一個基於一般性Fischer-Burmeister函數的NCP函數

全文

(1)On an NCP-function Based on the Generalized Fischer-Burmeister Function December 28, 2007. Abstract Over the past years, there has been much effort on the solution and analysis of the nonlinear complementarity problem (NCP) by means of a reformulation of the problem as an equivalent unconstrained optimization problem involving a merit function. In this thesis, we study a new merit function for the NCP based on the generalized Fischer-Burmeister function φp (a, b) = k(a, b)kp −(a+b). In particular, we list conditions under which the function provides a global error bound, and conditions under which the function has bounded level sets. Key Words. NCP, NCP-function, merit function, bounded level sets, global error bound.. 1. Introduction. The nonlinear complementarity problem (NCP) [12, 18] is to find a point x ∈ IRn such that x ≥ 0,. F (x) ≥ 0,. hx, F (x)i = 0,. (1). where h·, ·i is the Euclidean inner product, and F = (F1 , F2 , · · · , Fn )T is a map from IRn to IRn . We assume that F is continuously differentiable throughout this thesis. The NCP has attracted much attention because of its applications in economics, engineering, and operations research [6, 12], to name a few. Many methods have been proposed to solve the NCP [10, 12, 18]. One of the most powerful and popular ways studied intensively recently is to reformulate the NCP as a system of nonlinear equations [16, 19, 23], or an unconstrained minimization problem [7, 8, 9, 11, 13, 14, 20, 22]. The objective function that can constitute an equivalent unconstrained minimization problem is called a merit function. That is, a merit function 1.

(2) is a function whose global minima are coincident with the solutions of the original NCP. To construct a merit function, a class of functions called NCP-functions and defined below plays a significant role. Definition 1.1 A function φ : IR2 → IR is called an NCP-function if it satisfies φ(a, b) = 0. ⇐⇒. a ≥ 0, b ≥ 0, ab = 0.. The Fischer-Burmeister function is a well-known NCP-function defined as √ φFB (a, b) = a2 + b2 − (a + b).. (2). (3). Thus, the NCP is equivalent to a system of nonsmooth equations:      ΦFB (x) =     . φFB (x1 , F1 (x)) · · · φFB (xn , Fn (x)).       = 0.    . (4). Then the function ΨFB : IRn → IR+ defined by n 1 1X ΨFB (x) := kΦFB (x)k2 = φ (xi , Fi (x))2 2 2 i=1 FB. (5). is a merit function for the NCP; that is, the NCP can be rewritten as an unconstrained minimization problem: minn ΨFB (x). (6) x∈IR. We define the generalized FB function φp : IR2 → IR by φp (a, b) := k(a, b)kp − (a + b),. (7). where p is a fixed real number in the interval (1, ∞), and k(a, b)kp denotes the p-norm q p of (a, b); that is, k(a, b)kp = |a|p + |b|p . In other words, in the function φp , we replace the 2-norm of (a, b) in the FB function (3) by a more general p-norm with p ∈ (1, ∞). The function φp is still an NCP-function. Besides, we define ψp : IR2 → IR+ by 1 ψp (a, b) := |φp (a, b)|2 . 2 2. (8).

(3) For every given p > 1, the function ψp is a nonnegative NCP-function, and smooth on IR2 . Like ΦFB , the function Φp : IRn → IRn given as      Φp (x) =     . φp (x1 , F1 (x)) · · · φp (xn , Fn (x)).          . (9). yields a family of merit functions Ψp : IRn → IR for the NCP n n X 1 1X φp (xi , Fi (x))2 = ψp (xi , Fi (x)). Ψp (x) := kΦp (x)k2 = 2 2 i=1 i=1. (10). Moreover, the natural residual ΨNR : IRn → IRn is a merit function defined by ΨNR (x) :=. n X i=1. φ2NR (xi , Fi (x)),. (11). where φNR : IR2 → IR is an NCP-function given by φNR (a, b) = min{a, b}. The natural residual is very important in our subsequent analysis. In this thesis, we investigate the following merit function Ψα,p : IRn → IR for the NCP: n Ψα,p (x) :=. X. ψα,p (xi , Fi (x)),. (12). i=1. where ψα,p : IR2 → IR is a nonnegative function given by µ. ¶2. α 1 ψα,p (a, b) = (ab)+ 2 + k(a, b)kp − (a + b) 2 2. ,. (13). and α ≥ 0 is a real parameter. Note that for every real number z, (z)+ equals z if z ≥ 0, and equals 0 if z < 0. When α = 0, the function ψα,p reduces to ψp . Hence, ψα,p is an extension of ψp . We show that ψα,p enjoys many favorable properties of ψp . Additionally, when α > 0, ψα,p has a significant property ψp does not have. In particular, we give conditions under which Ψα,p provides a global error bound for the NCP, and conditions 3.

