A new smoothing method for solving nonlinear complementarity problems

Jianguang Zhu 1  and Binbin Hao 2
  • 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, 266590, Qingdao, China
  • 2 College of Science, China University of Petroleum, 266555, Qingdao, China
Jianguang Zhu
  • Corresponding author
  • College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China
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and Binbin Hao

Abstract

In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

1 Introduction

Consider the following nonlinear complementarity problems (denoted by NCP),

x0,F(x)0,xTF(x)=0,

where F := (F1, F2, …, Fn)T, and F : ℜn → ℜn is continuously differentiable function.

NCP has been extensively studied due to its various important applications in operations research, engineering design, and economic equilibrium. Many algorithms have been developed for solving the Problem (1.1) [1, 2, 3, 4, 5, 6, 7, 8, 9]. Recently, smoothing method has attracted much attention because that it is a very efficient algorithm to solve NCP. It’s main idea is to use a smoothing function to approximate NCP via a family of parameterized smooth equations and solve the smooth equations approximately at each iteration. By reducing the parameter to zero, we have hoped that a solution of the original problem can be found. It is obvious that smoothing functions play a very important role in smoothing methods. Up to now, a large number of smoothing functions have been proposed, Fischer-Burmeister smoothing function [10] and CHKS smoothing function [11] are the most famous ones. Based on the smoothing functions, scholars proposed a number of smoothing algorithms. In order to solve systems of nonsmooth equations, Chen et al. [7] proposed a smoothing Newton method and proved that the algorithm is globally convergent and locally supperlinearly convergent; Yang et al.[12] proposed a smoothing trust region algorithm by using trust region technique instead of line search strategy, but they had to solve complicated quadratic programming subproblems in its current version. Recently, smoothing Levenberg-Marquardt method has attracted much attention. Based on the trust region technique, a smoothing Levenberg-Marquardt method is proposed for the extended linear complementarity problems in[13]. By employing Fischer-Burmeister smoothing function, a smoothing Levenberg-Marquardt method is proposed for solving nonlinear complementarity problems with P0 function in [14]. In [14], a smoothing parameter τ as an independent variable was introduced. In order to ensure the strict positivity of the smoothing parameter, a relatively complicated subproblems have to be solved at each iteration. Based on a partially smoothing function, Wan et al.[15] proposed a partially smoothing Jacobian method for solving the nonlinear complementarity problems. Like most Jacobian smoothing methods, they still have to assume the function F is a P0-function. Under the condition that the level set of a merit function is bounded, they proved the proposed algorithm is globally convergent and superlinearly convergent.

In this paper, motivated by the above work, we propose an improved smoothing Newton method for solving the Problem (1.1). First, based on a one-parametric class of smoothing function, the Problem (1.1) can be reformulated to a system of smoothing equations, and an improved smoothing method is proposed for solving the smoothing equations. Different from the processing in [12, 14], we solve a system of linear equations instead of a quadratic programming problem for each inner iteration. Moreover, when the iteration point is close enough to the solution point of NCP, the algorithm always takes the full steps. Without strict complementarity conditions and the assumption of P0 property, we prove that the proposed smoothing method possess the global and local quadratic convergence properties. Compared with previous smoothing methods, our method has some other good properties. Especially,

  1. Compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function;
  2. Without requiring strict complementarity conditions and the assumption of P0 property, proposed smoothing method is also proved to possess global and local quadratic convergence rate.

We introduce the following notations. Throughout this paper, all vectors are column vectors, the superscript T denotes transpose of a matrix and a vector, ∥x∥ stands for the 2-norm of vector xn.+n(++n) denotes the nonnegative (positive) orthant in ℜn. For a continuously differentiable function Φτ(x): ℜn → ℜn, we denote the Jacobian of Φτ(x) at x ∈ ℜn by Φτ(x).

The rest of paper is organized as follows. In section 2, we investigate a one-parametric class of smoothing function and discuss its properties, and recall some preliminary result used in the subsequent. The algorithm model is stated in section 3. In section 4, the global and local quadratic convergence of the new algorithm is established. Some numerical test results are reported in section 5, which show that the proposed algorithm is efficient.

2 Smoothing function and its properties

In this section, we investigate a parameter smoothing function and discuss its properties. Based on this smoothing function, the equivalent smoothing reformulation of NCP is given. Firstly, we recall a class of NCP function, which was defined in [16],

φθ(a,b):=a+bθ(ab)2+(1θ)(a2+b2),θ[0,1],(a,b)2.

It has the following characterizations

φθ(a,b)=0a0,b0andab=0.

Using the function φθ, we can reformulate Problem (1.1) as the following system of nonlinear equations

Φ(x)=0,

where Φ : ℜn → ℜn is defined by

Φ(x):=φθ(x1,F1(x))φθ(xn,Fn(x)).

Then the natural merit function Ψ : ℜn → [0, +∞) is defined by

Ψ(x)=12Φ(x)2.

Therefore, the following equivalence relation is established,

x solves Problem(1.1)x solves Φ(x)=0x solves minxnΨ(x) with Ψ(x)=0.

