• 沒有找到結果。

# New smoothing functions for solving a system of equalities and inequalities

N/A
N/A
Protected

Share "New smoothing functions for solving a system of equalities and inequalities"

Copied!
27
0
0

(1)

to appear in Paciﬁc Journal of Optimization, January, 2016

### New smoothing functions for solving a system of equalities andinequalities

Jein-Shan Chen 1 Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

Chun-Hsu Ko

Department of Electrical Engineering I-Shou University

Kaohsiung 840, Taiwan E-mail: chko@isu.edu.tw

Yan-Di Liu

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: 60140026S@ntnu.edu.tw

Sheng-Pen Wang 2

Department of Industrial and Business Management Chang Gung University

Taoyuan 333, Taiwan E-mail: wangsp@mail.cgu.edu.tw

September 22, 2014 (revised on November 20, 2014)

Abstract In this paper, we propose a family of new smoothing functions for solving a system of equalities and inequalities, which is a generalization of . We then investigate an algorithm based on a new reformation bH with less dimensionality and show, as in , that it is globally and locally convergent under suitable assumptions. Numerical evidence

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce. The author’s work is supported by Ministry of Science and Technology, Taiwan.

2Corresponding author.

(2)

shows the better performance of the algorithm in the sense that some unsolved examples in  can be solved by our proposed method. Moreover, the involved parameters in the family of new smoothing functions does not have inﬂuence in the algorithm, which is a new discovery to the literature.

Keywords. Smoothing function, System of equations and inequalities, Convergence

### 1Introduction and Motivation

The target problem of this paper is the following system of equalities and inequalities:

{ fI(x) ≤ 0

fE(x) = 0 (1)

where I = {1, 2, · · · , m} and E = {m + 1, m + 2, · · · , n}. In other words, the function fI : IRn→ IRm is given by

fI(x) =



 f1(x) f2(x)

... fm(x)





where fi : IRn → IR for i = {1, 2, · · · , m}; and the function fE : IRn → IRn−m is given by

fE(x) =





fm+1(x) fm+2(x)

... fn(x)





where fj : IRn → IR for j = {m + 1, m + 2, · · · , n}. For simplicity, throughout this paper, we denote f : IRn → IRn as

f (x) :=

[ fI(x) fE(x)

]

=













f1(x) f2(x)

... fm(x) fm+1(x) fm+2(x)

... fn(x)













and assume that f is continuously diﬀerentiable. When E is empty set, the system (1) reduces to a system of inequalities; whereas it reduces to a system of equations when I

(3)

is empty.

Problems in form of (1) arise in real applications, including data analysis, computer- aided design problems, image reconstructions, and set separation problems, etc.. Many optimization methods have been proposed for solving the system (1), for instance, non- interior continuation method , smoothing-type algorithm [6, 11], Newton algorithm , and iteration methods [4, 7, 8, 10]. In this paper, we consider the similar smoothing- type algorithm studied in [6, 11] for solving the system (1). In particular, we propose a family of smoothing functions, investigate its properties, and report numerical perfor- mance of an algorithm in which this family of new smoothing functions is involved.

As seen in [6, 11], the main idea of smoothing-type algorithm for solving the system (1) is to reformulate system (1) as a system of smoothing equations via projection function, More speciﬁcally, for any x = (x1, x2,· · · , xn)∈ IRn, one deﬁnes

(x)+:=



max{0, x1} ... max{0, xn}

 .

Then, the system (1) is equivalent to the following system of equations:

{ (fI(x))+ = 0

fE(x) = 0. (2)

Note that the function (fI(x))+ in the reformulation (2) is nonsmooth, the classical Newton methods cannot be directly applied to solve (2). To conquer this, a smooth- ing algorithm was considered in [6, 11], in which the following smoothing function was employed:

ϕ(µ, t) =





t if t≥ µ,

(t+µ)2

if −µ < t < µ, 0 if t≤ −µ,

(3)

where µ > 0.

In this paper, we propose a family of new smoothing functions, which include the function ϕ(µ, t) given as in (3) as a special case, for solving the reformulation (2). More speciﬁcally, we consider the family of smoothing functions as below:

ϕp(µ, t) =





t if t≥ p−1µ ,

µ p−1

[(p−1)(t+µ)

]p

if −µ < t < p−1µ ,

0 if t≤ −µ,

(4)

where µ > 0 and p≥ 2. Note that ϕp reduces to the smoothing function studied in 

when p = 2. The graphs of ϕp with diﬀerent values of p and various µ are depicted as in Figures 1-3.

(4)

Proposition 1.1. Let ϕp be deﬁned as in (4). For any (µ, t)∈ IR++× IR, we have (a) ϕp(., .) is continuously diﬀerentiable at any (µ, t)∈ IR++× IR.

(b) ϕp(0, t) = (t)+.

(c) ∂ϕp∂t(µ,t) ≥ 0 for any (µ, t) ∈ IR++× IR.

(d) limp→∞ϕp(µ, t) → (t)+.

