).

(5) In fact, from Theorem 2 of [12], the MCP (1) is also equivalent to the famous variational inequality problem (VIP) which is to find a vector x∗ ∈ [l, u] such that hF (x∗ ), x − x∗ i ≥ 0 ∀x ∈ [l, u].. (4). In the rest of this paper, we assume the mapping F to be continuously differentiable. It is well-known that NCP functions play an important role in the design of algorithms for the MCP (1). With an NCP function φ, the MCP (1) can be reformulated as a nonsmooth system of equations Φ(x) = 0, and consequently nonsmooth Newton methods or smoothing Newton methods can be applied for solving the system Φ(x) = 0. Among others, the latter is based on a smooth approximation of φ. In the past two decades, many smooth approximation functions and Newton-type methods using smoothing NCP functions for complementarity problems have been developed [3, 4, 9, 10, 19, 20, 23]. Most of these methods focus on the Chen-Mangasarian class of smoothing functions of the minimum NCP function or the smoothing function of the Fischer-Burmeister NCP function. Recently, an extension of the Fischer-Burmeister (FB) NCP function was considered in [5, 6, 7] by two of the authors. Specifically, they define the generalized FB function by φp (a, b) := k(a, b)kp − (a + b). ∀a, b ∈ IR,. (5). where p is an arbitrary fixed real number from the interval (1, +∞) and k(a, b)kp denotes p p the p-norm of (a, b), i.e., k(a, b)kp = |a|p + |b|p . In other words, in the function φp , they replace the 2-norm of (a, b) involved in the FB function by a more general p-norm. The function φp is still an NCP-function, that is, it satisfies the equivalence φp (a, b) = 0. ⇐⇒. a ≥ 0, b ≥ 0, ab = 0.. (6). Moreover, it turns out that φp possesses all favorable properties of the FB function; see [5, 6, 7]. For example, φp is strongly semismooth and its square is continuously differentiable everywhere on IR2 . In this paper, we are concerned with the smoothing Newton method [10] based on the generalized FB function. In Section 2, we review some definitions and preliminary results to be used in the subsequent analysis. In Section 3, we present a smooth approximation function of the generalized FB function, and studied some favorable properties for it, including the Jacobian consistency property. In Section 4, we make concluding remarks. Throughout this paper, IRn denotes the space of n-dimensional real column vectors and ei means a unit vector with ith component being 1 and the others being 0. For a differentiable mapping F , F 0 (x) and ∇F (x) denote the Jacobian of F at x and the transposed Jacobian of F , respectively. Given an index set I, the notation [F 0 (x)]II denotes 2.

(6) the submatrix consisting of the ith row and the jth column of F 0 (x) with i ∈ I and j ∈ I.. 2. Preliminary. In this section, we review some basic concepts and results that will be used in subsequent analysis. We start with introducing the concept of generalized Jacobian of a mapping. Let G : IRn → IRm be a locally Lipschitz continuous mapping. Then, G is almost everywhere differentiable by Rademacher’s Theorem (see [8]). In this case, the generalized Jacobian ∂G(x) of G at x (in the Clarke sense) is defined as the convex hull of the B-subdifferential

(7) . ∂B G(x) := V ∈ IRm×n

(8) ∃{xk } ⊆ DG : {xk } → x and G0 (xk ) → V , where DG is the set of differentiable points of G. In other words, ∂G(x) = conv∂B G(x). If m = 1, we call ∂G(x) the generalized gradient of G at x. The calculation of ∂G(x) is usually difficult in practice, so Qi proposed so-called C-subdifferential of G: ∂C G(x)T := ∂G1 (x) × ∂G2 (x) × · · · × ∂Gm (x). (7). which is easier to compute than the generalized Jacobian ∂G(x). Here, the right-hand side of (7) denotes the set of matrices in IRn×m whose i-th column is given by the generalized gradient of the i-th component function Gi . In fact, by Proposition 2.6.2 of [8], ∂G(x)T ⊆ ∂C G(x)T .. (8). In addition, we also need the P -functions and P -matrices in the subsequent sections. Definition 2.1 Let F = (F1 , F2 , . . . , Fn )T with Fi : IRn → IR for i = 1, 2, . . . , n. Then, (a) the mapping F is monotone if hx − y, F (x) − F (y)i ≥ 0 for all x, y ∈ IRn (b) the mapping F is strictly monotone if hx − y, F (x) − F (y)i > 0 for all x, y ∈ IRn and x 6= y (c) the mapping F is strong monotone with modulus µ > 0 if hx − y, F (x) − F (y)i ≥ µkx − yk2 for all x, y ∈ IRn (d) the mapping F is called a P0 -function if for all x, y ∈ IRn and x 6= y, there is an index i ∈ {1, 2, . . . , n} such that xi 6= yi and (xi − yi )(Fi (x) − Fi (y)) ≥ 0 3.