As we all known, the nonsmooth equation Φ(x) = 0 is very difficult to solve. In order to overcome the difficulty, Zhu et al.[22] introduced the following smoothing function for φθ:

φθ(τ,a,b)=a+bθ(ab)2+(1θ)(a2+b2)+2τ2,(τ,a,b)3,

where θ is a given constant with θ ∈ [0, 1]. It is obvious that when θ = 1, φθ reduces to the famous CHKS smoothing function, θ = 0, φθ reduces to the famous Fischer-Burmeister smoothing function. As we all know, these two smoothing functions and their variants have been widely used in designing smoothing-type methods for solving mathematical programming problems, such as the nonlinear complementarity problems (NCPs) [16, 17, 18, 19, 20, 21, 22], the second-order cone complementarity problems(SOCCPs)[23, 24, 25, 26, 27, 28, 29], the second-order cone programming (SOCP) [30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

Using the smoothing function (2.2), a smooth approximation of Φ is defined by Φτ : ℜn → ℜn

Φτ(x):=φθ(τ,x1,F1(x))φθ(τ,xn,Fn(x)),

and the corresponding merit function can also be defined by

Ψτ(x)=12Φτ(x)2.

So, to solve Problem (1.1), we only need to solve Φτ(x) = 0 and make τ ↓ 0.

Next, we review the concept of semismooth, which was first introduced by Mifflin [40] for functions and extended to vector-valued functions by Qi and Sun [41].

Definition 2.1

Suppose that F : ℜn → ℜn is locally Lipschitz function. The generalized Jacobian of F at x in the sense of Clark [42] denote by ∂F(x), then, F is said to be semismooth (or strongly semismooth) at x ∈ ℜn, if F is directionally differentiable at x ∈ ℜn and F(x + h) – F(x) – Vh = o(∥h∥) (or = O( ∥h2) ) holds for any V∂F(x + h).

Lemma 2.1

The function Φτ(x) satisfies the inequality

Φτ1(x)Φτ2(x)κ|τ1τ2|

for all x ∈ ℜn and τ1, τ2 ≥ 0, where κ = 2n. In particular, we have

Φτ(x)Φ(x)κτ

for all x ∈ ℜn and τ ≥ 0.

Proof

It is obvious to hold for τ1 = τ2 = 0. So, we suppose at least one of the perturbation parameters is positive. By simple calculation, we can have

|Φτ1,i(x)Φτ2,i(x)|=|ηθ(τ1,xi,Fi(x))ηθ(τ2,xi,Fi(x))|=2|τ12τ22||ηθ(τ1,xi,Fi(x))+ηθ(τ2,xi,Fi(x))|2|τ12τ22|2τ12+2τ12=2|τ1τ2|.

Obviously, for any x ∈ ℜn,

Φτ1(x)Φτ2(x)=i=1n|Φτ1,i(x)Φτ2,i(x)|2κ|τ1τ2|,

where κ = 2n.

Similar to the proof of Lemma 2.8 in [22], we have the following lemma.

Lemma 2.2

Let Φτ(x) be defined by (2.3). Then,

limτ0dist(Φτ(x),CΦ(x))=0,

where CΦ(x) is the C-subdifferential of Φ(x), and Φτ(x) is the Jacobian of Φτ(x).

Similar to the proof of Proposition 3.4 in [8], we have the following lemma.

Lemma 2.3

Let x ∈ ℜn be arbitrary but fixed. Assume that x is not a solution of NCP. Let us define the constants

γ(x):=maxiβ(x){xiei+Fi(x)Fi(x)}0

and

α(x):=maxiβ(x){xi2+Fi(x)2}>0,

where β(x) := {i|xi = Fi(x)}. Let δ > 0 be given, and define

τ¯(x,δ):=1,ifnγ(x)2δ2α(x)0,α(x)22(δ2nγ(x)2δ2α(x)),otherwise.

Then

dist(Φτ(x),CΦ(x))δ

for all τ such that 0 < ττ(x, δ).

3 Proposed Algorithm

In this section, we establish a new smoothing method for Problem(1.1) and prove that well-definiteness of the proposed algorithm.

Algorithm 3.1

Step 0 Given a starting point x0 ∈ ℜn, choose α, σ, η, ρ, γ ∈ (0, 1), δ ∈ (0, ∞), set β0 = ∥Φ(x0)∥, κ = 2n,τ0=α2κβ0,μ0=Φτ0(x0),k:=0.

  1. Step 1Find a solution dk ∈ ℜn of the linear system
    Φτk(xk)TΦτk(xk)+μkId=Φτk(xk)TΦτk(xk)
  2. Step 2If the condition
    Φτk(xk+dk)γΦτk(xk)
    is satisfied, then set xk+1 := xk+dk(we call this ‘fast step’) and go to Step 4.
  3. Step 3set λk = ρmk and xk+1 = xk+λk dk, where mk be the smallest nonnegative integer m such that
    Ψτk(xk+ρmdk)Ψτk(xk)σρmΨτk(xk)Tdk.
  4. Step 4If ∇Ψ(xk+1) = 0, stop.
  5. Step 5If
    Φ(xk+1)maxηβk,α1Φ(xk+1)Φτk(xk+1),
    then set
    βk+1:=Φ(xk+1)
    and choose τk+1 satisfied
    0<τk+1minα2κβk+12,τk2,τ¯(xk+1,δβk+1),
    where τ(⋅, ⋅) is defined in Lemma 2.4; otherwise, let βk+1 := βk and τk+1 := τk.
  6. Step 6Set μk+1 = ∥Φτk+1(xk+1)∥ and k := k+1. Go to Step 1.