(4) under which Ψα,p has bounded level sets. This thesis is organized as follows. In Section 2, we review some definitions and preliminary results to be used in the subsequent analysis. In Section 3, we show some important properties of the proposed merit function. In Section 4, we list a derivativefree descent method proposed in [24]. In Section 5, we make concluding remarks. Throughout this thesis, IRn denotes the space of n-dimensional real column vectors and T denotes transpose. For every differentiable function f : IRn → IR, ∇f (x) denotes the gradient of f at x. For every differentiable mapping F = (F1 , · · · , Fm )T : IRn → IRm , ∇F (x) = (∇F1 (x) · · · ∇Fm (x)) denotes the transpose Jacobian of F at x. We denote by kxkp the p-norm of x, and by kxk the Euclidean norm of x. In addition, unless otherwise stated, we always suppose that, in the sequel, p is a fixed real number in (1, ∞). The level set of a function Ψ : IRn → IR is denoted L(Ψ, c) := {x ∈ IRn | Ψ(x) ≤ c}.. 2. Preliminaries. In this section, we recall some definitions and preliminary results that will play an important role in the subsequent analysis. Definition 2.1 Let F : IRn → IRn . Then we have: (a) F is monotone if hx − y, F (x) − F (y)i ≥ 0 for all x, y ∈ IRn . (b) F is strictly monotone if hx − y, F (x) − F (y)i > 0 for all x, y ∈ IRn and x 6= y. (c) F is strongly monotone with modulus µ > 0 if hx − y, F (x) − F (y)i ≥ µkx − yk2 for all x, y ∈ IRn . (d) F is a P0 -function if 1≤i≤n max (xi − yi )(Fi (x) − Fi (y)) ≥ 0 for all x, y ∈ IRn and x 6= y. xi 6=yi. (e) F is a P -function if max (xi − yi )(Fi (x) − Fi (y)) > 0 for all x, y ∈ IRn and x 6= y. 1≤i≤n. 4.

(5) (f ) F is a uniform P -function with modulus µ > 0 if max (xi − yi )(Fi (x) − Fi (y)) ≥ 1≤i≤n. µkx − yk2 for all x, y ∈ IRn . (g) ∇F (x) is uniformly positive definite with modulus µ > 0 if dT ∇F (x)d ≥ µkdk2 for all x ∈ IRn and d ∈ IRn . (h) F is Lipschitz continuous if there exists a constant L > 0 such that kF (x)−F (y)k ≤ Lkx − yk for all x, y ∈ IRn . From the above definitions, it is easy to see that strongly monotone functions are strictly monotone, and strictly monotone functions are monotone. Moreover, F is a P0 -function if F is monotone, and F is a uniform P -function with modulus µ > 0 if F is strongly monotone with modulus µ > 0. In addition, when F is continuously differentiable, we have the following conclusions. 1. F is monotone if and only if ∇F (x) is positive semidefinite for all x ∈ IRn . 2. F is strictly monotone if ∇F (x) is positive definite for all x ∈ IRn . 3. F is strongly monotone if and only if ∇F (x) is uniformly positive definite. We also review some concepts to be used when we discuss boundedness of the solution set of the NCP. Definition 2.2 The NCP is said to be strictly feasible if there exists an xˆ > 0 such that F (ˆ x) > 0.. Lemma 2.1 [2, Lemma 4.2] If F is strongly monotone, then the NCP has at most one solution. The error bound can tell us how close an arbitrary point is to the solution set of the NCP. The following lemma shows that the natural residual provides a global error bound for the NCP under certain conditions. 5.

(6) Lemma 2.2 [17] Let ΨNR be defined by (11). Suppose that F is strongly monotone with modulus µ > 0, and Lipschitz continuous with constant L > 0 on some convex set S ⊆ IRn . Let x∗ be the unique solution of the NCP. Then the following inequality holds: kx − x∗ k ≤. 1 L+1 ΨNR (x) 2 . µ. Next, we list some important properties of ψFB defined by ψFB (a, b) =. ´2 1 ³√ 2 a + b2 − (a + b) . 2. (14). Lemma 2.3 [11, 21] The function ψFB defined by (14) has the following properties: (a) ψFB is an NCP-function; that is, it satisfies (2). (b) ψFB (a, b) ≥ 0 for all (a, b) ∈ IR2 . (c) ψFB is continuously differentiable everywhere. (d) ∇a ψFB (a, b) · ∇b ψFB (a, b) ≥ 0 for all (a, b) ∈ IR2 . The equality holds if and only if φFB (a, b) = 0. (e) ∇a ψFB (a, b) = 0 ⇐⇒ ∇b ψFB (a, b) = 0 ⇐⇒ φFB (a, b) = 0. (f ). 1 (2 2. −. √. 2)2 min{a, b}2 ≤ ψFB (a, b) ≤ 12 (2 +. √. 2)2 min{a, b}2 for all (a, b)T ∈ IR2 .. Thus, it is obvious that ψFB , ∇a ψFB , and ∇b ψFB are all NCP-functions. Important results about the merit function ΨFB defined by (5) are summarized in the following theorem. We want to point out that most results in Lemma 2.2 (e.g. part(a)-(e)) and Theorem 2.1 (e.g. part(a)-(b), (d)) are also true for φp , ψp , and Ψp . For more detailed information and proofs, please refer to [3, 4, 5]. Theorem 2.1 [11, 21] The function ΨFB defined by (5) has the following properties: (a) If the NCP has at least one solution, then the NCP is equivalent to the unconstrained optimization problem minx∈IRn ΨFB (x). 6.