(9) (e) the mapping F is called a P -function if for all x, y ∈ IRn and x 6= y, there is an index i ∈ {1, 2, . . . , n} such that (xi − yi )(Fi (x) − Fi (y)) > 0 (f ) the mapping F is called a uniform P -function with modulus µ > 0 if there is an index i ∈ {1, 2, . . . , n} such that (xi − yi )(Fi (x) − Fi (y)) ≥ µkx − yk2. for all x, y ∈ IRn .. From above definition, we know that F is strong monotone ⇒ F is strictly monotone ⇒ F is monotone ⇓ ⇓ ⇓ F is uniform P -function ⇒ F is P -function ⇒ F is P0 -function Definition 2.2 A matrix M ∈ IRn×n is called an (a) P0 -matrix if each of its principal minors is nonnegative. (b) P -matrix if each of its principal minors is positive. From above definition, we know that M is P -matrix ⇒ M is P0 -matrix From Definition 2.1 and 2.2, we see that a continuously differentiable mapping F is a P0 -function if and only if ∇F (x) is P0 -matrix for all x ∈ IRn . For the P0 -matrix, we also have the following important property. Lemma 2.1 A matrix M ∈ IRn×n is a P0 -matrix if and only if for every nonzero vector x, there exists an index i such that xi 6= 0 and xi (M x)i ≥ 0. Next we present some favorable properties of φp whose proofs can be found in [5, 6, 7]. Lemma 2.2 Let φp : IR × IR → IR be defined by (5). Then, the following results hold. (a) φp is a strongly semismooth NCP-function. (b) Given any point (a, b) ∈ IR2 , each element in the generalized gradient ∂φp (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 IR2 satisfying |ξ| p−1 + |ζ| p−1 ≤ 1. 4.

(10) (c) The square of φp is a continuously differentiable NCP function. (d) If {(ak , bk )} ⊆ IR2 satisfies (ak → −∞) or (bk → −∞) or (ak → ∞ and bk → ∞), then we have |φp (ak , bk )| → ∞ as k → ∞.. Lemma 2.3 Let φp : IR × IR → IR be defined by (5). Then, the following limits hold. (a). lim φp (xi − li , φp (ui − xi , −Fi (x))) = −φp (ui − xi , −Fi (x)).. li →−∞. (b) lim φp (xi − li , φp (ui − xi , −Fi (x))) = φp (xi − li , Fi (x)). ui →∞. (c). lim. lim φp (xi − li , φp (ui − xi , −Fi (x))) = −Fi (x).. li →−∞ ui →∞. Proof. Let {ak } ⊆ IR be any sequence converging to +∞ as k → ∞ and b ∈ IR be any fixed number. We will prove lim φp (ak , b) = −b, and part (a) then follows by continuity k→∞. arguments. Without loss of generality, assume that ak > 0 for each k. Then, φp (ak , b) =. |ak |p + |b|p. 1/p. − (ak + b) 1/p = ak 1 + (|b|/ak )p − ak − b " p 2p 1 |b| 1 − p |b| k = a 1+ + + ···+ p ak 2p2 ak np pn |b| (1 − p) · · · (1 − pn + p) |b| +o − ak − b n k n!p a ak 1 − p |b|2p (1 − p) · · · (1 − pn + p) |b|np 1 |b|p + + · · · + = p (ak )p−1 2p2 (ak )2p−1 n!pn (ak )np−1 pn (ak )|b|np o |b|/ak + k np −b (a ) (|b|/ak )pn. where the third equality is using the Taylor expansion of the function (1 + t)1/p and the |b|np notation o(t) means limt→0 o(t)/t = 0. Since ak → +∞ as k → ∞, we have k np−1 → 0 (a ) for all n. This together with the last equation implies limk→∞ φp (ak , b) = −b. This proves part (a). Part (b) and (c) are direct by part (a) and the continuity of φFB . 2. Lemma 2.4 [22, 1.3]Let x ∈ IRn and 1 < p1 < p2 . Then kxkp2 ≤ kxkp1 ≤ n(1/p1 −1/p2 ) kxkp2 .. 5.

