Journal of Computational and Applied Mathematics, vol. 232, pp. 455-471, 2009
A R-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function
Jein-Shan Chen 1 Department of Mathematics National Taiwan Normal University
Taipei, Taiwan 11677 E-mail: jschen@math.ntnu.edu.tw
Hung-Ta Gao
Department of Mathematics National Taiwan Normal University
Taipei, Taiwan 11677 E-mail: kleinmankao@gmail.com
Shaohua Pan
School of Mathematical Sciences South China University of Technology
Guangzhou 510640, China E-mail: shhpan@scut.edu.cn
July 10, 2008
Abstract. In the paper [4], the authors proposed a derivative-free descent algorithm for the nonlinear complementarity problems (NCPs) by the generalized Fischer-Burmeister merit function: ψp(a, b) = 12[∥(a, b)∥p − (a + b)]2, and observed that the choice of the parameter p has a great influence on the numerical performance of the algorithm. In this paper, we analyze the phenomenon theoretically for a derivative-free descent algorithm which is based on a penalized form of ψp and uses a different direction from [4]. More specifically, we show that the algorithm proposed is globally convergent and has a locally R-linear convergence rate, and furthermore, its convergence rate will become worse when the parameter p decreases. Numerical results are also reported for the test problems from MCPLIB, which further verify the obtained theoretical results.
1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.
Key Words. Nonlinear complementarity problem, NCP-function, merit function, global error bound, convergence rate.
1 Introduction
The nonlinear complementarity problem (NCP) is to find a point x∈ IRn such that
x≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩ = 0, (1)
where ⟨·, ·⟩ is the Euclidean inner product and F = (F1, . . . , Fn)T is a map from IRn to IRn. We assume that F is continuously differentiable throughout this paper. The NCP has attracted much attention because of its wide applications in the fields of economics, engineering, and operations research [5, 12], to name a few.
Many methods have been proposed to solve the NCP; see [9, 12, 18] and the references therein. One of the most powerful and popular methods is to reformulate the NCP as a system of nonlinear equations [16, 19, 24], or an unconstrained minimization problem [6, 7, 8, 10, 13, 14, 20, 23]. The objective function that can constitute an equivalent unconstrained minimization problem is called a merit 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. (2)
The Fischer-Burmeister (FB) function is a well-known NCP-function defined as ϕFB(a, b) =√
a2+ b2− (a + b), (3)
by which the NCP can be reformulated as a system of nonsmooth equations:
ΦFB(x) =
ϕFB(x1 , F1(x))
·
·
·
ϕFB(xn , Fn(x))
= 0. (4)
Thus, the function ΨFB : IRn → IR+ defined as below is a merit function for the NCP:
ΨFB(x) := 1
2∥ΦFB(x)∥2 =
∑n i=1
ψFB(xi, Fi(x)), (5)
where ψFB : IR2 → IR+ is the square of ϕFB, i.e., ψFB(a, b) = 1
2
√
a2+ b2− (a + b) 2. (6) Consequently, the NCP is equivalent to an unconstrained minimization problem:
xmin∈IRnΨFB(x). (7)
Recently, an extension of FB function was considered in [2, 3, 4] by the authors. More specifically, we define the generalized FB function ϕp : IR2 → IR by
ϕp(a, b) :=∥(a, b)∥p− (a + b), (8) where p > 1 is an arbitrary fixed real number and ∥(a, b)∥p denotes the p-norm of (a, b), i.e., ∥(a, b)∥p = p
√|a|p+|b|p. In other words, in the function ϕp, we replace the 2-norm of (a, b) in the FB function by a more general p-norm. The function ϕp is still an NCP- function, which naturally induces another NCP-function ψp : IR2 → IR+ given by
ψp(a, b) := 1
2|ϕp(a, b)|2. (9)
For any given p > 1, the function ψpis shown to possess all favorable properties of the FB function ψFB; see [2, 3, 4]. For example, ψp is also continuously differentiable everywhere on IR2. Like ϕFB, the operator Φp : IRn→ IRn defined as
Φp(x) =
ϕp(x1 , F1(x))
·
·· ϕp(xn , Fn(x))
(10)
yields a family of merit functions Ψp : IRn → IR+ for the NCP Ψp(x) := 1
2∥Φp(x)∥2 =
∑n i=1
ψp(xi , Fi(x)). (11)
In this paper, we study the following merit function Ψα,p: IRn → IR for the NCP:
Ψα,p(x) :=
∑n i=1
ψα,p(xi , Fi(x)), (12)
where ψα,p : IR2 → IR+ is an NCP-function defined by ψα,p(a, b) := α
2(max{0, ab})2+ ψp(a, b) = α
2(ab)2++ 1
2(∥(a, b)∥p− (a + b))2 (13)
with α ≥ 0 being a real parameter. When α = 0, the function ψα,p reduces to ψp. Hence, ψα,p is an extension of ψp. Besides, ψα,p also extends the function ψα studied in [25] by Yamada, Yamashita, and Fukushima which corresponds to p = 2. Indeed, ψα,p was ever studied in [3] by one of the authors (see ψ4 therein), but there was no investigation on property of error bound. In this paper, we present more favorable properties of ψα,p, and particularly, the conditions under which Ψα,p provides a global error bound for the NCP.