(7) (b) If F is a P0 -function, every stationary point of ΨFB is a solution of the NCP. (c) Let ΨNR be the natural residual defined by (11). Then the following inequalities hold: √ √ 1 1 (2 − 2)2 ΨNR (x) ≤ ΨFB (x) ≤ (2 + 2)2 ΨNR (x). 2 2 Moreover, if F is strongly monotone, and Lipschitz continuous, then ΨFB provides a global error bound for the NCP. (d) If F is a uniform P -function, then the level set L(ΨFB , c) is bounded for all c ∈ IR.. 3. A New Merit Function and Its Properties. In this section, we propose a new merit function, and show that it has many favorable properties. P. Let Ψα,p : IRn → IR be defined by (12) and (13); that is, Ψα,p (x) := ni=1 ψα,p (xi , Fi (x)), µ ¶2 α 1 2 2 where ψα,p : IR → IR is given by ψα,p (x) = (ab)+ + k(a, b)kp − (a + b) , and α ≥ 0 2 2 is a real parameter. The function ψα,p enjoys many favorable properties of ψp . Moreover, when α > 0, ψα,p has an important property ψp does not have.. Lemma 3.1 Let α ≥ 0. Then the function ψα,p defined by (13) has the following properties: (a) ψα,p is an NCP-function. (b) ψα,p ≥ 0 for all (a, b)T ∈ IR2 . (c) ψα,p is continuously differentiable everywhere. In particular, ∇a ψα,p (0, 0) = ∇b ψα,p (0, 0) = 0; for (a, b) 6= (0, 0), we have Ã. !. sgn(a) · |a|p−1 ∇a ψα,p (a, b) = αb(ab)+ + − 1 φp (a, b), p−1 ! Ã k(a, b)kp sgn(b) · |b|p−1 − 1 φp (a, b). ∇b ψα,p (a, b) = αa(ab)+ + k(a, b)kp−1 p 7. (15).

(8) (d) ∇a ψα,p (a, b) · ∇b ψα,p (a, b) ≥ 0 for all (a, b)T ∈ IR2 . The equality holds if and only if ψα,p (a, b)=0. (e) ∇a ψα,p (a, b) = 0 ⇐⇒ ∇b ψα,p (a, b) = 0 ⇐⇒ ψα,p (a, b) = 0. If α > 0, then the function ψα,p enjoys the following additional property: (f ) If a → −∞ or b → −∞ or ab → ∞, then ψα,p (a, b) → ∞. Proof. Since (a)-(c) are immediate consequences of Lemma 2.1, and (f) can be easily shown, we only show part (d) and (e). If (a, b) = (0, 0), (d) holds obviously. Suppose (a, b) 6= (0, 0). Then, by formula (15), we know ∇a ψα,p (a, b) · ∇b ψα,p (a, b) Ã ! p−1 sgn(a) · |a| 2 − 1 φp (a, b) = α2 ab(ab)+ + αa(ab)+ k(a, b)kpp−1 ! Ã sgn(b) · |b|p−1 − 1 φp (a, b) +αb(ab)+ k(a, b)kp−1 p Ã !Ã ! sgn(a) · |a|p−1 sgn(b) · |b|p−1 + −1 − 1 φ2p (a, b). k(a, b)kp−1 k(a, b)kpp−1 p. (16). Since ab(ab)+ 2 ≥ 0,. sgn(a) · |a|p−1 − 1 ≤ 0, k(a, b)kp−1 p. and. sgn(b) · |b|p−1 − 1 ≤ 0, k(a, b)kp−1 p. (17). it suffices to show that the second and third term of (16) are both nonnegative for all (a, b) 6= (0, 0) ∈ IR2 . To see these, we first claim that Ã. αa(ab)+. !. sgn(a) · |a|p−1 − 1 φp (a, b) ≥ 0. k(a, b)kp−1 p. (18). If a = 0, then (18) holds obviously. If a < 0, then φp (a, b) > 0, which together with the second inequality in (17) implies that (18) holds. If a > 0 and b > 0, then φp (a, b) < 0, implying (18) holds by a similar reason. If a > 0 and b ≤ 0, then (ab)+ = 0, and hence (18) holds. Thus, (18) holds for all (a, b) 6= (0, 0) ∈ IR2 . Similarly, Ã. αb(ab)+. !. sgn(b) · |b|p−1 − 1 φp (a, b) ≥ 0. k(a, b)kp−1 p 8.