(11) 3. The smoothing function and its properties. For convenience, in the rest of this paper, we adopt the following index sets: Il Iu Ilu If. := := := :=. {i ∈ {1, 2, . . . , n} {i ∈ {1, 2, . . . , n} {i ∈ {1, 2, . . . , n} {i ∈ {1, 2, . . . , n}. | | | |. − ∞ < li − ∞ = li − ∞ < li − ∞ = li. < ui < ui < ui < ui. = +∞} , < +∞} , < +∞} , = +∞} .. With the generalized FB function, we define a operator Φp : IRn  φp (xi − li , Fi (x)) if    −φp (ui − xi , −Fi (x)) if Φp,i (x) := φ (x − l , φ (u − x , −F (x))) if  i p i i i   p i −Fi (x) if. (9). → IRn componentwise as i ∈ Il , i ∈ Iu , i ∈ Ilu , i ∈ If ,. (10). where the minus sign for i ∈ Iu and i ∈ If is motivated by Lemma 2.3. In fact, all results of this paper would be true without the minus sign. Using the equivalence (6), it is not hard to verify that the following result holds. Proposition 3.1 x∗ ∈ IRn is a solution of the MCP (1) if and only if x∗ solves the nonlinear system of equations Φp (x) = 0. Since φp is not differentiable at the origion, the system Φp (x) = 0 is nonsmooth. In this paper, we will find a solution of nonsmooth system Φp (x) = 0 by solving a sequence of smooth approximations Ψp (x, ε) = 0, where ε > 0 is a smoothing parameter and the operator Ψp : IRn × (0, ∞) → IRn is defined componentwise as  ψp (xi − li , Fi (x), ε) if i ∈ Il ,    −ψp (ui − xi , −Fi (x), ε) if i ∈ Iu , (11) Ψp,i (x, ε) := ψp (xi − li , ψp (ui − xi , −Fi (x), ε), ε) if i ∈ Ilu ,    −Fi (x) if i ∈ If , with ψp (a, b, ε) :=. p p |a|p + |b|p + |ε|p − (a + b).. (12). In what follows, we concentrate on the favorable properties of the smoothing function ψp and the operator Ψp . First, let us state the favorable properties of ψp . Lemma 3.1 Let ψp : IR3 → IR be defined by (12). Then, the following result holds. (a) For any fixed ε > 0, ψp (a, b, ε) is continuously differentiable for all (a, b) ∈ IR2 with −2 <. ∂ψp (a, b, ε) ∂ψp (a, b, ε) < 0, −2 < < 0. ∂a ∂b 6. (13).

(12) (b) For any fixed (a, b) ∈ IR2 , ψp (a, b, ε) is continuously differentiable, strictly increasing and convex with respect to ε > 0. Moreover, for any 0 ≤ ε1 ≤ ε2 , 0 ≤ ψp (a, b, ε2 ) − ψp (a, b, ε1 ) ≤ ε2 − ε1 .. (14). In particular, |ψp (a, b, ε) − φp (a, b)| ≤ ε for all ε ≥ 0. 2. (c) For any fixed (a, b) ∈ IR , let. ψp0 (a, b). := lim ε↓0. ∂ψp (a, b, ε) ∂ψp (a, b, ε) , ∂a ∂b. . Then,. φp (a + h1 , b + h2 ) − φp (a, b) − ψp0 (a + h1 , b + h2 )T h lim = 0. h=(h1 ,h2 )→(0,0) khk (d) For any given ε > 0, if ψp (a, b, ε) = 0, then a > 0, b > 0, min{a, b} ≤ √ p In particular, if ψp (a, b, ε) = 0, then a > 0, b > 0, ab ≤. ε 2p. −2. .. ε2 when p ≥ 2. 2. Proof. (a) Using an elementary calculation, we immediately obtain that ∂ψp (a, b, ε) = p ∂a p ∂ψp (a, b, ε) = p ∂b p. sgn(a)|a|p−1 |a|p. +. |b|p. +. |ε|p. sgn(b)|b|p−1 |a|p. +. |b|p. +. |ε|p. p−1 − 1, p−1 − 1.. (15). ∂ψp (a, b, ε) ∂ψp (a, b, ε) and are continuous at every (a, b) ∈ IR2 , ∂a ∂b we have that ψp (a, b, ε) is continuously differentiable for all (a, b) ∈ IR2 . Noting that For any fixed ε > 0, since.