Remark

Differently from the algorithm in [7, 15], our proposed algorithm is well-defined without the assumption that smoothing function Φτk(xk) is nonsingular.

For proving that the algorithm 3.1 is well-defined and has the global convergence property, we assume that algorithm 3.1 does not terminate in a finite number of iterations.

Define the index set

K:={0}k|Φ(xk)max{ηβk1,α1Φ(xk)Φτk1(xk)},k=1,2,

It follows from Lemma 2.1 and Step 5 of the algorithm 3.1 that

Φ(xk)Φτk(xk)<αΦ(xk),

which indicates ∥Φτk(xk)∥ ≠ 0 for all kK. By the definition τ(⋅, ⋅) in Lemma 2.3 and the updating rule (3.5), it is not difficult to find that

dist(Φτk(xk),CΦ(xk))δΦ(xk)

holds, for any kK with k ≥ 1.

The following proposition shows the well-definiteness of the proposed algorithm.

Proposition 3.1

Suppose that xk is the sequence generated by algorithm 3.1. Then Algorithm 3.1 is well-defined.

Proof

  1. If dk = 0. Obviously, there exists a finite nonnegative mk such that (3.3) always holds.
  2. If dk ≠ 0. From the above analysis, we know that μk = ∥Φτk(xk)∥ is positive, then Φτk(xk)TΦτk(xk)+μkI is always positive definite. Since, Ψτk(xk)=Φτk(xk)TΦτk(xk). By the construction of Algorithm 3.1, we have
    Ψτk(xk)Tdk=(dk)T(Φτk(xk)TΦτk(xk)+μkI)(dk)<0,
    moreover
    Ψτk(xk+λdk)Ψτk(xk)=λΨτk(xk)Tdk+o(λ)=σλΨτk(xk)Tdk+(1σ)λΨτk(xk)Tdk+o(λ).
    It follows from (3.9) and σ ∈ (0, 1) that
    (1σ)λΨτk(xk)Tdk<0.
    By (3.10) and (3.11), we can obtain the following inequality
    Ψτk(xk+λdk)Ψτk(xk)σλΨτk(xk)Tdk.
    So, there exists a finite nonnegative integer mk such that
    Ψτk(xk+ρmkdk)Ψτk(xk)σρmkΨτk(xk)Tdk.
    Therefore, Algorithm 3.1 is well-defined. □

4 Convergence of the proposed algorithm

In this section, we will give the global and local quadratic convergence of the proposed algorithm.

Theorem 4.1

Suppose that {xk} is a sequence generated by algorithm 3.1. If there exists at least an accumulation point in the sequence {xk}, then the index set K defined by (3.6) is infinite, and

limkτk=0,limkΦτk(xk)=0,limkΦ(xk)=0.

Proof

Assume that x* is an arbitrary accumulation point of the sequence {xk}, and {xk}kK be a subsequence converging to x*. We first show that the index set K is infinite. By contradiction, we assume that the set K is finite. Let be the largest number in K, then for any k, wehave τk = τ and βk = β.

Denote

τ^=τk^,β^=βk^,q(x)=Φ(x)Φτ^(x),

then for all k, we have

Φ(xk)>maxηβ^,q(xk)α

and

Φ(xk)=Φτ^(xk)+q(xk).

In the following paragraphs, we will show that the following equation holds.

Ψτ^(x)=0.

Now, we consider two cases for the sequence {μk}:

case 1: If μk → 0(kK), then, we can obtain ∥Φτ̂(xk)∥ → 0 (kK). It follows from the boundness of {Φτ^(xk)} that

Ψτ^(xk)=Φτ^(xk)Φτ^(xk)0(kK),

which implies ∇Ψτ̂(x*) = 0.

case 2: In this case, there exists a constant ξ such that μkξ > 0 for all kK. Next, we show that ∇Ψτ̂(x*) = 0. Suppose to the contrary that ∇Ψτ̂(x*) ≠ 0.

It follows from (3.1), we have

dkΦτ^(xk)TΦτ^(xk)(Φτ^(xk)TΦτ^(xk)+μkI,

and

dk=(Φτ^(xk)TΦτ^(xk)+μkI)1Φτ^(xk)TΦτ^(xk)(Φτ^(xk)TΦτ^(xk)+μkI)1Φτ^(xk)TΦτ^(xk)Φτ^(xk)TΦτ^(xk)ξ.

From the continuity of Φτ̂(xk) and the upper semicontinuity of the generalized Jacobian, we deduce that for a constant δ > 0,

Φτ^(xk)TΦτ^(xk)+μkIδfor allkK.