With these results, we propose a derivative-free descent algorithm based on ϕα,p and establish its global convergence and local R-linear convergence rate. Moreover, we also analyze the influence of p on the convergence rate of the proposed algorithm theoretically and obtain the conclusion that the convergence rate of the algorithm will become worse when the value of p decreases. Thus, this paper can be viewed as a follow-up of [3] and [4].
This paper is organized as follows. In Section 2, we review some definitions and prelim- inary 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 propose a derivative-free al- gorithm associated with Ψα,p, prove its global convergence and the R-linear convergence rate, and analyze the influence of p on the convergence rate. Some numerical experiments are reported in Section 5, and we make concluding remarks in Section 6.
Throughout this paper, 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, . . . , Fn)T : IRn → IRn,
∇F (x) = (∇F1(x) . . . ∇Fn(x)) denotes the transpose Jacobian of F at x. We denote by∥x∥p the p-norm of x and by∥x∥ the Euclidean norm of x. The level set of a function Ψ : IRn→ IR is denoted by L(Ψ, c) := {x ∈ IRn | Ψ(x) ≤ c}. In addition, we also use the natural residual merit function ΨNR : IRn→ IR+ defined by
ΨNR(x) := 1 2
∑n i=1
ϕ2
NR(xi , Fi(x)), (14)
where ϕNR : IR2 → IR denotes the minimum NCP-function min{a, b}. Unless otherwise stated, in the sequel, we always suppose that p is a fixed real number in (1,∞).
2 Preliminaries
This section mainly recalls some concepts about the mapping F that will be used later.
Definition 2.1 Let F = (F1, . . . , Fn)T with Fi : IRn→ IR for i = 1, . . . , n. We say that (a) F is monotone if ⟨x − y, F (x) − F (y)⟩ ≥ 0 for all x, y ∈ IRn.
(b) F is strongly monotone with modulus µ > 0 if ⟨x − y, F (x) − F (y)⟩ ≥ µ∥x − y∥2 for all x, y ∈ IRn.
(c) F is a P0-function if max
1≤i≤n xi̸=yi
(xi− yi)(Fi(x)− Fi(y))≥ 0 for all x, y ∈ IRn and x̸= y.
(d) F is a uniform P -function with modulus µ > 0 if max
1≤i≤n(xi − yi)(Fi(x)− Fi(y)) ≥ µ∥x − y∥2 for all x, y∈ IRn.
(e) ∇F (x) is uniformly positive definite with modulus µ > 0 if dT∇F (x)d ≥ µ∥d∥2 for all x∈ IRn and d∈ IRn.
(f ) F is Lipschitz continuous if there exists a constant L > 0 such that∥F (x) − F (y)∥ ≤ L∥x − y∥ for all x, y ∈ IRn.
From Definition 2.1, it is easy to see that F is a uniform P -function with modulus µ > 0 if F is strongly monotone with modulus µ > 0, and F is a P0-function if F is monotone. In addition, when F is continuously differentiable, the following results hold:
1. F is monotone if and only if ∇F (x) is positive semidefinite for all x ∈ IRn. 2. F is strongly monotone if and only if ∇F (x) is uniformly positive definite.