(13)

(14)

(15)

(16)

(17)

(18) sgn(a)|a|p−1

(19)

(20)

(21) p p−1

(22) < 1 and

(23) p

(24) |a|p + |b|p + εp

(25)

(26).

(27)

(28)

(29)

(30)

(31)

(32) sgn(b)|b|p−1

(33)

(34)

(35) p p−1

(36) < 1,

(37) p

(38) |a|p + |b|p + εp

(39)

(40). we readily get the inequality (13). (b) For any ε > 0, from an elementary calculation, we have that εp−1 ∂ψp (a, b, ε) = p p−1 > 0, ∂ε p p p p |a| + |b| + ε ∂ 2 ψp (a, b, ε) (p − 1)εp−2 εp = p 1− p ≥ 0. ∂ε2 |a| + |b|p + εp ( p |a|p + |b|p + εp )p−1 7.

(41) Therefore, for any fixed (a, b) ∈ IR2 , ψp (a, b, ε) is continuously differentiable, strictly increasing and convex with respect to ε > 0. By the mean-value theorem, for any 0 < ε1 ≤ ε2 , there exists some ε0 ∈ (ε1 , ε2 ) such that ∂ψp ψp (a, b, ε2 ) − ψp (a, b, ε1 ) = (a, b, ε0 )(ε2 − ε1 ). ∂ε p Together with ∂ψ (a, b, ε0 ) ≤ 1, we have that (14) holds for all 0 < ε1 ≤ ε2 . Letting ∂ε ε1 ↓ 0, the desired result then follows. (c) Using the formula (15), it is easy to calculate that  sgn(a)|a|p−1   p p−1 − 1 if (a, b) 6= (0, 0), ∂ψp (a, b, ε) p p p lim = |a| + |b| ε↓0  ∂a  −1 if (a, b) = (0, 0);  sgn(b)|b|p−1   p−1 − 1 if (a, b) 6= (0, 0), p ∂ψp (a, b, ε) p p p = (16) lim |a| + |b| ε↓0  ∂b  −1 if (a, b) = (0, 0). (a,b) ∂φp (a,b) From this, we see that ψp0 (a, b) = ∂φp∂a , ∂b at (a, b) 6= (0, 0). Therefore, we only need to check the case (a, b) = (0, 0). The desired result follows by φp (h1 , h2 ) − φp (0, 0) − ψp0 (h1 , h2 )T h p |h1 |p + |h2 |p = p |h1 |p + |h2 |p − p ( p |h1 |p + |h2 |p )p−1 p p = p |h1 |p + |h2 |p − p |h1 |p + |h2 |p = 0. (d) From the definition of ψp (a, b, ε), clearly, ψp (a, b, ε) = 0 implies a + b ≥ 0, and hence a ≥ 0 or b ≥ 0. In addition, from the monotonicity of p-norm, if a ≥ 0, b ≤ 0 or a ≤ 0, b ≥ 0, we have p p p |a|p + |b|p + εp > p |a|p + |b|p ≥ max{|a|, |b|} ≥ a + b, which implies ψp (a, b, ε) > 0. The two sides show that for any given ε > 0, ψp (a, b, ε) = 0 implies a > 0 and b > 0. Without loss of generality, we let 0 < a ≤ b. For any fixed a > 0, consider the function f (t) = (t + a)p − tp − ap − εp (t ≥ 0). It is easy to verify that f is strictly increasing on [0, +∞). Moreover, since ψp (a, b, ε) = 0, we have f (b) = 0. Hence ε ε f (a) = (2p − 2)ap − εp ≤ 0, we get that a ≤ √ . Therefore, min{a, b} ≤ √ . p p p p 2 −2 2 −2 Moreover, if p ≥ 2, let x = (a, b, ε) ∈ IR3 , by lemma 2.4 we have kxkp ≤ kxk2 . Hence p p a + b = p |a|p + |b|p + εp ≤ |a|2 + |b|2 + ε2 ⇒ (a + b)2 ≤ a2 + b2 + ε2 ε2 ⇒ ab ≤ . 2 8.