Proof. (a) First, we calculate ∂ϕp∂t(µ,t) and ∂ϕp∂µ(µ,t) as below:

∂ϕp(µ, t)

∂t =





1 if t≥ p−1µ ,

[(p−1)(t+µ)

]p−1

if −µ < t < p−1µ ,

0 if t≤ −µ,

∂ϕp(µ, t)

∂µ =





0 if t p−1µ ,

[(p−1)(t+µ)

]p−1(t+µ−pt)

if −µ < t < p−1µ ,

0 if t ≤ −µ,

Then, we see that ∂ϕp∂t(µ,t) ∈ C1 because

lim

tp−1µ

∂ϕp(µ, t)

∂t = lim

tp−1µ

[(p− 1)(p−1µ + µ)

]p−1

= 1,

t→−µlim

∂ϕp(µ, t)

∂t = lim

t→−µ

[(p− 1)(−µ + µ)

]p−1

= 0.

and ∂ϕp∂µ(µ,t) ∈ C1 since

lim

tp−1µ

∂ϕp(µ, t)

∂µ = lim

tpµ−1

[(p− 1)(p−1µ + µ)

]p−1

(p−1µ + µ− pp−1µ )

= 0,

t→−µlim

∂ϕp(µ, t)

∂µ = lim

t→−µ

[(p− 1)(−µ + µ)

]p−1 (−µ + µ − p(−µ))

= 0.

The above veriﬁcations imply that ϕp(., .) is continuously diﬀerentiable.

(b) From the deﬁnition of ϕp(µ, t), it is clear that

ϕp(0, t) =

{ t if t≥ 0

0 if t≤ 0 = (t)+ which is the desired result.

(5)

(c) When −µ < t < p−1µ , we have t + µ > 0. Hence, from the expression of ∂ϕp∂t(µ,t), it is obvious that

[(p−1)(t+µ)

]p−1

≥ 0, which says ∂ϕp∂t(µ,t) ≥ 0.

(d) Part(d) is clear from the deﬁnition. 2

The properties of ϕp in Proposition 1.1 can be veriﬁed via the graphs. In particular, in Figures 1-2, we see that when µ → 0, ϕp(µ, t) goes to (t)+ which veriﬁes Proposition 1.1(b).

−2.50 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

0.5 1 1.5 2 2.5

t φ p(µ,t)

p=2, µ=0.1 p=2, µ=0.5 p=2, µ=1 p=2, µ=2

Figure 1: Graphs of ϕp(µ, t) with p = 2 and µ = 0.1, 0.5, 1, 2.

Figure 3 says that for ﬁxed µ > 0, ϕp(µ, t) approaches to (t)+ as p → ∞. This also veriﬁes Proposition 1.1(d).

Next, we will form another reformulation for problem (1). To this end, we deﬁne

F (z) :=

fI(x)− s fE(x) Φp(µ, s)

 with Φp(µ, s) :=



ϕp(µ, s1) ... ϕp(µ, sm)

 and z = (µ, x, s) (5)

where Φp is a mapping from IR1+m → IRm. Then, in light of Proposition 1.1(b), we see that

F (z) = 0 and µ = 0 ⇐⇒ s = fI(x), s+= 0, fE(x) = 0.

(6)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

t φ p(µ,t)

p=10, µ=0.1 p=10, µ=0.5 p=10, µ=1 p=10, µ=2

Figure 2: Graphs of ϕp(µ, t) with p = 10 and µ = 0.1, 0.5, 1, 2.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0

0.05 0.1 0.15 0.2 0.25

t φ p(µ,t)

p=2, µ=0.2 p=3, µ=0.2 p=10, µ=0.2 p=20, µ=0.2

Figure 3: Graphs of ϕp(µ, t) with p = 2, 3, 10, 20 and µ = 0.2.

(7)

This, together with Proposition 1.1(a), indicates that one can solve system (1) by ap- plying Newton-type methods to solve F (z) = 0 by letting µ ↓ 0. Furthermore, by introducing an extra parameter p, we deﬁne a function H : IR1+n+m → IR1+n+m by

H(z) :=



µ

fI(x)− s + µxI

fE(x) + µxE

Φp(µ, s) + µs



 (6)

where xI = (x1, x2,· · · , xm), xE = (xm+1, xm+2,· · · , xn), s ∈ IRm, x := (xI, xE) ∈ IRn and functions ϕp and Φp are deﬁned as in(4) and (5), respectively. Thereby, it is obvious that if H(z) = 0, then µ = 0 and x solves the system (1). It is not diﬃcult to see that, for any z ∈ IR++× IRn× IRm, the function H is continuously diﬀerentiable. Let H denote the Jacobian of the function H. Then, for any z ∈ IR++× IRn× IRm, we have

H(z) =

1 O1×n O1×m

x1

... xm

m×1

A

−1 · · · 0 ... . .. ... 0 · · · −1

m×m

xm+1

... xn

(n−m)×1

B

0 · · · 0 ... . .. ... 0 · · · 0

(n−m)×m

s1+∂µ ϕ(µ, s1) ...

sm+∂µ ϕ(µ, sm)

m×1

0 · · · 0 ... 0 ... 0 · · · 0

m×n

∂sϕ(µ, s1) + µ · · · 0 ... . .. ... 0 · · · ∂sϕ(µ, sm) + µ

m×m

where

A =



∂f1(x1)

∂x1 + µ · · · 0

... . .. ... 0m×(n−m) 0 · · · ∂fm∂x(xmm) + µ



m×n

and

B =



∂fm+1(xm+1)

∂xm+1 + µ · · · 0

0(n−m)×m ... . .. ...