We now note that

0<mdk,for some positivem.

In fact, if {∥dk∥}K → 0, by (4.5) and (4.7), we can deduce {Φτ^(xk)TΦτ^(xk)}K0, thus contradicting the assumption ∇Ψτ̂(x*) ≠ 0. On the other hand, we denote the bound of {Φτ^(xk)TΦτ^(xk)} as M, it follows from (4.6) and (4.8) that

0<mdkMξ.

Since μk = ∥Φτ̂(xk)∥ > 0, for all kK, Φτ^(xk)TΦτ̂(xk)+μkI is positive definite, this together with (3.1) and (4.9), implies that

Ψτ^(xk)Tdk=(dk)T(Φτ^(xk)TΦτ^(xk)+μkI)dk<0.

Next, we consider two cases for the sequence {infkλk}:

  1. If infkKλk = λ* > 0 for all k. By (3.3) and (4.10), we can obtain that
    Ψτ^(xk+1)Ψτ^(xk)σλkΨτ^(xk)TdkσλΨτ^(xk)Tdk<0.
    Using Ψτ̂(xk+1) − Ψτ̂(xk) → 0 yields that
    {Ψτ^(xk)Tdk}K0.
    By (3.1), we have
    Ψτ^(xk)Tdk=Ψτ^(xk)T(Φτ^(xk)TΦτ^(xk)+μkI)1Ψτ^(xk)k.
    Since ∂Φτ̂(x) is a nonempty compact set for any x ∈ ℜn, {Φτ^}K is bounded. Without loss of generality, assume that {Φτ^}KΦ*. Considering that the set-valued mapping x∂Φτ̂(x) is closed and {xk}Kx*, we have Φ*∂Φτ̂(x*). In addition, since Φτ̂(x*) ≠ 0, we have μkμ* with μ* = ∥Φτ̂(x*)∥ > 0. Thus, {Φτ^(xk)TΦτ^(xk)+μkI}KΦTΦ+μI0. This together with (4.13) and the continuity of ∇Ψτ̂, implies that ∇Ψτ̂(xk)T dk has a nonzero limit as k → + ∞, which contradicts (4.12). Hence ∇Ψτ̂(x*) = 0.
  2. If infkKλk = 0. In this case, without loss of generality, we assume that {λk}kK → 0. By (4.9), there exists a subsequence {dk}kKd ≠ 0.From the line search rule (3.3), it is clear that for all k,
    Ψτ^(xk+ρmk1dk)Ψτ^(xk)ρmk1>σΨτ^(xk)Tdk.
    On K, by taking the limit k → ∞, we obtain from (4.14),
    Ψτ^(x)Td¯σΨτ^(x)Td¯.
    Since, σ ∈ (0, 1), then, by (4.15) we have ∇Ψτ̂(x*)T d ≥ 0. On the other hand, d is the solution of (ΦTΦ+μI)d¯=Ψτ^(x), which implies ∇Ψτ̂(x*)Td < 0, (since (ΦTΦ+μI)0.) Thus, we get a contradiction. Then, ∇Ψτ̂(x*) = 0.To sum up by case (1) and case (2), we have ∇Ψτ̂(x*) = 0.On the other hand, in view of Lemma 2.1 and (4.3), we have Ψτ̂(xk) → Ψτ̂(x*) = 0, then, there exists such that for all kL with k,
    Φτ^(xk)(1α)ηβ^.
    In view of (4.1) and (4.2), we deduce that for all kK with k,
    Φτ^(xk)(1α)Φ(xk)(1α)(Φτ^(xk)+q(xk)),
    i.e.,
    Φτ^(xk)(11α)q(xk).
    Therefore,
    Φ(xk)Φτ^(xk)+q(xk)<q(xk)α
    which in contradiction to (4.1). Hence the set K is infinite.By the updating rule of τk, we have {τk} → 0. Similar to the proof of Proposition 4.2 in [8], we deduce that
    Φ(xk)rj(1+α)Φ(x0),askjk<kj+1.
    where r=max{12,η}.Since the set K is infinite, it follows from (3.8) and (4.16) that
    limkΦτk(xk)=0,limkΦ(xk)=0.
    As a consequence of Theorem 4.1, we can obtain the following global convergence result.

Theorem 4.2

Let {xk} be a sequence generated by Algorithm 3.1, then every accumulation point of the sequence {xk} is a solution of the Problem(1.1).

In order to obtain the local convergent result, we introduce the following lemma.

Lemma 4.1

Let {xk}kK converge to x*. If VCΦ(x*) is nonsigular, then there exist two constants M1 > 0, M2 > 0 and k̂ such that for all kK with k,

Φτk(xk)M1,(Φτk(xk)TΦτk(xk))1M2.

Proof

By the condition, we know that the matrix V is nonsingular. Notice that CΦ(x) is compact for all x ∈ ℜn. Therefore, there exists VkCΦ(xk) such that

dist(Φτk(xk),CΦ(xk))=Φτk(xk)Vk.

Combining with (3.8), we can deduce that for all kK,

Φτk(xk)Vkδβk=δΦ(xk).