3 Properties of the Merit Function
In this section, we study some favorable properties of the merit function ψα,pwhich will be used in the subsequent analysis, and then present some mild conditions under which the merit function Ψα,phas bounded level sets and provides a global error bound, respectively.
The following lemma states that ψα,p enjoys many favorable properties as ψp holds.
Furthermore, when α > 0, it has an important property that ψp does not have (see Lemma 3.1(f)). Although most results of the lemma were investigated in [3, Prop. 3.3]
where only p being integer was considered, we here provide more detailed arguments for general case where p is any real number greater than one.
Lemma 3.1 The function ψα,p defined by (13) has the following favorable properties:
(a) ψα,p is an NCP-function and ψα,p ≥ 0 for all (a, b) ∈ IR2.
(b) ψα,p is continuously differentiable everywhere, and moreover, if (a, b) ̸= (0, 0),
∇aψα,p(a, b) = αb(ab)++
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b),
∇bψα,p(a, b) = αa(ab)++
(sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b);
(15)
and otherwise ∇aψα,p(0, 0) =∇bψα,p(0, 0) = 0.
(c) For p≥ 2, the gradient of ψα,p is Lipschitz continuous on any nonempty bounded set S, i.e., there exists L > 0 such that for any (a, b), (c, d)∈ S,
∥∇ψα,p(a, b)− ∇ψα,p(c, d)∥ ≤ L∥(a, b) − (c, d)∥.
(d) ∇aψα,p(a, b) · ∇bψα,p(a, b) ≥ 0 for any (a, b) ∈ IR2, and furthermore, the equality holds if and only if ψα,p(a, b)=0.
(e) ∇aψα,p(a, b) = 0⇐⇒ ∇bψα,p(a, b) = 0⇐⇒ ψα,p(a, b) = 0.
(f ) Suppose that α > 0. If a→ −∞ or b → −∞ or ab → ∞, then ψα,p(a, b)→ ∞.
Proof. Parts (a), (b) and (f) directly follow from the definition of ψα,p and Proposition 3.2 (a)–(c) and Lemma 3.1 of [4]. It remains to show parts (c)–(e).
(c) Notice that the functions a(ab)+ and b(ab)+for any a, b∈ IR are Lipschitz continuous on any nonempty bounded set S, whereas ϕp(a, b) is Lipschitz continuous on IR2 by [4, Proposition 3.1 (e)]. Therefore, by the expression of ∇ψα,p(a, b) and the boundedness of
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
)
and
(sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1
)
,
it is not hard to verify that the gradient ∇ψα,p(a, b) is Lipschitz continuous on S for p≥ 2.
(d) If (a, b) = (0, 0), part (d) clearly holds. Now suppose that (a, b)̸= (0, 0). Then,
∇aψα,p(a, b)· ∇bψα,p(a, b) =
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
) (sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1
)
ϕ2p(a, b) +α2ab(ab)+2+ αa(ab)+
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b) +αb(ab)+
(sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b). (16)
Since
ab(ab)+2 ≥ 0, sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1 ≤ 0, and sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1 ≤ 0, (17) it suffices to show that the last two terms of (16) are nonnegative. We next claim that
αa(ab)+
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b)≥ 0, ∀ (a, b) ̸= (0, 0). (18)
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, which implies (18) by a similar reason.
If a > 0 and b≤ 0, then (ab)+ = 0, and hence (18) holds. Similarly, we have that αb(ab)+
(sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b)≥ 0, ∀ (a, b) ̸= (0, 0).
Consequently, ∇aψα,p(a, b)· ∇bψα,p(a, b) ≥ 0. From (16), ∇aψα,p(a, b)· ∇bψα,p(a, b)=0 if and only if {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} and {ab=0}. Thus, ∇aψα(a, b)·∇bψα,p(a, b) = 0 if and only if ψα,p(a, b) = 0.
(e) If ψα,p(a, b) = 0, then ab = 0 and ϕp(a, b) = 0 by part (a), which in turn implies that ∇aψα,p(a, b) = 0 and ∇bψα,p(a, b) = 0. Next, we claim that∇aψα,p(a, b) = 0 implies ψα,p(a, b) = 0. Suppose that ∇aψα,p(a, b) = 0. Then,
αb(ab)+=−
(sgn(a)· |a|p−1
∥(a, b)∥pp−1 − 1
)
ϕp(a, b). (19)
We can verify that the equality (19) implies b = 0, a≥ 0 or b > 0, a = 0. Under the two cases, we both have ψα,p(a, b) = 0. Similarly, ∇bψα,p(a, b) = 0 also implies ψα,p(a, b) = 0.