0 · · · ∂f∂xn(xnn) + µ



(n−m)×n

With the above, we can simplify the matrix H(z) as

H(z) =



1 0n 0m

xI fI(x) + µU −Im

xE fE (x) + µV 0(n−m)×m s + Φµ(µ, s) 0m×n Φs(µ, s) + µIm



 (7)

where

U := [Im 0m×(n−m)], V := [0(n−m)×m In−m],

(8)

s + Φµ(µ, s) =



s1+ ∂µ ϕ(µ, s1) ...

sm+ ∂µ ϕ(µ, sm)



m×1

,

Φs(µ, s) + µIm =



∂sϕ(µ, s1) + µ · · · 0

... . .. ...

0 · · · ∂sϕ(µ, sm) + µ



m×m

.

Here, we use 0I to denote the I-dimensional zero vector and 0l×q to denote the l × q zero matrix for any positive integers l and q. Thus, we might apply some Newton-type methods to solve the system of smooth equations H(z) = 0 at each iteration by letting µ > 0 and H(z) → 0 so that a solution of (1) can be found. This is the main idea of smoothing approach for solving system (1).

Alternatively, one may have another smoothing reformulation for system (1) without introducing the extra variable s. More speciﬁcally, we can deﬁne bH : IR1+n → IR1+n as

H(µ, x) :=b

µ

fE(x) + µxE Φp(µ, fI(x)) + µxI

 (8)

The Jacobian of bH(µ, x) is similar to H(z) and indeed is a bit tedious, so we omit its presentation here. The reformulation of bH(µ, x) = 0 has less dimension than H(z) = 0, whereas the expression of bH(µ, x) is more tedious than H(z). Both smoothing ap- proaches can lead to the solution to system (1). The numerical results based on H(z) = 0 and bH(µ, x) = 0 are compared in this paper. Moreover, we also investigate how the pa- rameter p aﬀect the numerical performance when diﬀerent ϕp is employed. Proposing the new family of smoothing functions as well as the above two aspects of numerical points are the main motivation and contribution of this paper.

### 2A smoothing-type algorithm

In this section, we consider a non-monotone smoothing-type algorithm whose similar framework has been discussed in [6, 11]. In particular, we correct a ﬂaw in Step 5 in  and show that only this modiﬁcation can really make the algorithm well-deﬁned.

Moreover, for bH(µ, x), a new reformulation of H(z) with lower dimensionality, we will use the function ψ(·) := ∥H(z)∥2 or ψ(·) := ∥ bH(µ, x)∥2 alternatively. Below are the details of the algorithm.

Algorithm 2.1. (A Nonmonotone Smoothing-Type Algorithm)

(9)

Step 0 Choose δ ∈ (0, 1), σ ∈ (0, 1/2), β > 0. Take τ ∈ (0, 1) such that τβ < 1. Let µ0 = β and (x0, s0) ∈ IRn+m be an arbitrary vector. Set z0 := (µ0, x0, s0). Take e0 := (1, 0,· · · , 0) ∈ IR1+n+m, R0 :=∥H(z0)2 = ψ(z0) and Q0 = 1.

Choose ηmin and ηmax such that 0≤ ηmin ≤ ηmax < 1. Set θ(z0) := τ min{1, ψ(z0)} and k := 0.

Step 1 If ∥H(zk)∥ = 0, stop.

Step 2 Compute △zk := (△µk,△xk,△sk)∈ IR × IRn× IRm by using

H△zk =−H(zk) + βθ(zk)e0 (9) Step 3 Let αk be the maximum of the values 1, δ, δ2,· · · such that

ψ(zk+ αk△zk)≤ [1 − 2σ(1 − τβ)αk] Rk (10) Step 4 Set zk+1 := zk+ αk△zk. If ∥H(zk+1)∥ = 0, stop.

Step 5 Choose ηk∈ [ηmin, ηmax]. Set

Qk+1 := ηkQk+ 1 θ(zk+1) := min{

τ, τ ψ(zk+1), θ(zk)}

(11) Rk+1 := kQkRk+ ψ(zk+1))

Qk+1 and k := k + 1. Go to Step 2

In Algorithm 2.1, a nonmonotone line search technique is adopted. It is easy to see that Rk+1 is a convex combination of Rk and ψ(zk+1). Since R0 = ψ(z0), it follows that Rk is a convex combination of the function values ψ(z0), ψ(z1),· · · , ψ(zk). The choice of ηk controls the degree of nonmonotonicity. If ηk = 0 for every k, then the line search is the usual monotone Armijo line search. The scheme of Algorithm 2.1 is not exactly the same as the one in . In particular, θ(zk+1) := min{

τ, τ ψ(zk+1), θ(zk)}

which is diﬀerent from θ(zk+1) := min{

τ, τ ψ(zk), θ(zk)}

given in . Only this modiﬁcation can really make the algorithm well-deﬁned as shown in the following Theorem 2.1. For convenience, we denote

f(x) :=

[ fI(x) fE (x)

]

and make the following assumption.

Assumption 2.1. f(x) + µIn is invertible for any x ∈ IRn and µ∈ IR++.

Some basic properties of Algorithm 2.1 are stated in the following lemma. Since the proof arguments are almost the same as those in , they are thus omitted.

(10)

Lemma 2.1. Let the sequence {Rk} and { zk}

be generated by Algorithm 2.1. Then, the following hold.

(a) The sequence {Rk} is monotonically decreasing.

(b) The function ψ(zk)≤ Rk for all k ∈ J .

(c) The sequence θ(zk) is monotonically decreasing.

(d) βθ(zk)≤ µk for all k ∈ J .

(e) µk > 0 for all k∈ J and the sequence {µk} is monotonically decreasing.

