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A SMOOTHING NEWTON METHOD FOR NCP BASED ON A NEW CLASS OF SMOOTHING FUNCTIONS†
A SMOOTHING NEWTON METHOD FOR NCP BASED ON A NEW CLASS OF SMOOTHING FUNCTIONS†
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 211-225
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 08, 2013
  • Accepted : April 20, 2013
  • Published : January 28, 2014
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JIANGUANG ZHU
BINBIN HAO

Abstract
A new class of smoothing functions is introduced in this paper, which includes some important smoothing complementarity functions as its special cases. Based on this new smoothing function, we proposed a smoothing Newton method. Our algorithm needs only to solve one linear system of equations. Without requiring the nonemptyness and bounded-ness of the solution set, the proposed algorithm is proved to be globally convergent. Numerical results indicate that the smoothing Newton method based on the new proposed class of smoothing functions with θ ∈ (0, 1) seems to have better numerical performance than those based on some other important smoothing functions, which also demonstrate that our al-gorithm is promising. AMS Mathematics Subject Classification : 65K05, 90C33.
Keywords
1. Introduction
Consider the following nonlinear complementarity problem (NCP): to find a vector
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is continuously differentiable with F := ( F 1 , F 1 , . . . , Fn ) T . The NCP has been studied extensively due to its many applications in operation research, engineering and economics(see, for example, [1 , 2] ).
For the NCPs, many solution methods, such as interior point methods [3 , 4] , smoothing methods [5 , 6 , 7] . In this paper, we are interested in smoothing Newton methods for solving NCP. This method is to reformulate NCP as a system of smoothing equations by using smoothing function, and to solve the equation at each iteration by Newton method. Smoothing function plays an important role in smoothing Newton algorithms. Up to now, many smoothing functions have been proposed: the Kanzow smoothing function [8] , Chen-Harker-Kanzow-Smale smoothing function [5] , Chen-Mangasarian smoothing function [9] , Huang-Han-Chen smoothing function [10] , and so on. Generally, the construction of a smoothing function is based on a so-called NCP-function: An NCP-function is a mapping
PPT Slide
Lager Image
having the property
PPT Slide
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Many NCP-functions have been studied. Among them, the Fischer-Burmeister function and the minimum function are the most prominent NCP-functions, which are defined respectively by
PPT Slide
Lager Image
By smoothing the symmetric perturbed Fischer-Burmeister function, Huang, Han, Xu and Zhang [11] proposed the following smoothing function:
PPT Slide
Lager Image
By smoothing the symmetric perturbed minimum function, Huang et. al. [10] proposed the following smoothing function:
PPT Slide
Lager Image
Recently, by combining the Fischer-Burmeister function and the minimum function, Liu and Wu [12] proposed the following function:
PPT Slide
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Motivated by [10 , 11 , 12] , we introduce in this paper the following smoothing function:
PPT Slide
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where θ is a given constant with θ ∈ [0,1]. It is easy to see that when θ = 1, ϕθ reduces to the smoothing function defined by (1.3); and when θ = 0, ϕθ reduces to smoothing function defined by (1.2). Thus, the class of smoothing functions defined by (4) contains the smoothing function (1.2) and (1.3) as special cases.
Motivated by the above mentioned work, by using the symmetric perturbed technique and the idea of convex combination, we propose a new class of smoothing functions. We also investigate a smoothing Newton method to solve the NCP based on a new class of smoothing functions. Our algorithm has the following nice properties: (a) Our algorithm needs only to solve one linear system of equations and perform one line search per iteration. (b) Here we give the boundedness of the level set and hence the iteration sequence is bounded and thus there exists at least one accumulation point. We do not need to assume the nonemptyness and boundedness of the solution set of NCP (1.1), although this assumption is widely used in the literature. (c) The function we use is a parametric class of smoothing functions containing some important smoothing complementarity functions as its special cases. We can adjust the two parameter to get better effect in practice. The numerical experiments implicate that the algorithm is efficient and promising.