2
Notice that ab → ∞ does not necessarily imply ψp(a, b) → ∞ which means ψp does not share Lemma 3.1(f). In fact, for α = 0, the lemma needs to be modified as “if (a → ∞) or (b → ∞) or (a → ∞ and b → ∞), then ψα,p(a, b) → ∞”. As we will see later, Lemma 3.1(f) is useful for proving that the level sets of Ψα,p are bounded. Besides, by Lemma 3.1(a), we immediately have the following theorem.
Theorem 3.1 Let Ψα,p be defined as in (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.
Theorem 3.1 indicates that the NCP can be recast as the unconstrained minimization:
xmin∈IRnΨα,p(x). (20)
In general, it is hard to find a global minimum of Ψα,p. Therefore, it is important to know under what conditions a stationary point of Ψα,p is a global minimum. Using Lemma 3.1(d) and the same proof techniques as in [10, Theorem 3.5], we can estabish that each stationary point of Ψα,p is a global minimum only if F is a P0-function.
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.
From the following theorem, we see that the unconstrained minimization problem (20) has a stationary point under rather weak conditions of the mapping F . Since similar results and analogous analysis can be found in [3, Proposition 4.1], [10, Theorem 3.8] and [15, Theorem 4.1], we here omit the proof.
Theorem 3.3 The function Ψα,p has bounded level sets L(Ψα,p, c) for all c∈ IR, if F is monotone and the NCP is strictly feasible (i.e., there exists ˆx > 0 such that F (ˆx) > 0) when α > 0, or F is a uniform P -function when α≥ 0.
In what follows, we will show that the merit functions Ψp, ΨNR and Ψα,phave the same order on every bounded set. For this purpose, we need the following crucial technical lemma, which generalizes the important property of ϕFB proved by Tseng in [22].
Lemma 3.2 Let ϕp : IR2 → IR be defined as in (8). Then for any p > 1 we have
(2− 21p)| min{a, b}| ≤ |ϕp(a, b)| ≤ (2 + 21p)| min{a, b}|. (21) Proof. Without loss of generality, 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)| ≥ ∥(a, b)∥p ≥ |b| = | min{a, b}| ≥ (2 − 2p1)| min{a, b}|. (22) On the other hand, since a≥ b and a + b ≤ 0, we have |b| ≥ |a|. Then
|ϕp(a, b)| ≤ ∥(a, b)∥p− 2b = (2 + 2p1)|b| = (2 + 2p1)| min{a, b}|. (23) Case(2): a + b > 0. If ab=0, then (21) clearly holds. Thus, we discuss by two subcases:
(i) ab < 0. In this subcase, we have a > 0, b < 0, and |a| > |b|. Consequently,
ϕp(a, b)≤ |a| + |b| − (a + b) = −2b = 2| min{a, b}| ≤ (2 + 21p)| min{a, b}|, (24) and
ϕp(a, b)≥ ∥(a, b)∥∞− (a + b) = −b = | min{a, b}| ≥ (2 − 21p)| min{a, b}|. (25) (ii) ab > 0. Now we have a≥ b > 0. Since for any p > 1 there holds that
0≥ ϕp(a, b) ≥ ∥(a, b)∥∞− (a + b) = a − (a + b) = −b = − min{a, b}, we immediately obtain that
|ϕp(a, b)| ≤ | min{a, b}| ≤ (2 + 21p)| min{a, b}|. (26)
On the other hand, since ϕp(a, b)≤ 0, it follows that
|ϕp(a, b)| = a + b − ∥(a, b)∥p = b
[(a b + 1
)
−((a b
)p
+ 1
)1/p]
.
Let f (t) = t + 1− (tp + 1)1/p for t≥ 1. Then
f′(t) = 1−( tp tp + 1
)p−1p
.