Lemma 2.2. Suppose A∈ IRn×nwhich is partitioned as A =

[ A11 A12 A21 A22

]

where A11and A22 are square matrices. If A12 or A21 is zero matrix, then det(A) = det(A11)· det(A22).

Proof. This a well known result in matrix analysis, which is a special case of Fischer’s ineqiality [1, 5]. Please refer to [9, Theorem 7.3] for a proof. 2

Theorem 2.1. Suppose that f is a continuously diﬀerentiable function and Assumption 2.1 is satisﬁed. Then Algorithm 2.1 is well deﬁned.

Proof. Applying Lemmas 2.1-2.2 and mimicking the arguments as in , it is easy to achieve the desired result. However, we point it out again that θ(zk+1) in step 5 is diﬀerent from the one in . Only this modiﬁcation can really make the algorithm well-deﬁned. 2

### 3Convergence analysis

In this section, we analyze the convergence of the algorithm proposed in previous section.

To this end, the following assumption is needed which was introduced in .

Assumption 3.1. For an arbitrary sequence {(µk, xk)} with lim

k−→∞∥x∥ = +∞ and the sequence k} ⊂ IR+ bounded, then either

(i) there is at least an index i0 such that lim sup

k−→∞ {fi0(xk) + µkxki0} = +∞; or (ii) there is at least an index i0 such that lim sup

k−→∞ k(fi0(xk) + µkxki0)} = −∞.

(11)

It can be seen that many functions satisfy Assumption 3.1, see . The global con- vergence of Algorithm 2.1 is stated as follows. In fact, with Proposition 1.1, the main idea for the proof is almost the same as that in [11, Theorem 4.1], only a few technical parts are diﬀerent. Thus, we omit the details.

Theorem 3.1. Suppose that f is a continuously diﬀerentiable function and Assumptions 2.1 and 3.1 are satisﬁed. Then, the sequence{zk} generated by Algorithm 2.1 is bounded.

Moreover, any accumulation point of xk is a solution to (1).

Next, we analyse the convergence rate for Algorithm 2.1. Before presenting the re- sult, we introduce some concepts that will used in the subsequent analysis as well as a technical lemma.

A locally Lipschitz function F : IRn → IRm, which has the generalized Jacobian

∂F (x), is said to be semismooth (or strongly semismooth) at x∈ IRn if F is directionally diﬀerentiable at x and

F (x + h)− F (x) − V h = o(∥h∥) (or = O(∥h∥2)

holds for any V ∈ ∂F (x + h). It is well known that convex functions, smooth functions, and piecewise linear functions are examples of semismooth functions. The composition of (strongly) semismooth functions is still a (strongly) semismooth function. It can be veriﬁed that the function ϕp deﬁned by (4) is strongly semismooth on IR2. Thus, f being continuously diﬀerentiable implies that the function H deﬁned by (6) and bH deﬁned by (8)are semismooth (or strongly semismooth if f is Lipschitz continuous on IRn.

Lemma 3.1. For any α, β ∈ IR++, α = O(β) represents that αβ is uniformly bounded, and α = o(β) denotes αβ → 0 as β → 0. Then, we have

(a) O(β)± O(β) = O(β);

(b) o(β)± o(β) = o(β);

(c) If c̸= 0 then O(cβ) = O(β), o(cβ) = o(β);

(d) O(o(β)) = o(β), o(O(β)) = o(β);

(e) O(β1)O(β2) = O(β1β2), O(β1)o(β2) = o(β1β2), o(β1)O(β2) = o(β1β2).

(f ) If α = O(β1) and β1 = o(β2), then α = o(β2).

Proof. For parts (a)-(e), please refer to  for a proof. Part (f) can be veriﬁed straightforwardly. 2

With Proposition 1.1 and Lemma 3.1, mimicking the arguments as in [11, Theorem 5.1] gives the following theorem.

(12)

Theorem 3.2. Suppose that f is a continuously diﬀerentiable function and Assumptions 2.1 and 3.1 are satisﬁed. Let z = (µ, x, s) be an accumulation point of{zk} generated by Algorithm 2.1. If all V ∈ ∂H(z) are nonsingular, then the following hold.

(a) αk ≡ 1 for all zk suﬃciently close to z; (b) the whole sequence {zk} converges to z;

(c) ∥zk+1− z∥ = o(∥zk− z∥) (or ∥zk+1− z∥ = O(∥zk− z2)) provided f is Lipschitz continuous on IRn );

(d) µk+1 = o(µk) (or µk+1 = O(µ2k) if f is Lipschitz continuous on IRn).

### 4Numerical Results

In this section, we present our test problems and report numerical results. In this paper, the function f is assumed to be a mapping from IRn to IRn, which means the dimension of x is exactly the same as the total number of inequalities and equalities. In reality, this may not be the case. In other words, there may have a system like this:

{ fI(x) ≤ 0, I = {1, 2, · · · , m}

fE(x) = 0, E ={m, m + 1, · · · , l} (12) This says f could be a mapping from IRn to IRl. When l ̸= n, the scheme in Algorithm 2.1 cannot be applied to the system (12) because the dimension of x is not equal to the total number of inequalities and equalities. To make system (12) solvable under the proposed algorithm, as remarked in [11, Sec. 6], some additional inequality or variable needs to be added. For example,

(i) if l < n, we may add a trivial inequality like

n i=1

x2i ≤ M

where M is suﬃciently large, into system (12) so that Algorithm 2.1 can be applied.