The organization of this paper is as follows. In section 2, we recall some useful definitions and give some properties of new smoothing function. In section 3, we propose a smoothing Newton algorithm. Convergence results are analyzed in section 4. Some preliminary computational results are reported in section 5. Some words about notation are needed. All vectors are column vectors.
PPT Slide
Lager Image
denote the nonnegative and positive orthants of
PPT Slide
Lager Image
respectively. We define N = {1, 2, . . . , n }.
2. Preliminaries
In this section, we recall some useful definitions and give some properties of the new smoothing function defined by (4).
Definition 2.1. A matrix
PPT Slide
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is said to be a P 0 -matrix if all its principal minors are non-negative.
Definition 2.2. A function
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is said to be a P 0 -function if for all
PPT Slide
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there exists an index i 0 N such that
PPT Slide
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The following lemma gives some properties of the smoothing function ϕθ (·, ·, ·) defined by (4). Its proof is obviously.
Lemma 2.3. Let
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be defined by (4). Then,
(i) ϕθ (0, a, b ) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0.
(ii) ϕθ ( μ, a, b ) is continuously differentiable for all points in
PPT Slide
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different from (0, c, c ) for arbitrary
PPT Slide
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In particular, ϕθ ( μ, a, b ) is continuously differentiable for arbitrary ( μ, a, b ) ∈
PPT Slide
Lager Image
with μ ≠ 0.
(iii) ϕθ ( μ, a, b ) is semismooth on
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
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By (5) and Lemma 2.1, we known that solving NCP (1) is equivalent to solve H ( z ) = 0.
Define merit function
PPT Slide
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PPT Slide
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We also know that the NCP (1) is equivalent to the following equation:
PPT Slide
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For simplicity, we denote
PPT Slide
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Lemma 2.4. Let
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be defined by (5) and (6), respectively. Then:
(i) Փ θ is continuously differentiable at any
PPT Slide
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(ii) H is continuously differentiable at any
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with its Jacobian
PPT Slide
Lager Image
where
PPT Slide
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with
PPT Slide
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If F is a P 0 −function, then the matrix H ′( z ) is nonsingular on
PPT Slide
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Proof . It is easy to see that Փ θ is continuously differentiable at any
PPT Slide
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Next we prove (ii). It follows from (i) and F is continuously differentiable that H is continuously differentiable at any
PPT Slide
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From the definition of H ( z ) (5), it follows that (9) holds. For all i N ,
PPT Slide
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By the above equation, we have
PPT Slide
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Since
PPT Slide
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which together with (2.6), we have
PPT Slide
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Thus,
PPT Slide
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which imply that D 1 ( z ) and D 2 ( z ) are positive diagonal matrices for any
PPT Slide
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Since F is a P 0 -function, then F ′( x ) is a P 0 -matrix for any
PPT Slide
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by Lemma 5.4 in [13] . In view of the fact that D 2 ( z ) is a positive diagonal matrix, by a straightforward calculation we have that all principal minors of the matrix D 2 ( z ) F ′( x ) are nonnegative. By Definition 2.1, we know that the matrix D 2 ( z ) F ′( x ) is a P 0 -matrix. Hence, by Theorem 3.1 in [14] , the matrix D 1 ( z ) + D 2 ( z ) F ′( x ) is obviously nonsingular, which implies that H ′( z ) is nonsingular.
3. Algorithm
In this section we shall present a smoothing Newton method for NCP and prove that the proposed algorithm is well defined.
Algorithm 3.1. ( Smoothing Newton algorithm)
S0 Choose
PPT Slide
Lager Image
Take γ ∈ (0, 1) such that
PPT Slide
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Let
PPT Slide
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be an arbitrary vector,
PPT Slide
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S1 Termination criterion. If ∥ H ( zk )∥ = 0, stop.
S2 Compute
PPT Slide
Lager Image
PPT Slide
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where βk = β ( zk ) is defined by β ( z ) := γmin {1, h ( z )}.