(42) The proof is thus complete.. 2. Using Lemma 3.1 and the expression of Ψp , we readily obtain the following result. Proposition 3.2 Let Ψp : IRn × (0, ∞) → IRn be defined by (11). Then, (a) for any fixed ε > 0, Ψp (x, ε) is continuously differentiable on IRn with ∇x Ψp (x, ε) = Da (x, ε) + ∇F (x)Db (x, ε), where Da (x, ε) and Db (x, ε) are n × n diagonal matrices with the diagonal elements (Da )ii (x, ε) and (Db )ii (x, ε) defined as follows: (a1) For i ∈ Il , (Da )ii (x, ε) =. sgn(xi − li )|xi − li |p−1 − 1, k(xi − li , Fi (x), ε)kp−1 p. (Db )ii (x, ε) =. sgn(Fi (x))|Fi (x)|p−1 − 1. k(xi − li , Fi (x), ε)kpp−1. (a2) For i ∈ Iu , (Da )ii (x, ε) =. sgn(ui − xi )|ui − xi |p−1 − 1, k(ui − xi , Fi (x), ε)kp−1 p. (Db )ii (x, ε) =. −sgn(Fi (x))|Fi (x)|p−1 − 1. k(ui − xi , Fi (x), ε)kpp−1. (a3) For i ∈ Ilu , (Da )ii (x, ε) = ai (x, ε) + bi (x, ε)ci (x, ε) and (Db )ii (x, ε) = bi (x, ε)di (x, ε) with ai (x, ε) =. sgn(xi − li )|xi − li |p−1 − 1, k(xi − li , ψp (ui − xi , −Fi (x), ε), ε)kp−1 p. bi (x, ε) =. sgn(ψp (ui − xi , −Fi (x), ε))|ψp (ui − xi , −Fi (x), ε)|p−1 − 1, k(xi − li , ψp (ui − xi , −Fi (x), ε), ε)kp−1 p. ci (x, ε) = − di (x, ε) =. sgn(ui − xi )|ui − xi |p−1 + 1, k(ui − xi , Fi (x), ε)kp−1 p. sgn(Fi (x))|Fi (x)|p−1 + 1. k(ui − xi , Fi (x), ε)kp−1 p 9.

(43) (a4) For i ∈ If , (Da )ii (x, ε) = 0 and (Db )ii (x, ε) = −1. Moreover, −2 < (Da )ii (x, ε) < 0 and −2 < (Db )ii (x, ε) < 0 for all i ∈ Il ∪ Iu and −6 < (Da )ii (x, ε) < 0 and −4 < (Db )ii (x, ε) < 0 for all i ∈ Ilu . (b) For any given ε1 ≥ 0 and ε2 ≥ 0, we have kΨp (x, ε2 ) − Ψp (x, ε1 )k ≤. √ √ p n( 2 + 1)|ε2 − ε1 |,. ∀x ∈ IRn .. Particularly, for any given ε ≥ 0, kΨp (x, ε) − Φp (x)k ≤. √. √ p n( 2 + 1)ε,. ∀x ∈ IRn .. To show that the smoothing operator Ψp satisfies the Jacobian consistency property, we need the following characterization of the generalized Jacobian ∂C Φp (x), which is direct by Lemma 2.2 (b). Proposition 3.3 For any given x ∈ IRn , we have ∂C Φp (x)T = {Da (x) + ∇F (x)Db (x)}, where Da (x), Db (x) are n × n diagonal matrices whose diagonal elements are given below: (a) For i ∈ Il , if (xi − li , Fi (x)) 6= (0, 0), then sgn(xi − li ) · |xi − li |p−1 − 1, k(xi − li , Fi (x))kp−1 p sgn(Fi (x)) · |Fi (x)|p−1 − 1, (Db )ii (x) = k(xi − li , Fi (x))kpp−1 (Da )ii (x) =. and otherwise n o p p ((Da )ii (x), (Db )ii (x)) ∈ (ξ − 1, ζ − 1) ∈ IR2 | |ξ| p−1 + |ζ| p−1 ≤ 1 . (b) For i ∈ Iu , if (ui − xi , −Fi (x)) 6= (0, 0), then sgn(ui − xi ) · |ui − xi |p−1 − 1, k(ui − xi , −Fi (x))kp−1 p sgn(Fi (x)) · |Fi (x)|p−1 (Db )ii (x) = − − 1, k(ui − xi , −Fi (x))kp−1 p (Da )ii (x) =. and otherwise n o p p ((Da )ii (x), (Db )ii (x)) ∈ (ξ − 1, ζ − 1) ∈ IR2 | |ξ| p−1 + |ζ| p−1 ≤ 1 . 10.