(ii) if l > n and m≥ 1, we may add a variable xn+1 into the inequalities so that fi(x)≤ 0 → fi(x) + x2n+1 ≤ 0.

(iii) if l > n and m = 0, we may add a trivial inequality like

n+2 i=1

x2i ≤ M

where M is suﬃciently large, into system (12) so that Algorithm 2.1 can be applied.

(13)

In real implementation, the H(z) given as in (6) is replaced by

H(z) :=



µ

fI(x)− s + cµxI

fE(x) + cµxE Φp(µ, s) + cµs



 (13)

where c is a given constant. Likewise, the bH(µ, x) given as in (8) is replaced by

H(µ, x) :=b

µ

fE(x) + cµxE Φp(µ, fI(x)) + cµxI

 . (14)

Adding such a constant c is helpful when coding the algorithm because µ approaches to zero eventually. The theoretical results will not be aﬀected in any case. In practice, in order to obtain an interior solution x for inequalities (fI(x) < 0), the following system

{ fI(x) + εe≤ 0 fE(x) = 0

is considered, where ε is a small number and e is the vector of all ones. Now, we list the test problems which are employed from [6, 11].

Example 4.1. Consider f (x) =

[ f1(x) f2(x)

]

with x∈ IR2 where

f1(x) = x21+ x22− 1 + ε ≤ 0,

f2(x) = −x21− x22+ (0.999)2+ ε≤ 0.

Example 4.2. Consider f (x) =







f1(x) f2(x) f3(x) f4(x) f5(x) f6(x)







with x∈ IR2 where

f1(x) = sin(x1) + ε≤ 0, f2(x) = − cos(x2) + ε≤ 0, f3(x) = x1− 3π + ε ≤ 0, f4(x) = x2 π

2 − 2 + ε ≤ 0, f5(x) = −x1− π + ε ≤ 0, f6(x) = −x2 π

2 + ε≤ 0.

(14)

Example 4.3. Consider f (x) =

[ f1(x) f2(x)

]

with x∈ IR2 where f1(x) = sin(x1) + ε≤ 0, f2(x) = − cos(x2) + ε≤ 0.

Example 4.4. Consider f (x) =





f1(x) f2(x) f3(x) f4(x) f5(x)





with x∈ IR5 where

f1(x) = x1+ x3− 1.6 + ε ≤ 0, f2(x) = 1.333x2+ x4− 3 + ε ≤ 0, f3(x) = −x3 − x4 + x5+ ε≤ 0, f4(x) = x21+ x23− 1.25 = 0, f5(x) = x1.52 + 1.5x4− 3 = 0.

Example 4.5. Consider f (x) =

f1(x) f2(x) f3(x)

 with x ∈ IR3 where

f1(x) = x1+ x2e0.8x3 + e1.6+ ε≤ 0, f2(x) = x21+ x22+ x23− 5.2675 + ε ≤ 0, f3(x) = x1+ x2+ x3− 0.2605 = 0.

Example 4.6. Consider f (x) =

f1(x) f2(x) f3(x)

 with x ∈ IR2 where

f1(x) = 0.8− ex1+x2 + ε≤ 0, f2(x) = 1.21ex1 + ex2 − 2.2 = 0, f3(x) = x21+ x22 + x2− 0.1135 = 0.

Example 4.7. Consider f (x) =

[ f1(x) f2(x)

]

with x∈ IR2 where f1(x) = x1− 0.7 sin(x1)− 0.2 cos(x2) = 0 f2(x) = x2− 0.7 cos(x1) + 0.2 sin(x2) = 0

Moreover, in light of the aforementioned discussions, there have corresponding mod- iﬁed problems for Example 4.2’, Example 4.6’, and Example 4.7’, which are stated as

(15)

below. The other examples are kept unchanged. In other words, Example 4.1 and Exam- ple 4.1’ are the same , so are Example 4.3 and Example 4.3’, Example 4.4 and Example 4.4’, Example 4.5 and Example 4.5’.

Example 4.2’. Consider f (x) =







f1(x) f2(x) f3(x) f4(x) f5(x) f6(x)







with x∈ IR6 where

f1(x) = sin(x1) + ε≤ 0 f2(x) = − cos(x2) + ε≤ 0 f3(x) = x1− 3π + x23+ ε≤ 0 f4(x) = x2 π

2 − 2 + x24+ ε≤ 0 f5(x) = −x1− π + x25+ ε≤ 0 f6(x) = −x2 π

2 + x26+ ε≤ 0

Example 4.6’. Consider f (x) =

f1(x) f2(x) f3(x)

 with x ∈ IR3 where

f1(x) = 0.8− ex1+x2 + x23+ ε≤ 0, f2(x) = 1.21ex1 + ex2 − 2.2 = 0, f3(x) = x21+ x22 + x2− 0.1135 = 0.

Example 4.7’. Consider f (x) =

f1(x) f2(x) f3(x)

 with x ∈ IR3 where

f1(x) = x21+ x22+ x23− 10000 + ε ≤ 0, f2(x) = x1− 0.7 sin(x1)− 0.2 cos(x2) = 0, f3(x) = x2− 0.7 cos(x1) + 0.2 sin(x2) = 0.

The numerical implementations are coded in Matlab. In the numerical reports, x0 is the stating point, NI is the total number of iterations, NF denotes the number of function evaluations for H(zk) or bH(µk, xk), and SOL means the solution obtained from the algorithm. The parameters used in the algorithm are set as

ε = 0.00001, δ = 0.3, σ = 0.001, β = 1.0, µ0 = 1.0, Q0 = 1.0.