(9) Consequently, ∇a ψα,p (a, b) · ∇b ψα,p (a, b) ≥ 0. From (16), ∇a ψα,p (a, b) · ∇b ψα,p (a, b)=0 if and only if {ab=0} and {a = 0 or (a ≥ 0 and b = 0) or φp (a, b)=0} and {b = 0 or (b ≥ 0 and a=0) or φp (a, b) = 0}. Thus, ∇a ψα (a, b) · ∇b ψα,p (a, b) = 0 if and only if ψα,p (a, b) = 0. Now, we consider (e). If ψα,p (a, b) = 0, then ab = 0 and φp (a, b) = 0, implying ∇a ψα,p (a, b) = 0 and ∇b ψα,p (a, b) = 0. Next, we claim that ∇a ψα,p (a, b) = 0 implies ψα,p (a, b) = 0. Suppose ∇a ψα,p (a, b) = 0. Then Ã. !. sgn(a) · |a|p−1 αb(ab)+ = − − 1 φp (a, b). k(a, b)kp−1 p. (19). Assume both sides of (19) do not vanish. Then b 6= 0. If b < 0, then the left-hand side of (19) is nonnegative, and the right-hand side of (19) is negative, which is a contradiction. If b > 0, then a > 0. Thus, the left-hand side of (19) is positive, and the right-hand side of (19) is negative, which is a contradiction, too. Therefore, both sides of (19) equal to zero, and ψα,p (a, b) = 0. Similarly, ∇b ψα,p (a, b) = 0 implies ψα,p (a, b) = 0. 2 Note that ab → ∞ does not necessarily imply ψp (a, b) → ∞. Lemma 3.1(f) is useful for proving that the level sets of Ψα,p are bounded. By Lemma 3.1, we can deduce the following theorem. Theorem 3.1 Let Ψα,p be defined by (12). Then Ψα,p (x) ≥ 0 for all x ∈ IRn and Ψα,p (x) = 0 if and only if x solves the NCP. Moreover, if the NCP has at least one solution, then x is a global minimizer of Ψα,p if and only if x solves the NCP. By Theorem 3.1, the NCP can be recast as the unconstrained minimization problem: min Ψα,p (x).. x∈IRn. (20). In general, it is hard to find a global minimum of Ψα,p . Hence, it is important to know conditions under which a stationary point of Ψα,p is a global minimum. Theorem 3.2 Let F be a P0 -function. Then x∗ ∈ IRn is a global minimum of the unconstrained optimization problem (20) if and only if x∗ is a stationary point of Ψα,p . 9.

(10) Proof. The theorem can be proven by using (d) of Lemma 3.1, and the same proof technique at that of Theorem 3.5 in [11]. 2 We will show that the functions Ψα,p and ΨNR have the same order on every bounded set. Before doing so, we need a preliminary result. Proposition 3.1 Let φp : IR2 → IR be given by (7). Then (a) for p ∈ (1, 2), we have 1. 1. (2 − 2 p )| min{a, b}| ≤ |φp (a, b)| ≤ (2 + 2 p )| min{a, b}|;. (21). (b) for p ∈ [2, ∞), we have 1. 1. (2 − 2 2 )| min{a, b}| ≤ |φp (a, b)| ≤ (2 + 2 2 )| min{a, b}|.. (22). Proof. Without loss of generality, we suppose a ≥ b. We will prove the desired results by considering the following two cases: (1) a + b ≤ 0 and (2) a + b > 0. Case(1): a + b ≤ 0. In this case, we have |φp (a, b)| ≥ k(a, b)kp ≥ |b| = | min{a, b}|.. (23). On the other hand, since a ≥ b and a + b ≤ 0, we have |b| ≥ |a|. Then 1. 1. |φp (a, b)| ≤ k(a, b)kp − 2b = (2 + 2 p )|b| = (2 + 2 p )| min{a, b}|.. (24). Case(2): a + b > 0. If ab=0, then (21), and (22) hold obviously. Thus, we discuss two subcases: (i) ab < 0 and (ii) ab > 0. (i) We have a > 0, b < 0, and |a| > |b|. Consequently, φp (a, b) ≤ |a| + |b| − (a + b) = −2b = 2| min{a, b}|,. (25). φp (a, b) ≥ k(a, b)k∞ − (a + b) = |a| − (a + b) = −b = | min{a, b}|.. (26). and. (ii) We have a ≥ b > 0. For all p ∈ (1, 2), since 0 ≥ φp (a, b) ≥ φFB (a, b), 10.

(11) 1. |φp (a, b)| ≤ φFB (a, b) ≤ (2 + 2 2 )| min{a, b}|.. (27). Also, for all p ∈ [2, ∞), since 0 ≥ φp (a, b) ≥ k(a, b)k∞ − (a + b) = a − (a + b) = −b = | min{a, b}|, |φp (a, b)| ≤ | min{a, b}|.. (28). Moreover, φp (a, b) ≤ φFB (a, b) ≤ 0 for all p ∈ [2, ∞). Thus, 1. |φp (a, b)| ≥ |φFB (a, b)| ≥ (2 − 2 2 )| min{a, b}| for all p ∈ [2, ∞).. (29). Furthermore, since φp (a, b) ≤ 0, "µ. |φp (a, b)| = a + b − k(a, b)kp = b. ¶. a +1 − b. ¶1/p #. µµ ¶p. a b. +1. .. Let f (t) = t + 1 − (tp + 1)1/p for t ≥ 1. Then 1 f (t) = 1 − p. Ã. 0. √ p. t. !p−1. tp + 1. .. 1. Since f 0 (t) > 0 for t ≥ 1, and f (1) = 2 − 2 p , 1. |φp (a, b)| ≥ (2 − 2 p )| min{a, b}| for all p ∈ (1, 2). Consequently, it follows from (23)-(30) that (21), and (22) hold.. 2. Corollary 3.1 Let ψp : IR2 → IR be given by (8). Then (a) for p ∈ (1, 2), we have 1 ´2 1 ´2 1³ 1³ 2 − 2 p min{a, b}2 ≤ ψp (a, b) ≤ 2 + 2 p min{a, b}2 ; 2 2. (b) for p ∈ [2, ∞), we have ´ ´ 1 2 1 2 1³ 1³ 2 − 2 2 min{a, b}2 ≤ ψp (a, b) ≤ 2 + 2 2 min{a, b}2 . 2 2. 11. (30).