S3 Let mk is the smallest nonnegative integer such that
PPT Slide
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Let λk := δmk .
S4 Set zk+1 = zk + λkΔzk and k := k + 1. Go to S1.
The following theorem proves that Algorithm 3.1 is well-defined and generates an infinite sequence. Define the set
PPT Slide
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Theorem 3.1. Suppose F is a continuously differentiable P 0 - function. Then, Algorithm 3.1 is well-defined and generates infinite sequence { zk = ( μk , xk )} with
PPT Slide
Lager Image
Proof . If μk > 0, since F is a continuously differentiable P 0 -function, then it follows from Lemma 2.2 that the matrix H ′( zk ) is nonsingular. Hence, step S2 is well-defined at the k −th iteration. By (11) we have
PPT Slide
Lager Image
which implies
PPT Slide
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where the second inequality follows from
PPT Slide
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Hence, by the first equation of (3.1), we can get
PPT Slide
Lager Image
From (2.1) and (2.4), we have
PPT Slide
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Let Rk ( α ) = h ( zk + α Δ zk )− h ( zk )− αh ′( zk zk . It is easy to see that R ( α ) = o ( α ). When
PPT Slide
Lager Image
PPT Slide
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Then by (3.1), (3.2), (3.4) and (3.5), we have
PPT Slide
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Since
PPT Slide
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For α sufficiently small, we can get
PPT Slide
Lager Image
this shows that step S3 is well-defined at the k -th iteration. Therefore, Algorithm 3.1 is well-defined and generates an infinite sequence
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Next, we will prove zk Ω for k ≥ 0. This can be obtained by inductive method. Firstly, it is evident from the choice of the starting point z 0 Ω . Secondly, suppose that zk Ω , then by (13) we have
PPT Slide
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then
PPT Slide
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4. Convergence of Algorithm 3.1
In this section, we discuss the global convergence and local superlinear convergence of Algorithm 3.1. We need the following Lemma 4.1 which can be founded in [15] .
Lemma 4.1. Let ε > 0 and the function
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be defined by
PPT Slide
Lager Image
Let
PPT Slide
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be any two sequences such that ak, bk → +∞ or ak → −∞ or bk → −∞. Then | ϕ ( ak, bk )| → +∞.
Lemma 4.2. Let
PPT Slide
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be defined by
PPT Slide
Lager Image
Assume that
PPT Slide
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be any two sequences such that ak, bk → +∞ or ak → −∞ or bk → −∞. Then
PPT Slide
Lager Image
Proof . (i) Suppose that ak → −∞. If { bk } is bounded, then the result holds obviously; else if bk → +∞, we have − ak > 0 and bk > 0 for all k sufficiently large, and hence,
PPT Slide
Lager Image
which, together with − ak → +∞, implies that
PPT Slide
Lager Image
(ii) For the case of bk → −∞. By using the symmetry of function
PPT Slide
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about ak , bk , we know the result holds.
(iii) Suppose that ak → +∞ and bk → +∞. Thus, for sufficiently large k ,
PPT Slide
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hence,
PPT Slide
Lager Image
By Lemma 4.1, we know that
PPT Slide
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Lemma 4.3. Let F be a continuous P 0 - function and Փ θ ( μ, x ) be defined by (6). For any μ > 0 and c > 0, define the level set
PPT Slide
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Then, for any 0 < μ 1 μ 2 and c > 0 , the set
PPT Slide
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is bounded.
Proof . Suppose, to the contrary, that Lμ ( c ) is unbounded. Then for some fixed c > 0, we can find a sequence {( μk , xk )} such that μ 1 μk μ 2 and ∥Փ θ ( μk , xk )∥ ≤ c , ∥ xk ∥ → ∞.