Notice that f′(t) > 0 for t≥ 1, and f(1) = 2 − 21p, and hence we obtain that
|ϕp(a, b)| ≥ (2 − 21p)b = (2− 21p)| min{a, b}| for any p > 1. (27) All the aforementioned inequalities (22)-(27) imply that (21) holds. 2
Proposition 3.1 Let Ψp, ΨNR and Ψα,p be defined as in (11), (14) and (12), respectively.
Let S be an arbitrary bounded set. Then, for any p > 1, we have
(
2− 21p)2ΨNR(x)≤ Ψp(x)≤(2 + 2p1)2ΨNR(x) for all x∈ IRn (28) and
(
2− 21p)2ΨNR(x)≤ Ψα,p(x)≤(αB2+ (2 + 21p)2)ΨNR(x) for all x∈ S, (29)
where B is a constant defined by B = max
1≤i≤n
{
sup
x∈S{max {|xi|, |Fi(x)|}}
}
<∞.
Proof. The inequality in (28) is direct by Lemma 3.2 and the definitions of Ψp and ΨNR. In addition, from Lemma 3.2 and the definition of Ψα,p, it follows that
Ψα,p(x)≥(2− 2p1)2ΨNR(x) for all x∈ IRn.
We next prove the inequality on the right hand side of (29). We claim that, for each i, (xiFi(x))+≤ B| min{xi, Fi(x)}| for all x ∈ S. (30) Without loss of generality, suppose Fi(x)≥ xi. If Fi(x)≥ xi ≥ 0, it follows that
(xiFi(x))+= xiFi(x) = Fi(x)| min{xi, Fi(x)}| ≤ B| min{xi, Fi(x)}|.
If Fi(x)≥ 0 ≥ xi, then (xiFi(x))+= 0. If 0 ≥ Fi(x)≥ xi, it follows that (xiFi(x))+ =|xiFi(x)| ≤ |xi|2 ≤ B| min{xi, Fi(x)}|.
Thus, (30) holds for all x∈ S. By Lemma 3.2 and (30), for all i = 1, . . . , n and x ∈ S, ψα,p(xi, Fi(x)) ≤{αB2+ (2 + 21p)2}min{xi, Fi(x)}2
holds for any p > 1. The proof is then complete by the definition of Ψα,p and ΨNR. 2
From Proposition 3.1, we immediately obtain the following result.
Corollary 3.1 Let Ψp and Ψα,p be defined by (12) and (11), respectively, and S be any bounded set. Then, for any p > 1 and all x∈ S, we have the following inequalities:
(2− 21p)2
(
αB2+ (2 + 21p)2)
Ψα,p(x)≤ Ψp(x)≤ (2 + 21p)2 (2− 2p1)2
Ψα,p(x)
where B is the constant defined as in Proposition 3.1.
Since Ψp, ΨNR and Ψα,p have the same order on a bounded set, one will provide a global error bound for the NCP as long as the other one does. As below, we show that Ψα,p provides a global error bound without the Lipschitz continuity of F when α > 0.
Theorem 3.4 Let Ψα,p be defined as in (12). Suppose that F is a uniform P -function with modulus µ > 0. If α > 0, then there exists a constant κ1 > 0 such that
∥x − x∗∥ ≤ κ1Ψα,p(x)14 for all x∈ IRn;
if α = 0 and S is any bounded set, there exists a constant κ2 > 0 such that
∥x − x∗∥ ≤ κ2
(
max
{
Ψα,p(x),
√
Ψα,p(x)
})1
2 for all x∈ S;
where x∗ = (x∗1,· · · , x∗n) is the unique solution for the NCP.
Proof. Since F is a uniform P -function, the NCP has the unique solution, and moreover, µ∥x − x∗∥2 ≤ max
1≤i≤n(x− x∗)(Fi(x)− Fi(x∗))
= max
1≤i≤n{xiFi(x)− x∗iFi(x)− xiFi(x∗) + x∗iFi(x∗)}
= max
1≤i≤n{xiFi(x)− x∗iFi(x)− xiFi(x∗)}
≤ max
1≤i≤nτi{(xiFi(x))++ (−Fi(x))++ (−xi)+}, (31) where τi := max{1, x∗i, Fi(x∗)}. We next prove that for all (a, b) ∈ IR2,
(−a)+2
+ (−b)+2 ≤ [∥(a, b)∥p− (a + b)]2. (32)
Without loss of generality, suppose a ≥ b. If a ≥ b ≥ 0, then (32) holds obviously. If a≥ 0 ≥ b, then ∥(a, b)∥p− (a + b) ≥ −b ≥ 0, which in turn implies that
(−a)+2
+ (−b)+2
= b2 ≤ [∥(a, b)∥p− (a + b)]2. If 0≥ a ≥ b, then (−a)+2
+ (−b)+2
= a2+ b2 ≤ [∥(a, b)∥p− (a + b)]2. Hence, (32) follows.