(12) Proposition 3.2 Let Ψα,p and ΨNR be defined by (12) and (11) respectively, and α ≥ 0. Then, for every bounded set S, the following inequalities hold: (a) for all x ∈ S and p ∈ (1, 2), we have µ. ¶. µ. ¶. 1 ´2 1 1³ α 2 1 2 − 2 p ΨNR (x) ≤ Ψα,p (x) ≤ B + (2 + 2 p )2 ΨNR (x); 2 2 2. (b) for all x ∈ S and p ∈ [2, ∞), we have ´ 1 2 1 α 2 1 1³ 2 − 2 2 ΨNR (x) ≤ Ψα,p (x) ≤ B + (2 + 2 2 )2 ΨNR (x), 2 2 2 ½. ¾. where B is a constant defined by B = sup max {max {|xi |, |Fi (x)|}} < ∞. x∈S. 1≤i≤n. Proof. By Corollary 3.1, and the definition of Ψα,p , we have 1 1 Ψα,p (x) ≥ (2 − 2 p )2 ΨNR (x) for all x ∈ IRn if p ∈ (1, 2), 2. and. 1 1 Ψα,p (x) ≥ (2 − 2 2 )2 ΨNR (x) for all x ∈ IRn if p ∈ [2, ∞). 2 Now, we claim that, for each i, the following inequality holds for all x ∈ S:. (xi Fi (x))+ ≤ B| min{xi , Fi (x)}|.. (31). Without loss of generality, we suppose Fi (x) ≥ xi . If Fi (x) ≥ xi ≥ 0, it follows that (xi Fi (x))+ = xi Fi (x) = Fi (x)| min{xi , Fi (x)}| ≤ B| min{xi , Fi (x)}|. If Fi (x) ≥ 0 ≥ xi , then (xi Fi (x))+ = 0. If 0 ≥ Fi (x) ≥ xi , it follows that (xi Fi (x))+ = |xi Fi (x)| ≤ |xi |2 ≤ B| min{xi , Fi (x)}|. Thus, (31) holds for all x ∈ S. Then, by Corollary 3.1 and (31), for 1 ≤ i ≤ n and all x ∈ S, we obtain: ½. ψα,p (xi , Fi (x)) ≤. ¾. 1 α 2 1 B + (2 + 2 p )2 min{xi , Fi (x)}2 2 2. 12. if p ∈ (1, 2),.

(13) and. ½. ¾. 1 α 2 1 B + (2 + 2 2 )2 min{xi , Fi (x)}2 if p ∈ [2, ∞). 2 2 Consequently, the proof is complete from the definition of Ψα,p and ΨNR . 2. ψα,p (xi , Fi (x)) ≤. Since Ψα,p and ΨNR have the same order on a bounded set, from Lemma 2.1, Ψα,p provides a global error bound for the NCP if F is strongly monotone and Lipschitz continuous. In fact, we can show that Ψα,p provides a global error bound for the NCP without assuming that F is Lipschitz continuous when α > 0. Theorem 3.3 Let Ψα,p be given by (12), and α > 0. Suppose that F is a uniform P -function with modulus µ > 0. Then there exists a constant ρ such that 1. kx − x∗ k ≤ ρΨα,p (x) 4 , where x∗ = (x∗1 , · · · , x∗n ) is the unique solution for the NCP. Proof. Since F is a uniform P -function, we have µkx − x∗ k2 ≤ max (x − x∗ )(Fi (x) − Fi (x∗ )) 1≤i≤n. = max {xi Fi (x) − x∗i Fi (x) − xi Fi (x∗ ) + x∗i Fi (x∗ )} 1≤i≤n. = max {xi Fi (x) − x∗i Fi (x) − xi Fi (x∗ )} 1≤i≤n. ≤ max τi {(xi Fi (x))+ + (−Fi (x))+ + (−xi )+ }, 1≤i≤n. (32). where τi := max{1, x∗i , Fi (x∗ )}. Now, we claim that there exists a positive constant τ such that {(ab)+ + (−a)+ + (−b)+ }2 ≤ τ ψα,p (a, b) for all (a, b) ∈ IR2 .. (33). We need the following inequality. (−a)+ 2 + (−b)+ 2 ≤ {k(a, b)kp − (a + b)}2. (34). Without loss of generality, we suppose a ≥ b again. If a ≥ b ≥ 0, then (34) holds obviously. If a ≥ 0 ≥ b, then k(a, b)kp − (a + b) ≥ −b ≥ 0, implying (−a)+ 2 + (−b)+ 2 ≤ {k(a, b)kp − (a + b)}2 . If 0 ≥ a ≥ b, then k(a, b)kp − (a + b) ≥ k(a, b)kp , implying 13.