Since the sequence { xk } is unbounded, then the index set J := { i N :
PPT Slide
Lager Image
is unbounded } is nonempty. Without loss of generality, we can assume that
PPT Slide
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be defined by
PPT Slide
Lager Image
Then,
PPT Slide
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is bounded. Note that F is a P 0 -function, by Definition 2.2, we have
PPT Slide
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where j is one of the indices for which the max is attained, and j is assumed, without loss of generality, to be independent of k , we obtained
PPT Slide
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We consider the following two cases:
case 1:
PPT Slide
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In this case, since
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is bounded by the continuity of Fj , we deduce from Equation (4.3)
PPT Slide
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we have
PPT Slide
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By Lemma 4.2, we know that
PPT Slide
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case 2:
PPT Slide
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In this case, since
PPT Slide
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is bounded by the continuity of Fj , we deduce from Equation (4.3)
PPT Slide
Lager Image
for any k . Since μ 1 μk μ 2 , we have
PPT Slide
Lager Image
which, together with Lemma 4.2, gives
PPT Slide
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In either case, we obtained ∥Փ θ ( μk , xk )∥ → +∞, which contradicts with ∥Փ θ ( μk , xk )∥ ≤ c . This completes the proof.
Corollary 4.3 Suppose that F is a P 0 -function and μ > 0. Then the function ∥Փ θ ( μ , x )∥ is coercive, i.e., lim x∥→∞ ∥Փ θ ( μ , x )∥ = +∞.
Theorem 4.4. Suppose F is a continuously differentiable P 0 - function, and the sequence { zk = ( μk , xk )} is generated by Algorithm 3.1. Then the sequence { zk } is bounded and any accumulation point z = ( μ , x ) of the sequence { zk } is a solution of H ( zk ) = 0.
Proof . Since h ( zk ) is monotonically decreasing and bounded from below by zero, it then follows that the sequence ∥Փ θ ( zk )∥ is bounded. By Corollary 4.3, we immediately obtain { xk } is bounded. Note that the boundedness of { h ( zk )} implies the boundedness of μk . So { zk } is bounded. Without loss of generality, suppose zk z . Then h ( zk ) → h , β ( zk ) → β . If h ( zk ) = 0, we obtain the desired result. Now, we prove h = 0 by contradiction. In fact, if h ≠ 0, then h > 0, then β = γ min{1, h } > 0, and
PPT Slide
Lager Image
It follows from Lemma 2.2 that H ′( z ) is nonsingular. By the continuity of H ′( z ), there exists a closed neighborhood N ( z ) of z such that for any z N ( z ), we have
PPT Slide
Lager Image
is invertible. So, for all sufficiently large k , zk N ( z ) and H ′( zk ) is invertible. Let
PPT Slide
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be the unique solution of the following system:
PPT Slide
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It follows from the continuity of H and the definition of β (.) that { μk } and { βk } converge to μ and β , respectively. That together with (3.2), implies that
PPT Slide
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Thus, for sufficiently large k , the stepsize
PPT Slide
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does not satisfy (3.2), then
PPT Slide
Lager Image
which implies that
PPT Slide
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PPT Slide
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Taking limits on both sides of the inequalities (4.5), from (4.6) we have
PPT Slide
Lager Image
This indicates that
PPT Slide
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we have σ ≥ 1, which contradicts
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is a solution of H ( μ, x ) = 0.
Theorem 4.5. Suppose that F is a continuously differentiable P 0 - function. Let z be an accumulation point of the iteration sequence { zk } generated by Algorithm 3.1. If all V ∂H ( z ) are nonsingular, then:
(1) λk ≡ 1, for all zk sufficiently close to z;
(2) the whole sequence { zk } converges to z;
(3) z k+1 z ∥ = o (∥ zk z ∥)( or z k+1 z ∥ = O (∥ zk z 2 ) if F′ is Lipschitz continuous on n).
Proof . The proof is similar to the one given in [16] , Theorem 3.2.
5. Numerical experiments
In this section, we report some numerical results of Algorithm 3.1. All experiments are done using a PC with CPU of 1.6 GHz and RAM of 512 MB, and all codes are finished in MATLAB 7.5. Throughout our computational experiments, the parameters used in the algorithm are chosen as
PPT Slide
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In our implementation, we use ∥ H ( zk )∥ ≤ 10 -6 as the stopping rule.