Suppose that α > 0. Using the inequality (32), we then obtain that
[(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+
≤ 3[(ab)2++ (∥(a, b)∥p − (a + b))2]
≤ τ[α
2(ab)2++ 1
2(∥(a, b)∥p− (a + b))2]
= τ ψα,p(a, b) for all (a, b)∈ IR2, (33) where τ := max
{6 α, 6
}
> 0. Combining (33) with (31) and letting ˆτ = max
1≤i≤nτi, we get µ∥x − x∗∥2 ≤ max
1≤i≤nτi{τψα,p(xi, Fi(x))}1/2
≤ ˆττ1/2 max
1≤i≤nψα,p(xi, F (x))1/2
≤ ˆττ1/2
{ n
∑
i=1
{ψα,p(xi, Fi(x))
}1/2
= ˆτ τ1/2Ψα,p(x, F (x))1/2.
From this, the first desired result follows immediately by setting κ1 :=[ˆτ τ1/2/µ]1/2. Suppose that α = 0. From the proof of Proposition 3.1, the inequality (30) holds.
Combining with equations (31)–(32), it then follows that for all x∈ S, µ∥x − x∗∥2 ≤ max
1≤i≤nτi[B| min{xi, Fi(x)}| + (ψp(xi, Fi(x)))1/2]
≤ ˆτ max
1≤i≤n
[√
2 ˆBψp(xi, Fi(x)) + (ψp(xi, Fi(x)))1/2]
≤ √
2ˆτ ˆB
(
Ψp(x) +
√
Ψp(x)
)
≤ 4ˆτ ˆB max
{
Ψp(x),
√
Ψp(x)
}
= 4ˆτ ˆB max
{
Ψα,p(x),
√
Ψα,p(x)
}
where ˆB = B/(2− 21p) and the second inequality is from Lemma 3.2. Letting κ2 :=
2[τ ˆˆB/µ]1/2, we obtain the desired result from the above inequality. 2
The following lemma is needed for the proof of Proposition 3.2, which plays a crucial role in showing the convergence rate of the algorithm described in the next section.
Lemma 3.3 For all (a, b)̸= (0, 0) and p > 1, we have the following inequality:
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)2
≥(2− 21p)2.
Proof. If a = 0 or b = 0, the inequality holds obviously. Then we complete the proof by considering three cases: (i) a > 0 and b > 0, (ii) a < 0 and b < 0, and (iii) ab < 0.
Case (i): Without loss of generality, we suppose a≥ b > 0. Then
|a|p−1+|b|p−1
∥(a, b)∥pp−1 =
( ab )p−1
+ 1
(( ab )p
+ 1)1−
1 p
. (34)
Let f (t) := tp−1+ 1 (tp + 1)1−1p
for any t > 0. By computation, we have that
f′(t) = tp−2(p− 1)(1 − t)
(tp+ 1)2 ∀t > 0.
Since f′(t) < 0 for t≥ 1 and f(1) = 21p, it follows that f (t)≤ 21p for t ≥ 1. Therefore,
|a|p−1+|b|p−1
∥(a, b)∥pp−1 ≤ 21p for p > 1, which in turn implies that 2− |a|p−1+|b|p−1
∥(a, b)∥pp−1 ≥ 2 − 21p for p > 1. Squaring both sides then leads to the desired inequality.
Case (ii): By similar arguments as in case (i), we obtain 2− 21p ≤ 2 −|a|p−1+|b|p−1
∥(a, b)∥pp−1 ≤ 2 + |a|p−1+|b|p−1
∥(a, b)∥pp−1 for p > 1, from which the result follows immediately.