(14) (−a)+ 2 +(−b)+ 2 ≤ {k(a, b)kp − (a + b)}2 . Thus, (34) holds for all (a, b) ∈ IR2 . Therefore, for all (a, b) ∈ IR2 , we obtain [(ab)+ + (−a)+ + (−b)+ ]2 = (ab)2+ + (−b)2+ + (−a)2+ + 2(ab)+ (−a)+ + 2(−a)+ (−b)+ + 2(ab)+ (−b)+ ≤ (ab)2+ + (−b)2+ + (−a)2+ + (ab)2+ + (−a)2+ + (−a)2+ + (−b)2+ + (ab)2+ + (−b)2+ h. = 3 (ab)2+ + (−b)2+ + (−a)+ 2. i. h. i. ≤ 3 (ab)2+ + {k(a, b)kp − (a + b)}2 · ¸ α 1 2 2 ≤ τ (ab)+ + (k(a, b)kp − (a + b)) 2 2 = τ ψα,p (a, b), where τ := max{ α6 , 6} > 0. Consequently, letting τˆ := max1≤i≤n τi , and combining (32)-(33) yields µkx − x∗ k2 ≤ max τi {τ ψα (xi , Fi (x))}1/2 1≤i≤n. ≤ τˆτ 1/2 max ψα (xi , F (x))1/2 1≤i≤n. ≤ τˆτ. 1/2. ( n X. )1/2. {ψα,p (xi , Fi (x)). i=1. = τˆτ 1/2 Ψα,p (x, F (x))1/2 . 2 Next, we give conditions under which Ψα,p has bounded level sets. Theorem 3.4 Suppose that either of the following conditions holds: (a) α > 0, F is monotone, and the NCP is strictly feasible. (b) α ≥ 0, and F is strongly monotone. Then the level set L(Ψα,p , c) is bounded for all c ∈ IR. Proof. From [1], if F is a monotone function with a strictly feasible point, then the following condition holds: For every sequence {xk } such that kxk k → ∞, (−xk )+ < ∞, 14.

(15) and (−F (xk ))+ < ∞, we have maxi xk i + Fi (xk )+ → ∞. Next, we assume that there exists an unbounded sequence {xk } ⊆ L(Ψα,p , c) for some c ∈ IR. Since Ψα,p (xk ) ≤ c, there is no index i such that xk i → −∞ or Fi (xk ) → −∞. Hence, max (xki )+ (Fi (xk ))+ → ∞. 1≤i≤n. Also, there is an index j, and at least a subsequence {xkj } such that (xkj )+ (Fj (xk ))+ → ∞. However, this implies that Ψα,p (x) is unbounded, contracting to the assumption on level sets. 2 Finally, we state an important result for designing algorithms. To simplify notations, µ. ¶T. , and. we abbreviate the vectors ∇a ψα,p (x1 , F1 (x)), · · · , ∇a ψα,p (xn , Fn (x)) µ. ¶T. ∇b ψα,p (x1 , F1 (x)), · · · , ∇b ψα,p (xn , Fn (x)) as ∇a ψα,p (x, F (x)), and ∇b ψα,p (x, F (x)) respectively. Here, we need a preliminary lemma. Lemma 3.2 For all (a, b) 6= (0, 0) and p > 1, we have the following inequality: ³. 2−2. 1 p. ´2. Ã. ≤. !2. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 −2 k(a, b)kp−1 p. .. Proof. If a = 0 or b = 0, the inequality holds obviously. Then we consider three cases: (i) a > 0 and b > 0, (ii) a < 0 and b < 0, and (iii) ab < 0. (i) Without loss of generality, we suppose a ≥ b > 0. Then ³. ´p−1. | ab | +1 |a|p−1 + |b|p−1 = ³³ ´p ´1− 1 p−1 p k(a, b)kp | ab | + 1 From this fact, we denote f (t) =. tp−1 + 1 1. (tp + 1)1− p. , implying.  0. p−1. f (t) = t. (p − 1). . (t(tp + 1))−1 − tp−1 − 1   2. (tp + 1)1− p. 1. 1. .. Note that f 0 (t) < 0 for t ≥ 1 and f (1) = 2 p , implying 2 p ≥ f (t) for t ≥ 1. Therefore, |a|p−1 + |b|p−1 2 ≥ k(a, b)kp−1 p 1 p. 15. for all p > 1..

(16) Thus,. |a|p−1 + |b|p−1 for all p > 1, k(a, b)kp−1 p and squaring both sides leads to the desired inequality. (ii) Likewise, 1. 2 − 2p ≤ 2 −. 1. 2 − 2p ≤ 2 −. |a|p−1 + |b|p−1 |a|p−1 + |b|p−1 ≤ 2 + k(a, b)kp−1 k(a, b)kp−1 p p. (35). for all p > 1.. Thus, squaring both sides leads to the desired inequality. (iii) Without loss of generality, we suppose |a| ≥ |b|. Since 1. 2p ≥. |a|p−1 + |b|p−1 |a|p−1 − |b|p−1 ≥ k(a, b)kp−1 k(a, b)kp−1 p p. for all p > 1,. |a|p−1 − |b|p−1 for all p > 1. k(a, b)kp−1 p Again, squaring both sides, we have the desired inequality. 2 1. 2 − 2p ≤ 2 −. Theorem 3.5 Let α ≥ 0. Then the following inequalities hold for all x ∈ IRn : ³. 1. ´4. 1. ´4. (a) k∇a ψα,p (x, F (x)) + ∇b ψα,p (x, F (x))k2 ≥ 2 − 2 p ³. (b) k∇a ψα,p (x, F (x)) + ∇b ψα,p (x, F (x))k2 ≥ 2 − 2 2. ΨNR (x) for p ∈ (1, 2); ΨNR (x) for p ∈ [2, ∞).. Proof. Suppose (a, b) 6= (0, 0). It follows from (15) that (∇a ψα,p (a, b) + ∇b ψα,p (a, b))2 (. =. !)2. Ã. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 −2 α(a + b)(ab)+ + (k(a, b)kp − a − b) k(a, b)kp−1 p Ã. !2. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 = α (a + b) + (k(a, b)kp − a − b) −2 k(a, b)kp−1 p Ã ! sgn(a) · |a|p−1 + sgn(b) · |b|p−1 +2α(a + b)(ab)+ (k(a, b)kp − a − b) −2 . k(a, b)kpp−1 2. 2. (ab)2+. 2. Now, we claim that Ã. !. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 2α(a + b)(ab)+ (k(a, b)kp − a − b) −2 ≥0 k(a, b)kp−1 p 16. (36).