By theorem 4.1, we have {∥Φ(xk)∥} → 0. This together with the compactness and the upper semicontinuity of CΦ(x*), we obtain from the above inequality that the matrices Φτk(xk) and Φτk(xk)TΦτk(xk) are nonsingular and there exist M1 > 0, M2 > 0 such that (4.17) holds. □

The following lemma can be seen in Ref. [44].

Lemma 4.2

Assume that A, B ∈ ℜn × n and A is nonsingular. IfA−1B∥ < 1, then A+B is nonsingular and satisfies

(A+B)1A11A1B.

From Lemmas 4.1 and 4.2, we can prove that the following lemmas hold.

Lemma 4.3

Assume that the conditions of Lemma 4.1 hold. Then, for μk12M2,

(Φτk(xk)TΦτk(xk)+μkI)12M2.

Theorem 4.3

Assume that {xk} is a sequence generated by Algorithm 3.1. If x* is an accumulation point of the sequence {xk}, and for all VCΦ(x*) are nonsinglar. Then the sequence {xk} converges to x* quadratically.

Proof

It follows from Theorem 4.1 that x* is a solution of Φ(x) = 0. Notice that BΦ(x*) ⊆ CΦ(x*). From Proposition 2.5 in [45], it follows that there exists a neighbourhood of x* such that x* is the unique solution.

Since the sequence {xk} has an accumulation point x*, there exists a subsequence {xk}kK that converges to x*. By Lemma 4.1, there exist M1 > 0, M2 > 0 and kK,

Φτk(xk)M1,(Φτk(xk)TΦτk(xk))1M2.

Further, from Lemma 2.1, (3.5) and the Lipschitz continuity of Φ(x), we deduce that

βk=Φ(xk)=O(xkx),

τk=O(xkx2),

μk=Φτk(xk)κτk+Φ(xk)=O(xkx).

Then, it follows from (4.21) and Lemmas 4.2, 4.3 that

(Φτk(xk)TΦτk(xk)+μkI)12M2.

Therefore, for all k > and kK, we have

xk+dkx=(Φτk(xk)TΦτk(xk)+μkI)1Φτk(xk)TΦτk(xk)+xkx(Φτk(xk)TΦτk(xk)+μkI)1Φτk(xk)TΦτk(xk)(Φτk(xk)TΦτk(xk)+μkI)(xkx)(Φτk(xk)TΦτk(xk)+μkI)1Φτk(xk)T(Φτk(xk)Φτk(xk)(xkx))+μkxkx2M1M2Φτk(xk)T(Φτk(xk)Φτk(xk)(xkx))+μkxkx2M1M2μkxkx+(Φτk(xk)Φ(xk))(Φτk(xk)Vk)(xkx)+Φ(xk)Φ(x)Vk(xkx)2M1M2Φτk(xk)Φ(xk)+δβkxkx+Φ(xk)Φ(x)Vk(xkx)+μkxkx2M1M2κτk+δβkxkx+μkxkx+Φ(xk)Φ(x)Vk(xkx)

By Proposition 2.3 in [16], we known that Φ(x) is strongly semismooth, combining with Theorem 3.2 in [41], we have

Φ(xk)Φ(x)Vk(xkx)Σi=1nΦi(xk)Φi(x)Vki(xkx)=Oxkx2ask,kK,

where Vki denotes the i-th row of Vk. Hence, From (4.19)-(4.23), we can deduce

xk+dkx=O(xkx2)ask,kK.

The above equation together with (4.21) indicate that

Φτk+1(xk+dk)=O(xk+dkx)=O(xkx2)ask,kK.

It follows from triangular inequality and (3.8) that

Φτk(xk)=Φ(xk)(Φ(xk)Φτk(xk))Φ(xk)αΦ(xk)=(1α)Φ(xk).

The above equations (4.25) and (4.26) imply that

Φτk+1(xk+dk)Φτk(xk)Φτk+1(xk+dk)(1α)Φ(xk)=Oxkx2(1α)Oxkx0(askkK),

which means that there exists k such that kK and for any kk,

Φτk+1(xk+dk)γΦτk(xk).

That is, xk+1 = xk+dk for all kk and kK. This together with (4.24) implies that {xk} converges to x* quadratically. □

5 Numerical experiments

In this section, we implement algorithm 3.1 on some typical test problems for two purposes: one is to see the numerical behavior of algorithm 3.1; and the other is to investigate the behavior of these test problems for different θ ∈ [0, 1]. All the codes are finished in MATLAB 7.8 and done using a PC with Intel (R) Core (TM) i3-3240 CPU @ 3.40 GHz and RAM of 4 GB.

The parameters used in algorithm 3.1 are as follows:

α=0.95,σ=0.01,η=0.9,ρ=0.8,γ=0.9,δ=30.

We use

Ψ(xk)106

as the stop rule.

In the tables of experimental results, ST denotes the starting point; DIM denotes the dimension of the problem; θ denotes the values of θ; IT denotes the number of iteration; Fast denotes the number of ‘full step’ taken during the iteration; B denotes the number of backtracking steps; τ denotes the value of τ at the final iteration.