(44) (c) For i ∈ Ilu , (Da )ii (x) = ai (x) + bi (x)ci (x) and (Db )ii (x) = bi (x)di (x) , where if (xi − li , φp (ui − xi , −Fi (x))) 6= (0, 0), then sgn(xi − li ) · |xi − li |p−1 − 1, k(xi − li , φp (ui − xi , −Fi (x))kp−1 p sgn (φp (ui − xi , −Fi (x))) · |φp (ui − xi , −Fi (x))|p−1 bi (x) = − 1, k(xi − li , φp (ui − xi , −Fi (x))kp−1 p ai (x) =. and otherwise n o p p (ai (x), bi (x)) ∈ (ξ − 1, ζ − 1) ∈ IR2 | |ξ| p−1 + |ζ| p−1 ≤ 1 ; and if (ui − xi , −Fi (x)) 6= (0, 0), then −sgn(ui − xi ) · |ui − xi |p−1 + 1, k(ui − xi , −Fi (x))kpp−1 sgn (Fi (x)) · |Fi (x)|p−1 di (x) = + 1, k(ui − xi , −Fi (x))kp−1 p ci (x) =. and otherwise n o p p 2 p−1 p−1 (ci (x), di (x)) ∈ (ξ + 1, ζ + 1) ∈ IR | |ξ| + |ζ| ≤1 . (d) For i ∈ If , (Da )ii (x) = 0 and (Db )ii (x) = −1.. Proposition 3.4 Let Ψp be defined by (11). Then, for any fixed x ∈ IRn , lim dist(∇x Ψp (x, ε)T , ∂C Φp (x)) = 0. ε↓0. Proof. For the sake of notation, for any given x ∈ IRn , we define the index sets: β1 (x) := {i ∈ Il | (xi − li , Fi (x)) = (0, 0)}, β¯1 (x) := {i ∈ Il | (xi − li , Fi (x)) 6= (0, 0)}, β2 (x) := {i ∈ Iu | (ui − xi , Fi (x)) = (0, 0)}, β¯2 (x) := {i ∈ Iu | (ui − xi , Fi (x)) 6= (0, 0)}, β3 (x) := {i ∈ Ilu | (xi − li , φp (ui − xi , −Fi (x))) = (0, 0)}, β¯3 (x) := {i ∈ Ilu | (xi − li , φp (ui − xi , −Fi (x))) 6= (0, 0)}, β4 (x) := {i ∈ β¯3 (x) | (ui − xi , Fi (x)) = (0, 0)}, β¯4 (x) := {i ∈ β¯3 (x) | (ui − xi , Fi (x)) 6= (0, 0)}. We proceed the arguments by the cases i ∈ Il ∪ Iu , i ∈ Ilu and i ∈ If , respectively. 11. (17).