(17) Ã. !. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 for all (a, b) 6= (0, 0) ∈ IR2 . Note that − 2 ≤ 0 for k(a, b)kpp−1 all (a, b) 6= (0, 0) ∈ IR2 . If ab ≤ 0, (ab)+ = 0. If a > 0, and b > 0, then we have φp (a, b) ≤ 0. If a < 0, and b < 0, then we have φp (a, b) ≥ 0. Therefore, (36) holds for all (a, b) 6= (0, 0). Applying Lemma 3.2, we obtain the following inequalities. In particular, when p ∈ (1, 2), we have (∇a ψα,p (a, b) + ∇b ψα,p (a, b))2 Ã. 2. ≥ (k(a, b)kp − a − b) ≥ =. ³ ³. 1. ´2. 1. ´4. 2 − 2p 2 − 2p. !2. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 −2 k(a, b)kpp−1. ³. 1. min{a, b}2 2 − 2 p. (37). ´2. min{a, b}2 ,. for all (a, b) 6= (0, 0). In addition, when p ∈ [2, ∞), we have (∇a ψα,p (a, b) + ∇b ψα,p (a, b))2 Ã. 2. ≥ (k(a, b)kp − a − b) ≥ ≥. ³ ³. 1. ´2. 1. ´4. 2 − 22 2 − 22. !2. sgn(a) · |a|p−1 + sgn(b) · |b|p−1 −2 k(a, b)kpp−1. ³. 1. min{a, b}2 2 − 2 p. (38). ´2. min{a, b}2 .. for all (a, b) 6= (0, 0). Note that (37) and (38) hold obviously for (a, b) = (0, 0), and hence hold for all (a, b) ∈ IR2 . Consequently, the result follows from the definition of ΨNR . 2. With this theorem, no doubt, same convergence results including R-linear convergence rate in Section 4, and 5 of [24] can be established by mimicking the analysis therein. The original convergence results are listed in the appendix from which analogous results can be obtained.. 17.

(18) 4. Appendix. In this section, we list a derivative-free descent method in [24] based on the function ψα (a, b) =. α (ab)+ 2 2. µ. +. 1 2. ¶2. k(a, b)k − (a + b). . We consider the following search direction:. dk (β) := −∇b ψα (xk , F (xk )) − β∇a ψα (xk , F (xk )),. (39). where β is a parameter such that β ∈ (0, 1). Although this search direction is not P necessarily a descent direction of Ψα (x) = ni=1 ψα (xi , Fi (x)) at xk for every β ∈ (0, 1), we can choose β > 0 sufficiently small so that dk (β) is a descent direction of Ψα at xk , provided that the monotonicity assumption on F is fulfilled. Lemma 4.1 Suppose that F is monotone, and α ≥ 0. If xk is not a solution of the ˆ k ) ∈ (0, 1) such that for all β ∈ (0, β(x ˆ k )), the search direction NCP, then there exists β(x T dk (β) defined by (39) satisfies the descent condition ∇a ψα (xk ) dk (β) < 0. With the above lemma, we can propose the following descent algorithm. Algorithm 4.1 [24, Algorithm 4.1] Step1. Choose x0 ∈ IRn , ² ≥ 0, σ ∈ (0, 1), β ∈ (0, 1), and γ ∈ (0, 1). Set k = 0. Step2. If Ψα (xk ) ≤ ², stop. xk is an approximate solution of the NCP. Step3. Compute xk+1 := xk + γ lk dk (β lk , where dk (β lk ) := −∇b ψα (xk , F (xk )) − β lk ∇a ψα (xk , F (xk )) with lk being the smallest nonnegative integer l satisfying 2. Ψα (xk + γ l dk (β l )) − Ψα (xk ) ≤ −σ(γ l ) k∇a ψα (x, F (x)) + ∇b ψα (x, F (x))k2 . Step4. Set k := k + 1, and go to Step 2. Note that the above algorithm has no need to compute the gradients of Ψα , and therefore, there is no need to calculate the Jacobian of F . We are now ready to state the global convergence for Algorithm 4.1 under the monotonicity assumption on F . In the remainder of the thesis, we suppose that the parameter ² used in Algorithm 4.1 is set to be zero, and that Algorithm 4.1 generates an infinite sequence {xk }. 18.