Example 5.1. Kojima-Shindo Problem. This test problem was used by Pang and Gabriel [17] , Mangasarian and Solodov [18] , Kanzow [19] , and Jiang and Qi [20] with four variables. Let
PPT Slide
Lager Image
Table 1 gives the results for this example with starting points a 1 = (0, 0, 0, 1) T , a 2 = (1,−2, 1,−2) T , a 3 = (1, 2, 6, 8) T .
Numerical results for Examples 5.1 to 5.4
PPT Slide
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Numerical results for Examples 5.1 to 5.4
Example 5.2. Josephy Problem. This test problem was used by Dirkse and Ferris [22] with four variables. Let
PPT Slide
Lager Image
Table 1 gives the results for this example with starting points a 1 = (2,−2,−2,−2) T , a 2 = (2, 3, 4, 6) T , a 3 = (0, 2, 0, 6) T .
Example 5.3. Mathiesen Problem. This test problem was used by Pang and Gabriel [17] with four variables, which was also tested by Kanzow [19] . Let
PPT Slide
Lager Image
where α = 0.75, b 2 = 1, b 3 = 2. Table 1 gives the results for this example with starting points a 1 = (0.5, 0.5, 0.5, 2) T , a 2 = (2,−2,−2,−2) T , a 3 = (0,−2,−2, 0) T .
Example 5.4. HS 34 Problem. This test problem was from the book of Hock and Schittkowski [21] : Their Karush-Kuhn-Tucker (KKT) optimality conditions lead to complementarity problems of dimensions 8. Let
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Table 1 gives the results with starting points a 1 = (−1,−1,−1, 1, 1, 1, 1, 1) T , a 2 = (0, 0, 0, 1, 1, 1, 1, 1) T , a 3 = (1, 1, 1,−10,−10,−10,−10,−10) T .
In Table 1 , IT denotes the numbers of iteration; NF denotes the numbers of function value’s evaluation; CPU denotes the CPU time for solving the underlying problem in second; and − denotes the algorithm fails to find the optimizer in the sense that the iteration numbers are larger than 1000.
Table 1 shows that not all the best numerical results occur in the case of θ = 0(in this case, the smoothing function is proposed by Huang et. al. in [11] ) or θ = 1 (in this case, the smoothing function is proposed by Huang et. al. in [10] ). These demonstrate that the new smoothing function introduced in this paper is worth investigating. The Figures 1 and 2 below plot the corresponding convergence of merit function h ( zk ) versus the iteration number. From the two figures, when θ = 0.5 and θ = 0.75, h ( zk ) has a faster decrease than θ = 0 and θ = 1. These also demonstrate that the new smoothing function introduced in this paper is worth investigating. Numerical experiments also demonstrate the feasibility and efficiency of the new algorithm.This new proposed class of complementarity functions have great advantage because we can adjust the parameter θ to obtain an optimal solution to NCP.
PPT Slide
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Convergence behavior of Example 5.3 with the initial point a1
PPT Slide
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Convergence behavior of Example 5.3 with the initial point a3
BIO
Jianguang Zhu received his Ph.D. degree in applied mathematics from Xidian University, China, in 2011. Since 2011, he has been a lecture with Shandong University of Science and Technology, China. His research interests include optimization theory, and algorithm & applications in image processing.
School of Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China.
e-mail:jgzhu980@yahoo.com.cn
Binbin Hao received her Ph.D. degree in appliedmathematics from Xidian University, China, in 2009. From 2008 to 2009, she was a research assistant with the Department of Computing, The Hong Kong Polytechnic University, Hong Kong. Since 2010, she has been a lecture with China University of Petroleum, China. Her research interests include inverse problems in image processing, sparse signal representation, and super resolution image reconstruction.
School of Science, China University of Petroleum, Qingdao 266580, P. R. China.
e-mail: bbhao981@yahoo.com.cn
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