From Tables 1-3, It is easy to see that not all the best numerical results based on the proposed smoothing algorithm occur in the case of θ = 1 (in this case, the smoothing function is the famous CHKS smoothing function) or θ = 0 (in this case, the smoothing function is the famous Fischer-Burmeister smoothing function). On the other hand, from the column Fast, it can be seen that, in general, the number of the iteration in which ‘fast step’ are accepted occupied almost all the iterations. From Table 2, we can see that θ = 1 seems to be more suitable to find a non-degenerate solution. We also investigate the behavior of our proposed algorithm for solving large scale linear complementarity problems with θ = 1. The numerical results are listed in Tables 4 and 5. From Tables 4 and 5, we can see that our proposed method is efficient for solving large scale linear complementarity problems. From all the test results, we can see that our algorithm is promising.

Table 1

Numerical behavior for Example 5.1

STθITτFast∥∇Ψ(xk)∥B
(−2, −2, −2, −2)T0142.8e-1264.4e-716
0.25123.2e-1772.2e-711
0.5139.3e-1664.5e-714
0.7589.2e−2181.4e-90
1122.2e-14115.7e-72

(1, 4, 1, 4)T0194.5e−25131.4e-96
0.25172.2e-17121.3e-125
0.5159.5e−22101.1e-85
0.75152.3e−20102.3e-125
1224.6e−2082.3e-1014

(3, 3, 3,3)T0144.1e−23148.9e-110
0.25124.1e-17125.1e-70
0.5111.1e-17116.5e-80
0.75112.3e−20112.3e-100
1149.7e−2487.8e-116

Table 2

Numerical behavior for Example 5.2

STθITτFast∥∇Ψ(xk)∥Bsolution
(6, 6, 6, 6)T0213.7e-17163.2e-76Degenerate
0.25212.8e−28171.8e-126Degenerate
0.5163.3e−22165.2e-90Degenerate
0.75157.5e−24142.3e-102Degenerate
1237.7e−21222.8e-96Nondegenerate

(1, 2, 3, 4)T0123.4e-32124.9e-140Degenerate
0.25116.6e-32104.4e-142Degenerate
0.5115.7e−27101.3e-112Degenerate
0.75111.1e-1793.5e-76Degenerate
1218.7e−26203.4e-116Nondegenerate

(2, −3, −3, 2)T0131.6e−24133.2e-110Degenerate
0.25122.8e-16115.1e-72Degenerate
0.5111.3e−22103.1e-92Degenerate
0.75111.4e−28102.8e-123Degenerate
1251.5e−25211.3e-119Nondegenerate

Table 3

Numerical behavior for Example 5.3

STθITτFast∥∇Ψ(xk)∥B
(−1, −1, −1, −1, −1, −1, −1, −1)T0254.7e−18251.1e-80
0.25223.2e−15224.1e-70
0.5215.8e−20212.5e-90
0.75201.5e−19205.7e-90
1241.6e−15207.1e-713

(−1, −1, −1, −1, 1, 1, 1, 1)T0267.4e−17269.4e-80
0.25231.4e−17232.9e-80
0.5211.9e−17214.5e-80
0.75211.7e−19205.8e-94
1244.1e−17211.3e-717

(0, 0, 0,0,0, 0, 0,0)T0238.7e−16231.4e-70
0.25201.7e−17209.1e-70
0.5195.7e−20192.5e-90
0.75185.3e−21181.3e-90
1197.7e−18184.8e-81

Table 4

Numerical results for Example 5.4 with θ = 1

STDIMITτFast∥∇Ψ(xk)∥B
(−1, ⋯, −1)T500151.1e−23141.2e-91
1000197.5e−26151.5e−104
2000242.6e−22151.2e-89
3000286.2e−22152.2e-813

(0, ⋯, 0)T50084.3e−1982.4e-70
1000101.1e−26105.6e−110
2000121.5e−27122.9e−110
3000134.2e−21135.8e-80

(1, ⋯, 1)T50093.9e−2992.4e−120
1000102.2e−21102.5e-80
2000128.5e−22122.2e-80
3000149.1e−28142.8e−110

Table 5

Numerical results for Example 5.5 with θ = 1

STDIMITτFast∥∇Ψ(xk)∥B
(−1, ⋯, −1)T500111.1e−21111.9e-80
1000141.2e-32138.7e−141
2000171.1e−26151.1e−102
3000191.9e−20153.2e-84

(0, ⋯, 0)T50069.5e−2261.7e-80
100079.2e−2577.6e−100
200081.9e−2481.6e-90
300096.5e−2991.1e−110

(1, ⋯, 1)T500123.3e-33123.9e−140
1000152.9e-3394.6e−146
2000192.4e-3082.0e−1211
3000211.8e−2976.1e−1214

Example 5.1

Mathiesen Problem. This test problem was used by Jiang and Qi [46] with four variables, which was also tested by Pang and Gabriel [47]. Let

F1(x)=x2+x3+x4,F2(x)=x1α(b2x3+b3x4)/x2,F3(x)=b2x1(1α)(b2x3+b3x4)/x3,F4(x)=b3x1.whereα=0.75,b2=1,b3=2.