(16)

Table 1: Numerical performance when p = 2, stop criterion: 0.001.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 8 12 (-0.6188, 0.7853)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 Fail Fail Fail

Ex 4.3 (0, 0) 0.5 0.2 0.01 3 4 (-0.01516, 0.7207)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.5557, 1.324, 0.9703, 0.984, 1.156) Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 5 6 (-0.8301, -0.8662, 1.957) Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 7 8 (0.2743, -0.4975, 1.5e+006) Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 9 15 (0.5268, 0.5084, -100)

Ex 4.1’ (0, 5) 100 0.006 0.01 8 12 (-0.6188, 0.7853)

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 6 9 (-0.009654,1.428,2.846,1.28,1.639,1.666)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 3 4 (-0.01516, 0.7207)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.5557, 1.324, 0.9703, 0.984, 1.156) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 5 6 (-0.8301, -0.8662, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 5 7 (-0.09533, 0.09533, 0.3321) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 9 15 (0.5268, 0.5084, -100)

Note: Based on H(z) = 0 given as in (13).

In Table 1 and Table 2, we adapt the same x0, c, τ , η used as in  for p = 2.

Basically, in Table 1 and Table 2, the bottom half data for the modiﬁed problems are the same as those in , respectively. Below are our numerical observations and conclusions.

• From Table 1 and Table 2, we see that, when employing formulation H(z) = 0, solving the modiﬁed problems is more successful than solving the original problems.

• Table 3 indicates that the numerical results are the same for original problems and modiﬁed problems, when bH(µ, x) = 0 is employed. Hence, in Tables 4-11, we focus on the modiﬁed problems when formulation H(z) = 0 is employed, whereas we only test original problems whenever the implementations are based on bH(µ, x) = 0.

• From Table 5 (p = 2), we see that the algorithm based on bH(µ, x) = 0 can solve more problems than the one in  does. In view of the lower dimensionality of H(µ, x) = 0 and this performance, we can conﬁrm the merit of this new reformu-b lation.

• In Table 4 and Table 5, the initial point and other parameters are the same as those in . In Tables 6-7, we ﬁx the initial point x0 for all test problems. In Table 8 and Table 9, even x0, c, τ and η are all ﬁxed for all test problems. In Table 10 and Table 11, x0 is ﬁxed for all test problems and parts of c, τ and η are ﬁxed.

In general, we observe that the numerical performance based on the formulation H(µ, x) = 0 is better than the one based on H(z) = 0.b

(17)

Table 2: Numerical performance when p = 2, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 Fail Fail Fail

Ex 4.3 (0, 0) 0.5 0.2 0.01 4 5 (-0.01516, 0.7206)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 5 6 (0.5563, 1.326, 0.9698, 0.9822, 1.155) Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8299, -0.8663, 1.957) Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 7 8 (0.2743, -0.4975, 1.5e+006) Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

Ex 4.1’ (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.009276,1.429, 2.846,1.279,1.64, 1.667)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.01516, 0.7206)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 5 6 (0.5563, 1.326, 0.9698, 0.9822, 1.155) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8299, -0.8663, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 8 (-0.09533, 0.09533, 0.332) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

Note: Based on H(z) = 0 given as in (13).

• Moreover, the changing of parameter p seems have no inﬂuence on the numerical performance no matter bH(µ, x) = 0 or H(z) = 0 is adapted. This indicates that the smoothing approach may not be aﬀected when p is perturbed. This phenomenon is diﬀerent from the one for other approaches observed in [2, 3] and is a new discovery to the literature. We guess that the main reason comes from µ dominating the algorithm in the smoothing approach even various p is perturbed. This conjecture still needs further veriﬁcation and investigation.

In summary, the main contribution of this paper is to propose a new family of smooth- ing functions and correct a ﬂaw in an algorithm studied in , which is used to guarantee its convergence. We believe that the proposed new smoothing functions can be also em- ployed in other contexts where the projection function is involved. The related numerical performance can be investigated accordingly. We leave them as future research topics.

Acknowledgements. We are gratefully indebted to anonymous referees for their valu- able suggestions that help us to improve the presentation of the paper.

### References

 R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.

(18)

Table 3: Numerical performance when p = 2, stop criterion: 0.001.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 10 15 (0.5942, -0.8031)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 3 4 (-0.01407, 7.663e-006, 0, 0, 0, 0)

Ex 4.3 (0, 0) 0.5 0.2 0.01 3 4 (-0.01407, 7.663e-006)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 3 4 (0.5489, 2.066, 0.9741, 0.0204, 9.748e-007) Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 24 39 (0.5031, -1.7, 1.458)

Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 3 4 (-0.09533, 0.09533, 0.08515)

Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 3 4 (0.5271, 0.508, 0)

Ex 4.1’ (0, 5) 100 0.006 0.01 10 15 (0.5942, -0.8031)

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 3 4 (-0.01407, 7.663e-006, 0, 0, 0, 0)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 3 4 (-0.01407, 7.663e-006)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 3 4 (0.5489, 2.066, 0.9741, 0.0204, 9.748e-007) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 24 39 (0.5031, -1.7, 1.458)

Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 3 4 (-0.09533, 0.09533, 0.1698)

Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 3 4 (0.5271, 0.508, 0)

Note: Based on bH(µ, x) = 0 given as in (14).

 J-S. Chen and S-H. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, vol. 40, no. 3, pp. 389-404, 2008.