Case (iii): Again, we suppose |a| ≥ |b| and therefore have 21p ≥ |a|p−1+|b|p−1
∥(a, b)∥pp−1 ≥ |a|p−1− |b|p−1
∥(a, b)∥pp−1 for p > 1.
Thus 2− 21p ≤ 2 −|a|p−1− |b|p−1
∥(a, b)∥pp−1 for p > 1 and the desired result is also satisfied. 2
Proposition 3.2 Let ψα,p be given as in (13). Then, for all x∈ IRn and p > 1,
∥∇aψα,p(x, F (x)) +∇bψα,p(x, F (x))∥2 ≥ 2(2− 2p1)2Ψp(x), and particularly, for all x belonging to any bounded set S and p > 1,
∥∇aψα,p(x, F (x)) +∇bψα,p(x, F (x))∥2 ≥ 2(2− 21p)4
(
αB2+ (2 + 21p)2
)Ψα,p(x) where B is defined as in Proposition 3.1 and
∇aψα,p(x, F (x)) :=
(
∇aψα,p(x1, F1(x)), · · · , ∇aψα,p(xn, Fn(x))
)T
,
∇bψα,p(x, F (x)) :=
(
∇bψα,p(x1, F1(x)), · · · , ∇bψα,p(xn, Fn(x))
)T
. (35)
Proof. The second part of the conclusions is direct by Corollary 3.1 and the first part.
From the definition of ∇aψα,p(x, F (x)),∇bψα,p(x, F (x)) and Ψp(x), the first part of the conclusions is equivalent to proving that the following inequality
(∇aψα,p(a, b) +∇bψα,p(a, b))2 ≥ 2(2− 21p)2ψp(a, b) (36) holds for all (a, b)∈ IR2. When (a, b) = (0, 0), the inequality (36) clearly holds. Suppose (a, b)̸= (0, 0). Then, it follows from equation (15) that
(∇aψα,p(a, b) +∇bψα,p(a, b))2
=
{
α(a + b)(ab)++ (ϕp(a, b))
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)}2
= α2(a + b)2(ab)2++ (ϕp(a, b))2
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)2
+2α(a + b)(ab)+(ϕp(a, b))
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)
. (37)
Now, we claim that for all (a, b)̸= (0, 0) ∈ IR2, 2α(a + b)(ab)+(ϕp(a, b))
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)
≥ 0. (38)
If ab≤ 0, then (ab)+ = 0 and the inequality (36) is clear. If a, b > 0, then by noting that
(sgn(a)· |a|p−1+ sgn(b)· |b|p−1
∥(a, b)∥pp−1 − 2
)
≤ 0, ∀(a, b) ̸= (0, 0) ∈ IR2 (39) and ϕp(a, b) ≤ 0, the inequality (38) also holds. If a, b < 0, then ϕp(a, b) ≥ 0, which together with (39) then yields the inequality (38). Thus, we prove that the inequality (38) holds for all (a, b) ̸= (0, 0). Using Lemma 3.3 and equations (38)–(39), we readily obtain the inequality (36) holds for all (a, b)̸= (0, 0). The proof is thus complete. 2
4 A descent algorithm and convergence results
In this section, we propose a derivative-free descent algorithm based on the function Ψα,p. By Lemma 3.1 (d), it is easy to verify that ¯d :=−∇bψα,p(x, F (x)) is a descent direction for monotone nonlinear complementarity problems, i.e., the following result holds.
Lemma 4.1 Let Ψα,p be defined as in (12). If the mapping F is monotone, then ¯d :=
−∇bψα,p(x, F (x)) is a descent direction of Ψα,p at any x∈ IRn, i.e., ∇Ψα,p(x)Td < 0.¯ However, we observe that ¯d does not involve any information of ∇aψα,p(x, F (x)) and is lack of a certain symmetry, for which we can not find a constant c > 0 such that
∥ ¯d∥ ≥ cψα,p(x, F (x)).