We test this problem by using different starting points. The test results are listed in Table 1.

Example 5.2

Kojima-Shindo Problem. This problem was tested by Pang and Gabriel in [47], which was also tested by Mangasarian and Solodov [48], and Kanzow [49] with four variables. Let

F1(x)=3x12+2x1x2+2x22+x3+3x46,F2(x)=2x12+x1+x22+10x3+2x42,F3(x)=3x12+x1x2+2x22+2x3+9x49,F4(x)=x12+3x22+2x3+3x43.

This example has one degenerate solution (62,0,0,12)T and one nondegenerate solution (1, 0, 3, 0)T. We summarize the results in Table 2 by using different starting points.

Example 5.3

HS 66 Problem. This test problem is the NCP reformulation for 66th problem in the book of Hock and Schittkowski [50]: Let

F1(x)=0.8+x4ex1+x6,F2(x)=x4+x5ex2+x7,F3(x)=0.2x5+x8,F4(x)=x2ex1,F5(x)=x3ex2,F6(x)=100x1,F7(x)=100x2,F8(x)=10x3.

We summarize the test results in Table 3 by using different starting points.

Example 5.4

This problem is from Geiger and Kanzow [51], which was also tested by Kanzow [11]: Let F(x) = Mx+q, where

M=4100014100014000004100014,q=(1,1,,1)T.

We test this problem by using different starting points. The test results are listed in Table 4.

Example 5.5

This example was used by Ahn [52]. Let F(x) = Mx+q, where

M=4200014200014000004200014,q=(1,1,,1)T.

The test results for Example 5.5 are listed in Table 5 by using different starting points.

6 Conclusion

In this paper, we have presented a new improved smoothing algorithm for the NCP. Compared with the classical smoothing method, our proposed method needn’t nonsingular of the smoothing approximation function. We have established the global convergence and the local quadratic convergence for the developed algorithm without strict complementarity conditions and the assumption of P0 property. Numerical results have showed that the new algorithm works very well.

Competing interests

The authors declare that they have no competing interests.

Acknowledgement

This work was supported by the Training Program of the Major Research Plan of National Science Foundation of China under Grant(91746104), by the National Nature Science Foundation of China(Grant Nos.61101208, 11326186) Qindao Postdoctoral Science Foundation(Grant No.2016114), A Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J17KA166), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China and SDUST Research Fund(Grant No.2014TDJH102).

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Ferris M.C., Pang J.S., Engineering and economic applications of complementarity problems, SIAM Rev., 1997, 39, 669-713.

    • Crossref
    • Export Citation
  • [2]

    Hang Z.H., Sun J., A non-interior continuation algorithm for the P 0 or P * LCP, with strong global and local convergence properties, Appl. Math. Optim., 2005, 52, 237-262.

    • Crossref
    • Export Citation
  • [3]

    Tian Z.L. et al., An accelerated Jacobi gradient based iterative algorithm for solving, sylvester matrix equations, Filomat, 2017, 31(8), 2381-2390.

    • Crossref
    • Export Citation
  • [4]

    Zhu J.G., Hao B.B., A new noninterior continuation method for solving a system of equalities and inequalities, J. Appl. Math., 2014, Article ID 592540, 6 pages.

  • [5]

    Luca T.D., Fancchinei F., Kanzow C., A semismooth equation approach to the solution of nonlinear, complementarity problems, Math.Progam., 1996, 75, 407-439.

  • [6]

    Hao B.B., Zhu J.G., Fast L1 regularized iterative forward backward splitting with, adaptive parameter selection for image restoration, J. Visual Commun. Image Represent., 2017 44, 139-147.

    • Crossref
    • Export Citation
  • [7]

    Chen X., Qi L., Sun D., Global and superlinear convergence of the smoothing Newton method, and its application to general box constriained variational, inequalities, Math. Comput., 1999, 222, 519-540.

  • [8]

    Kanzow C., Pieper H., Jacobian smoothing methods for, nonlinear complementarity problems, SIAM J. Optim., 1999, 9,342-373.

    • Crossref
    • Export Citation
  • [9]

    Qi H.D., Liao L.Z., A smoothing Newton method for, general nonlinear complementarity problems, Comput. Optim. Appl., 2000, 17, 231-253.

    • Crossref
    • Export Citation
  • [10]

    Fischer A., A special Newton-type optimization methods, Optimization, 1992, 24, 269-284.

    • Crossref
    • Export Citation
  • [11]

    Kanzow C., Some, Noninterior Continuation Methods for Linear Complementarity, Problems, SIAM J. Matrix Anal. Appl., 1996, 17, 851-868.

    • Crossref
    • Export Citation
  • [12]

    Yang Y.F., Qi L., Smoothing trust region methods for nonlinear complementarity problems with P 0 functions, Ann. Oper. Res., 2005, 133, 99-117.