(45) Case 1: i ∈ Il ∪ Iu . When i ∈ β1 (x) ∪ β2 (x), it is easy to see that (Da )ii (x, ε) = −1 and (Db )ii (x, ε) = −1. From Proposition 3.2 (a1) and (a2), it then follows that ∇x Ψp,i (x, ε)T = −eTi − Fi0 (x) for all ε > 0. Since. n o p p 2 p−1 p−1 (−1, −1) ∈ (ξ − 1, ζ − 1) ∈ IR | |ξ| + |ζ| ≤1 ,. (18). by Proposition 3.3 (a) and (b) we get ∇x Ψp,i (x, ε)T ∈ ∂C Φp,i (x). When i ∈ β¯1 (x) ∪ β¯2 (x), lim (Da )ii (x, ε) = (Da )ii (x) and lim (Db )ii (x, ε) = (Db )ii (x), ε↓0. ε↓0. which by Proposition 3.2 (a1) and (a2) implies lim ∇x Ψp,i (x, ε)T = (Da )ii (x)eTi + (Db )ii (x)Fi0 (x) ∈ ∂C Φp,i (x). ε↓0. Since Il ∪ Iu = β1 (x) ∪ β2 (x) ∪ β¯1 (x) ∪ β¯2 (x), the last two subcases show that lim ∇x Ψp,i (x, ε)T ∈ ∂C Φp,i (x), ε↓0. ∀ i ∈ Il ∪ Iu .. (19). Case 2: i ∈ Ilu . When i ∈ β3 (x), clearly, ai (x, ε) = −1. Notice that φp (ui − xi , −Fi (x)) = 0 and xi − li = 0 imply ui − xi > 0 and Fi (x) = 0. Therefore, lim bi (x, ε) ε↓0. = lim ε↓0. sgn(ψp (ui − xi , −Fi (x), ε))|ψp (ui − xi , −Fi (x), ε)|p−1 −1 k(xi − li , ψp (ui − xi , −Fi (x), ε), ε)kp−1 p. ψp (ui − xi , 0, ε)p−1 −1 ε↓0 k(0, ψp (ui − xi , 0, ε), ε)kp−1 p 1 = lim s p !p−1 − 1 ε↓0 ε p 1+ ψp (ui − xi , 0, ε). = lim. = −1 where the last equality is by ε ε↓0 ψp (ui − xi , 0, ε) p ( p (ui − xi )p + εp )p−1 = lim ε↓0 εp−1 = ∞ lim. 12.

(46) which used L’Hospital’s rule and lim ci (x, ε) = 0, di (x, ε) = 1 and ci (x) = 0, di (x) = 1. ε↓0. From Proposition 3.2 (a3) and Proposition 3.3 (c) and (18), it follows that lim ∇x Ψp,i (x, ε)T = −eTi − Fi0 (x) ∈ ∂C Φp,i (x), ε↓0. i ∈ β3 (x).. When i ∈ β¯3 (x), we have limε↓0 ai (x, ε) = ai (x) and limε↓0 bi (x, ε) = bi (x). Also, ci (x, ε) = 1, di (x, ε) = 1. for i ∈ β4 (x). and lim ci (x, ε) = ci (x), lim di (x, ε) = di (x) for i ∈ β¯4 (x). ε↓0. ε↓0. Using Proposition 3.3 (c) and noting that n o p p (1, 1) ∈ (ξ + 1, ζ + 1) ∈ IR2 | |ξ| p−1 + |ζ| p−1 ≤ 1 , we get limε↓0 ∇x Ψp,i (x, ε)T ∈ ∂C Φp,i (x) for i ∈ β¯3 (x). Along with the above discussions, lim ∇x Ψp,i (x, ε)T ∈ ∂C Φp,i (x) for i ∈ Ilu . ε↓0. (20). Case 3: i ∈ If . By Proposition 3.2 (a4) and Proposition 3.3 (d), it is obvious that lim ∇x Ψp,i (x, ε)T ∈ ∂C Φp,i (x) for i ∈ If . ε↓0. Now the desired result follows from (19)–(21) and {1, 2, . . . , n} = If ∪ Il ∪ Iu ∪ Ilu .. (21) 2. In order to use Newton method, we need the Jacobian matrix of Ψp is nonsingular. Proposition 3.5 For any fixed ε > 0, the Jacobian matrix of Ψp at any x ∈ IRn is nonsingular if F is a P0 -function and the submatrix [F 0 (x)]If If is nonsingular. Particularly, if If = ∅, the Jacobian matrix of Ψp at any x ∈ IRn is nonsingular if and only if F is a P0 -function. Proof. For any given ε > 0, the Jacobian matrix of Ψp at any x ∈ IRn is ∇x Ψp (x, ε)T = Da (x, ε) + Db (x, ε)F 0 (x) where Da (x, ε) and Db (x, ε) are n×n diagonal matrices whose diagonal elements (Da )ii (x, ε) and (Db )ii (x, ε) are negative for i ∈ Il ∪ Iu ∪ Ilu , and (Da )ii (x, ε) = 0, (Db )ii (x, ε) = −1 for i ∈ If . Now suppose that ∇x Ψp (x, ε)T z = 0. Then, zi = −. (Db )ii (x, ε) 0 (F (x)z)i , (Da )ii (x, ε) 13. for i ∈ Il ∪ Iu ∪ Ilu. (22).