(19) Theorem 4.1 Suppose that F is monotone, and α ≥ 0. Then Algorithm 4.1 is well defined for every initial point x0 . Moreover, if x∗ is an accumulation point of the sequence {xk } generated by Algorithm 4.1, then x∗ is a solution of the NCP. Combing the above theorem, and Theorem 3.4, we obtain the following global convergence result. Corollary 4.1 Suppose that either of the following conditions holds: (a) α > 0, F is monotone, and the NCP is strictly feasible. (b) α ≥ 0, and F is strongly monotone. Then Algorithm 4.1 is well defined for every initial point x0 . Moreover, if x∗ is an accumulation point of the sequence {xk } generated by Algorithm 4.1, then x∗ is a solution of the NCP.. 5. Concluding Remarks. In this thesis, we propose a new merit function based on the generalized FischerBurmeister function, and improve some of the results in [24] to a more general situation. We consider Proposition 3.1 very important in that Proposition 3.2, and Theorem 3.5 are easily deduced. We also believe that Theorem 3.5 can be applied to designing algorithms, and obtaining some further consequences.. References [1] B. Chen, X. Chen, and C. Kanzow, A Penalized Fischer-Burmeister NCPfunction: Theoretical Investigation and Numerical Results, Mathematical Programming, vol. 88, pp. 211-216, 2000. [2] J.-S. Chen, The Semismooth-related Properties of a Merit function and a Descent Method for the Nonlinear Complementarity Problem, Journal of Global Optimization, vol. 24, pp. 565-580, 2006. 19.

(20) [3] J.-S. Chen, On Some NCP-functions Based on the Generalized Fischer-Burmeister function, Asia-Pacific Journal of Opertional Research, vol. 24, pp. 401-420, 2007. [4] J.-S. Chen and S. Pan, A Family of NCP-functions and a Descent Method for the Nonlinear Complementarity Problem, to appear in Computational Optimization and Applications, 2008. [5] J.-S. Chen and S. Pan, A Family of Penalized NCP-functions Based on the Generalized Fischer-Burmeister Functions: Theoretical Investigation and Numerical Results, submitted manuscript, 2007. [6] R.W. Cottle, J.-S. Pang and R.-E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992. [7] S. Dafermos, An Iterative Scheme for Variational Inequalities, Mathematical Programming, vol. 26, pp.40-47, 1983. [8] F. Facchinei and J. Soares, A New Merit Function for Nonlinear Complementarity Problems and a Related Algorithm, SIAM Journal on Optimization, vol. 7, pp. 225-247, 1997. [9] A. Fischer, A Special Newton-type Optimization Method, Optimization, vol. 24, pp. 269-284, 1992. [10] M. Fukushima Merit Functions for Varitional Inequality and Complementarity Problem, Nonlinear Optimization and Applications, edited by G Di Pillo and F. Giannessi, Pleneum Press, New York, pp. 155-170, 1996. [11] C. Geiger and C. Kanzow, On the Resolution of Monotone Complementarity Problems, Computational Optimization and Applications, vol. 5, pp. 155-173, 1996. [12] P. T. Harker and J.-S. Pang, Finite Dimensional Variational Inequality and Nonlinear Complementarity Problem: A Survey of Theory, Algorithms and Applications, Mathematical Programming, vol. 48, pp. 161-220, 1990. [13] H. Jiang, Unconstrained Minimization Approaches to Nonlinear Complementarity Problems, Journal of Global Optimization, vol. 9, pp. 169-181, 1996. 20.

(21) [14] C. Kanzow, Nonlinear Complementarity as Unconstrained Optimization, Journal of Optimization Theory and Applications, vol. 88, pp. 139-155, 1996. [15] C. Kanzow and N. Yamashita and M. Fukushima, New NCP-functions and Their Properties, Journal of Optimization Theory and Applications, vol. 94, pp. 115-135, 1997. [16] O. L. Mangasarian, Equivalence of the Complementarity Problem to a System of Nonlinear Equations, SIAM Journal on Applied Mathematics, vol. 31, pp. 89-92, 1976. [17] J.-S. Pang, A Posteriori Error Bounds for the Linearly-constrained Variational Inequality Problem, Mathematics of Operations Research, vol. 12, pp. 474-484, 1987. [18] J.-S. Pang, Complementarity problems, Handbook of Global Optimization, edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, pp. 271-338, 1994. [19] J.-S. Pang, Newton’s Method for B-differentiable Equations, Mathematics of Operations Research, vol. 15, pp. 311-341, 1990. [20] J.-S. Pang and D. Chan, Iterative Methods for Variational and Complemantarity Problems, Mathematics Programming, vol. 27, 99. 284-313, 1982. [21] P. Tseng, Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problem, Journal of Optimization Theory and Applications, vol. 89, pp. 17-37, 1996. [22] N. Yamashita and M. Fukushima, On Stationary Points of the Implicit Lagrangian for the Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, vol. 84, pp. 653-663, 1995. [23] N. Yamashita and M. Fukushima, Modified Newton Methods for Solving a Semismooth Reformulation of Monotone Complementarity Problems, Mathematical Programming, vol. 76, pp. 469-491, 1997. 21.

(22) [24] K. Yamada, N. Yamashita, and M. Fukushima, A New Derivative-free Descent Method for the Nonlinear Complementarity Problems, in Nonlinear Optimization and Related Topics edited by G.D. Pillo and F. Giannessi, Kluwer Academic Publishers, Netherlands, pp. 463-487, 2000.. 22.

(23)