    • Crossref
    • Export Citation
  • [13]

    Yu Z., Su K., Lin J., A smoothing Levenberg-Marquardt method for the extended, linear complementarity problem, Appl. Math. Model., 2009, 33, 3409-3420.

    • Crossref
    • Export Citation
  • [14]

    Zhang J.L., Zhang X.S., A Smoothing Levenberg-Marquardt method for NCP, Appl. Math. Comput., 2006, 178, 212-228.

    • Crossref
    • Export Citation
  • [15]

    Wan Z., Yuan M., Wang C., A partially smoothing Jacobian method for nonlinear, complementarity problems with P 0 function, J. Comput. Appl. Math., 2015, 286, 158-171.

    • Crossref
    • Export Citation
  • [16]

    Hu S.L., Huang Z.H., Chen J.S., Properties of a family of generalized NCP-functions and a, derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 2009, 230, 69-82.

    • Crossref
    • Export Citation
  • [17]

    Zhu J.G., Li K., Hao B.B., Image restoration by a mixed high-order total variation and, l1 regularization model, Math. Probl. Eng., 2018, 1-13.

  • [18]

    Sun L., He G., Wang Y., Zhou C., An accurate active set newton algorithm for large scale bound, constrained optimization, Appl. Math., 2011, 56(3), 297-314.

    • Crossref
    • Export Citation
  • [19]

    Sun L., He G., Wang Y., Fang L., An active set quasi-Newton method with projected search for bound, constrained minimization, Comput. Math. Appl., 2009, 58, 161-170.

    • Crossref
    • Export Citation
  • [20]

    Zhu J.G., Li K., Hao, B.B., Restoration of remote sensing images based on nonconvex, constrained high-order total variation regularization, J. Appl. Remote Sens., 2019, 13(2), 022006.

  • [21]

    Zhong B., Dong H., Kong L.C., Tao J.Y., generalized nonlinear complementarity problems with order and properties, Positivity, 2014, 18, 413-423.

    • Crossref
    • Export Citation
  • [22]

    Zhu J.G., Hao B.B., A new class of smoothing functions and a smoothing Newton method for complementarity problems, Optim. Lett., 2013, 7, 481-497.

    • Crossref
    • Export Citation
  • [23]

    Tang J.Y., He G.P., Dong L., Fang L., Zhou J.C., A smoothing Newton method for the second-order cone complementarity problem, Appl. Math., 2013, 58, 223-247.

    • Crossref
    • Export Citation
  • [24]

    Yu J., Li M., Wang Y. He G., A decomposition method for large-scale box constrained optimization, Appl. Math. Comput., 2014, 231, 9-15.

    • Crossref
    • Export Citation
  • [25]

    Li M., Kao X., Che H., Relaxed inertial accelerated algorithms for solving split equality feasibility problem, J. Nonlinear Sci. Appl., 2017, 10, 4109-4121.

    • Crossref
    • Export Citation
  • [26]

    Han C.Y., Zheng F.Y., Guo T.D., He G.P., Parallel algorithms for large-scale linearly constrained minimization problem, Acta Mathematicae Applicatae Sinica, 2014, 30, 707-720.

    • Crossref
    • Export Citation
  • [27]

    Sun L., Fang L., He G., An active set strategy based on the multiplier function or the gradient, Appl. Math., 2010, 55(4), 291-304.

    • Crossref
    • Export Citation
  • [28]

    Li Y., Tan T., Li X., A log-exponential smoothing method for mathematical programs with complementarity constraints, Appl. Math. Comput., 2012, 218, 5900-5909.

    • Crossref
    • Export Citation
  • [29]

    Fang L., Han C.Y., A new one-step smoothing Newton method for the second-order cone complementarity problem, Math. Method Appl. Sci., 2011, 34, 347-359.

    • Crossref
    • Export Citation
  • [30]

    Tang J.Y., He G.P., Dong L., Fang L., A smoothing Newton method for second-order cone optimization based on a new smoothing function, Appl. Math. Comput., 2011, 218, 1317-1329.

    • Crossref
    • Export Citation
  • [31]

    Feng Z.Z., Fang L., He G.P., An O ( n L ) $\begin{array}{} O(\sqrt{n}L) \end{array} $ iteration primal-dual path-following method based on wide neighbourhood and large update for second-order cone programming, Optimization, 2014, 63, 679-691.

    • Crossref
    • Export Citation
  • [32]

    Tang J.Y., He G.P., Dong L., Fang L., A new one-step smoothing newton method for second-order cone programming, Appl. Math., 2012, 57(4), 311-331.

    • Crossref
    • Export Citation
  • [33]

    Liu C.H., Shang Y.L., Han P., A New Infeasible-Interior-Point Algorithm for Linear Programming over Symmetric Cones, Acta Mathematicae Applicatae Sinica English Series, 2017, 33, 771-788.

    • Crossref
    • Export Citation
  • [34]

    Zheng F., Han C., Wang Y., Parallel SSLE algorithm for large scale constrained optimization, Appl. Math. Comput., 2011, 217(12), 5377-5384.

    • Crossref
    • Export Citation
  • [35]

    Fang L., He G.P., Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 2009, 228(1), 296-303.

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