This sets a big obstacle to establish the convergence rate of the derivative-free algorithm based on ¯d. In view of this, we follow the similar line as [25] to adopt the search direction of the following form:
dk(ρ) :=−∇bψα,p(xk, F (xk))− ρ∇aψα,p(xk, F (xk)), (40) where ρ is a parameter such that ρ ∈ (0, 1) and ∇aψα,p(x, F (x)), ∇bψα,p(x, F (x)) are defined as in (35). Although dk(ρ) for any ρ ∈ (0, 1) is not necessarily a descent direction of Ψα,p at the iterate xk, Lemma 4.1 implies that it is a descent one if ρ∈ (0, ¯ρk) where
¯
ρk:= 1 if ∇aψα,p(xk, F (xk))T∇Ψα,p(xk)≥ 0, and otherwise
¯
ρk := min
{
1,−∇bψα,p(xk, F (xk))T∇Ψα,p(xk)
∇aψα,p(xk, F (xk))T∇Ψα,p(xk)
}
.
Clearly, ¯ρk ∈ (0, 1) except that xk is a solution of the NCP. Thus, dkis a descent direction of Ψα,p at xk for monotone NCPs only if ρ is chosen sufficiently small. Similar to [25], we also determine an appropriate ρk by the backtracking search of Armijo-type instead of the value of ¯ρk, in our algorithm described as below.
Algorithm 4.1
(Step 0) Given real numbers p > 1 and α ≥ 0 and a starting point x0 ∈ IRn. Choose the parameters σ∈ (0, 1), β ∈ (0, 1), γ ∈ (0, 1) and ε ≥ 0. Set k := 0.
(Step 1) If Ψα,p(xk)≤ ε, then stop.
(Step 2) Let mk be the smallest nonnegative integer m satisfying
Ψα,p(xk+ βmdk(γm))≤ (1 − σβ2m)Ψα,p(xk), (41) where
dk(γm) :=−∇bψα,p(xk, F (xk))− γm∇aψα,p(xk, F (xk)).
(Step 3) Set xk+1 := xk+ βmkdk(γmk), k := k + 1 and go to Step 1.
We see that Algorithm 4.1 does not involve the computation of ∇Ψα,p and ∇F , and hence it is a derivative-free algorithm. In what follows, we establish the convergence results for Algorithm 4.1, and particularly, analyze its convergence rate under the strongly monotone assumption of F . To this end, we assume that the parameter ε in Algorithm 4.1 equals to zero and Algorithm 4.1 generates an infinite sequence {xk}.
Proposition 4.1 Suppose that F is monotone. Then Algorithm 4.1 is well-defined for any starting point x0. Furthermore, if x∗ is an accumulation point of the sequence {xk} generated by Algorithm 4.1, then x∗ is a solution of the NCP.
Proof. We first prove that Algorithm 4.1 is well-defined. From the construction of the algorithm, it suffices to show that Step 2 is well-defined. Assume to the contrary that there is no nonnegative integer m satisfying (41). Then, for any integer m≥ 0,
Ψα,p(xk+ βmdk(γm))− Ψα,p(xk) >−σβ2mΨα,p(xk).
Dividing the above inequality by βm and passing to the limit m→ +∞, we obtain that
mlim→+∞
Ψα,p(xk+ βmdk(γm))− Ψα,p(xk)
βm ≥ 0. (42)
Since Ψα,p is continuously differentiable, we have that Ψα,p is locally Lipschitz continuous at xk, which in turn implies that there exists L > 0 such that
∥Ψα,p(xk+ βmdk(γm))− Ψα,p(xk+ βmdk(0))∥ ≤ Lβm∥dk(γm)− dk(0)∥ for all sufficiently large m. Consequently,
mlim→+∞
Ψα,p(xk+ βmdk(γm))− Ψα,p(xk) βm
= lim
m→+∞
Ψα,p(xk+ βmdk(0))− Ψα,p(xk) βm
+ lim
m→+∞
Ψα,p(xk+ βmdk(γm))− Ψα,p(xk+ βmdk(0)) βm
≤ ∇Ψα,p(xk)Tdk(0).
This together with (42) yields that ∇Ψα,p(xk)Tdk(0) ≥ 0. However, by Lemma 4.1,
∇Ψα,p(xk)Tdk(0) < 0 which leads to a contradiction. Hence, Algorithm 4.1 is well- defined.
Next we prove that any accumulation point x∗ of {xk} is a solution of the NCP. Let {xk}k∈Kbe a subsequence converging to x∗. Notice that Ψα,pis continuously differentiable