(47) and (F 0 (x)z)i = 0,. for i ∈ If .. (23). Since F is a continuously differentiable P0 -function, F 0 (x) is a P0 -matrix. From Lemma 2.1, we get zi = 0 for i ∈ Il ∪ Iu ∪ Ilu . Substituting this into (23) then gives [F 0 (x)If If ]zIf = 0, where zIf is a vector consisting of zi with i ∈ If . This along with the nonsingularity of [F 0 (x)]If If implies zi = 0 for i ∈ If . Thus, we prove z = 0, and consequently the first part of the conclusions follows. The second part is implied by the above arguments. 2. Remark 3.1 We want to point out when p → +∞, the diagonal elements (Da )ii (x, ε) and (Db )ii (x, ε) for i ∈ Il ∪ Iu ∪ Ilu will tend to 0, though (Da )ii (x, ε) + (Db )ii (x, ε) < 0. This implies that for a larger p the nonsingularity of ∇Ψp (x, ε) actually requires stronger conditions than those given by Proposition 3.5. The boundedness of level sets of kΦp (x)k is also important since it ensures that the sequences generated by a descent method has at least one accumulation point. The following proposition is to prove that L(γ) := {x ∈ IRn | kΦp (x)k ≤ γ}. (24). are bounded. Proposition 3.6 The level sets L(γ) are bounded for all γ > 0 if one of the following two conditions is satisfied: (a) If li and ui are bounded for all i ∈ {1, 2, . . . , n}. (b) F is a uniform P -function. Proof. Under the condition (a), we have {1, 2, . . . , n} = Ilu . The result is clear by the definition of Φp and Lemma 2.2 (d). Next we prove the boundedness of L(γ) under the condition (b). Suppose that there exists some γ > 0 such that L(γ) is unbounded, i.e., there exists a sequence {xk } ⊆ L(γ) such that kxk k → ∞. Define the index set . J := i ∈ {1, 2, . . . , n} | {xki } is unbounded . Then J 6= ∅. We choose a bounded sequence y k with 0 if i ∈ J, k yi = xki otherwise. 14.

(48) Since F is a uniform P -function, there is a constant µ > 0 such that µkxk − y k k2 ≤ max (xki − yik )(Fi (xk ) − Fi (y k )) 1≤i≤n. = max(xki )(Fi (xk ) − Fi (y k )) ≤. i∈J |xkj0 ||Fj0 (xk ). − Fj0 (y k )|. where j0 is an index from {1, 2, · · · , n} for which the maximum is attained. Here we have, without loss of generality, assumed to be independent of k. Clearly, j0 ∈ J, which means that {xkj0 } is unbounded. Consequently, there exists a subsequence, assumed to be {xkj0 } without loss of generality, such that |xkj0 | → ∞. Notice that kxk − y k k2 ≥ |xkj0 − yjk0 |2 = |xkj0 |2. for each k.. Therefore, µ|xkj0 |2 ≤ |xkj0 ||Fj0 (xk ) − Fj0 (y k )| and µ|xkj0 | ≤ |Fj0 (xk ) − Fj0 (y k )| ≤ |Fj0 (xk )| + |Fj0 (y k )|, which implies |Fj0 (xk )| → ∞ as |xkj0 | → ∞. Thus, we prove that |xkj0 | → +∞ and |Fj0 (xk )| → +∞. Using the last equation and Lemma 2.2 (d), we have |Φp,j0 (xk )| → +∞ from the definition of Φp . This contradicts the fact that {xk } ⊆ L(γ). 2. 4. Conclusions. In this paper, we have studied the smoothing Newton method [10] based on the smooth approximation ψp of the generalized FB function, and smooth operator Ψp is shown to possess the Jacobian consistence. We also believe both Proposition 3.5 and Proposition 3.6 may be useful in general smoothing algorithms for MCP.. References [1] S. C. Billups, S. P. Dirkse and M. C. Soares, A comparison of algorithms for large scale mixed complementarity problems, Computational Optimization and Applications, vol. 7, pp. 3–25, 1997. [2] S. C. Billups, and M. C. Soares, QPCOMP: A quadratic programming based solver for mixed complementarity problems, Mathematical Programming, vol. 76, pp. 533–562, 1997. 15.

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