 J-S. Chen, S-H. Pan, and C-Y. Yang, Numerical comparison of two eﬀective methods for mixed complementarity problems, Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 667-683, 2010.

 J.W. Daniel, Newton’s method for nonlinear inequalities, Numerical Mathematics, vol. 21, pp. 381-387, 1973.

 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986.

 Z-H. Huang, Y. Zhang, and W. Wu, A smoothing-type algorithm for solving stsyem of inequalities, Journal of Computational and Applied Mathematics, vol. 220, pp. 355-363, 2008.

 D.Q. Mayne, E. Polak, and A.J. Heunis, Solving nonlinear inequalities in a ﬁnite number of iterations, Journal of Optimization Theory and Applications, vol.

33, pp. 207-221, 1981.

(19)

 M. Sahba, On the solution of nonlinear inequalities in a ﬁnite number of iterations, Numerical Mathematics, vol. 46, pp. 229-236, 1985.

 J.M. Schott, Matrix Analysis for Statistics, 2nd edition, John Wiley, New Jersey, 2005.

 H-X. Ying, Z-H. Huang, and L. Qi, The convergence of a Levenberg-Marquard method for the l2-norm solution of nonlinear inequalities, Numerical Functional Anal- ysis and Optimization, vol. 29, pp. 687-716, 2008.

 Y. Zhang and Z-H. Huang, A nonmonotone smoothing-type algorithm for solv- ing a system of equalities and inequalities, Journal of Computational and Applied Mathematics, vol. 233, pp. 2312-2321, 2010.

 J. Zhu and B. Hao, A new non-interior continuation method for solving a sys- tem of equalities and inequalities, Journal of Applied Mathematics, Artical Number 592540, 2014.

 Robert G. Bartle, The Elements of Real Analysis, Second Edition Jhon Wiley, New Jersey, 1976.

 S.L. HU, Z-H. Huang and P. Wang , A non-monotone smoothing Newton algorithm for solving nonlinear complementarity problems, Optimization Methods and Software , vol. 24, pp. 447-460, 2009

(20)

Table 4: Numerical performance with diﬀerent p for modiﬁed problems.

p = 2, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.009276,1.429,2.846,1.279,1.64,1.667)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.01516, 0.7206)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 5 6 (0.5563, 1.326, 0.9698, 0.9822, 1.155) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8299, -0.8663, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 8 (-0.09533, 0.09533, 0.332) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

p = 1.5, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.03532,1.428, 2.849, 1.278,1.63,1.666)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.0189, 0.7217)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 5 6 (0.5546, 1.329, 0.9708, 0.979, 1.135) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8288, -0.8673, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 8 (-0.09533, 0.09533, 0.3509) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

p = 3, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.007125,1.43,2.848,1.281, 1.641,1.667)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.01061, 0.72)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 6 7 (0.5589, 1.33, 0.9683, 0.9771, 1.17) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8313, -0.8648, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 7 (-0.09533, 0.09533, 0.4472) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

p = 8, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 5) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.00355,1.431,2.849,1.282,1.643,1.668)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.001338, 0.7196)

Ex 4.4’ (0.5, 2, 1, 0, 0) 5 0.02 0.8 6 7 (0.5622, 1.351, 0.9664, 0.9528, 1.18) Ex 4.5’ (-1, -1, 1) 0.5 0.2 0.8 6 7 (-0.8338, -0.8624, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 5 6 (-0.09533, 0.09533, 0.2985) Ex 4.7’ (0, 1, 0) 0.5 0.006 0.8 10 16 (0.5265, 0.5079, -100)

Note: Based on H(z) = 0 given as in (13).

(21)

Table 5: Numerical performance with diﬀerent p for original problems.

p = 2, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 12 20 (0.5821, -0.812)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 4 5 (-0.01407, 7.663e-006,0,0,0,0)

Ex 4.3 (0, 0) 0.5 0.2 0.01 4 5 (-0.01407, 7.663e-006)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.549,2.066,0.974,0.02039,9.747e-007) Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 25 40 (0.5029, -1.7, 1.458)

Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 4 5 (-0.09533, 0.09533, 0.08515)

Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 4 5 (0.5265, 0.5079, 0)

p = 1.5, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 12 20 (0.5821, -0.812)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 4 5 (-0.01739, 0.002797,0,0,0,0)

Ex 4.3 (0, 0) 0.5 0.2 0.01 4 5 (-0.01739, 0.002797)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.547,2.061,0.9751,0.02811,0.000359)

Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 Fail Fail Fail

Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 4 5 (-0.09533, 0.09533, 0.1086)

Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 4 5 (0.5265, 0.5079, 0)

p = 3, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 12 20 (0.5821, -0.812)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 4 5 (-0.009902, 6.814e-011,0,0,0,0)

Ex 4.3 (0, 0) 0.5 0.2 0.01 4 5 (-0.009902, 6.814e-011)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.5513, 2.071,0.9727,0.01239, 8.581e-012)

Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 Fail Fail Fail

Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 4 5 (-0.09533, 0.09533, 0.06131)

Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 4 5 (0.5265, 0.5079, 0)

p = 8, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1 (0, 5) 100 0.006 0.01 12 20 (0.5821, -0.812)

Ex 4.2 (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 4 5 (-0.002048, 6.022e-036, 0, 0, 0, 0)

Ex 4.3 (0, 0) 0.5 0.2 0.01 4 5 (-0.002048, 6.022e-036)

Ex 4.4 (0.5, 2, 1, 0, 0) 5 0.02 0.8 4 5 (0.557, 2.079,0.9694,0.001483,7.47e-037)

Ex 4.5 (-1, -1, 1) 0.5 0.2 0.8 Fail Fail Fail

Ex 4.6 (0, 0, 0) 0.5 0.02 0.8 4 5 (-0.09533, 0.09533, 0.01803)

Ex 4.7 (0, 1, 0) 0.5 0.006 0.8 4 5 (0.5265, 0.5079, 0)

Note: Based on bH(µ, x) = 0 given as in (14).

(22)

Table 6: Numerical performance with diﬀerent p for modiﬁed problems.

p = 1.5, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 0) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.03532,1.428,2.849,1.278,1.63,1.666)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.0189, 0.7217)

Ex 4.4’ (0, 0, 0, 0, 0) 5 0.02 0.8 Fail Fail Fail

Ex 4.5’ (0, 0, 0) 0.5 0.2 0.8 8 13 (-0.8355, -0.8607, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 8 (-0.09533, 0.09533, 0.3509)

Ex 4.7’ (0, 0, 0) 5 0.02 0.8 17 21 (0.5265, 0.5079, 100)

p = 2, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 0) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.009276,1.429,2.846,1.279,1.64,1.667)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.01516, 0.7206)

Ex 4.4’ (0, 0, 0, 0, 0) 5 0.02 0.8 Fail Fail Fail

Ex 4.5’ (0, 0, 0) 0.5 0.2 0.8 8 13 (-0.8356, -0.8606, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 8 (-0.09533, 0.09533, 0.332)

Ex 4.7’ (0, 0, 0) 5 0.02 0.8 17 21 (0.5265, 0.5079, 100)

p = 3, stop criterion: 1e− 006.

Problem x0 c τ η NI NF SOL

Ex 4.1’ (0, 0) 100 0.006 0.01 Fail Fail Fail

Ex 4.2’ (0, 0, 0, 0, 0, 0) 0.5 0.2 0.01 7 10 (-0.007125,1.43,2.848,1.281,1.641,1.667)

Ex 4.3’ (0, 0) 0.5 0.2 0.01 4 5 (-0.01061, 0.72)

Ex 4.4’ (0, 0, 0, 0, 0) 5 0.02 0.8 Fail Fail Fail

Ex 4.5’ (0, 0, 0) 0.5 0.2 0.8 8 13 (-0.8356, -0.8606, 1.957) Ex 4.6’ (0, 0, 0) 0.5 0.02 0.8 6 7 (-0.09533, 0.09533, 0.4472)

Ex 4.7’ (0, 0, 0) 5 0.02 0.8 17 21 (0.5265, 0.5079, 100)

Note: Based on H(z) = 0 given as in (13), x0 is ﬁxed.

Results We identified 19 new cases of oral syphilis (17 males, one female, and one case unknown sex) and described the clinical and histopathological features of this re-emerging

angular momentum is conserved. In the figure, the force F is always directed toward point O. Thus, the angular impulse of F about O is always zero, and angular momentum of

Replace this loading by an equivalent resultant force and specify where the resultant’s line of action intersects the column AB and boom

One of the main results is the bound on the vanishing order of a nontrivial solution u satisfying the Stokes system, which is a quantitative version of the strong unique

3.16 Career-oriented studies provide courses alongside other school subjects and learning experiences in the senior secondary curriculum. They have been included in the

The research proposes a data oriented approach for choosing the type of clustering algorithms and a new cluster validity index for choosing their input parameters.. The

Tseng, Growth behavior of a class of merit functions for the nonlinear comple- mentarity problem, Journal of Optimization Theory and Applications, vol. Fukushima, A new

In this paper, we build a new class of neural networks based on the smoothing method for NCP introduced by Haddou and Maheux  using some family F of smoothing functions.

Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, Journal of Global Optimization, vol.. Soares, A new

For different types of optimization problems, there arise various complementarity problems, for example, linear complemen- tarity problem, nonlinear complementarity problem

A derivative free algorithm based on the new NCP- function and the new merit function for complementarity problems was discussed, and some preliminary numerical results for

Qi (2001), Solving nonlinear complementarity problems with neural networks: a reformulation method approach, Journal of Computational and Applied Mathematics, vol. Pedrycz,

In summary, the main contribution of this paper is to propose a new family of smoothing functions and correct a flaw in an algorithm studied in , which is used to guarantee

For different types of optimization problems, there arise various complementarity problems, for example, linear complementarity problem, nonlinear complementarity problem,

The Hilbert space of an orbifold field theory  is decomposed into twisted sectors H g , that are labelled by the conjugacy classes [g] of the orbifold group, in our case

Abstract In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which.. 1

Recently, the paper  investigates a family of smoothing functions along with a smoothing-type algorithm to tackle the absolute value equation associated with second-order

Gu, Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones, Journal of

Type case as pattern matching on values Type safe dynamic value (existential types).. How can we

The MTMH problem is divided into three subproblems which are separately solved in the following three stages: (1) find a minimum set of tag SNPs based on pairwise perfect LD

“A Comprehensive Model for Assessing the Quality and Productivity of the Information System Function Toward a Theory for Information System Assessment.”,

A parallel route building algorithm for the vehicle routing and scheduling problem with time windows, European Journal of Operational Research, vol. A tabu search

For the case of with caisson chamber only but with no air turbine present, the 100% incident-wave energy was converted into reflected-wave energy (9.6%), the wave energy of MOWC