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A filled function method with one parameter for box constrained global optimization
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Applied Mathematics and Computation 194 (2007) 54–66 www.elsevier.com/locate/amc
A filled function method with one parameter for box constrained global optimization q Weixiang Wang a
a,d,* ,
Youlin Shang
b,c
, Liansheng Zhang
a
Department of Mathematics, Shanghai University, 99 Shangda Road, Baoshan Shanghai 200444, China b Department of Mathematics, Henan University of Science and Technology, Luoyang 471003, China c Department of Applied Mathematics, Tongji University, Shanghai 200092, China d Department of Mathematics, Shanghai Second Polytechnic University, Shanghai 201209, China
Abstract In this paper, a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed. First, a new definition of the filled function for box constrained minimization problem is given and under mild assumptions, this new auxiliary function is really a filled function. Then a new solution algorithm is proposed according to the theoretical analysis. And some numerical results demonstrate the efficiency of this method for box constrained global optimization. 2007 Elsevier Inc. All rights reserved. Keywords: Local minimizer; Global minimizer; Box constrained minimization; Filled function method
1. Introduction In many real world optimization problems it is essential or desirable to determine the global minimum of the objective function. In the past thirty years, methods that aim to find the global minimum of a function have attracted a lot of interest. Numerous algorithms have been studied and can be found in the surveys by Horst and Pardalos [7], Ge [2] as well as in the introductory textbook by Horst et al. [4]. These global optimization methods can be classified into two groups: stochastic (see e.g., [6]) and deterministic methods (see, e.g., [2,4,5]). Stochastic methods employ random factors, for example points are chosen randomly. Thus every run of such an algorithm could yield a different result. However, results obtained by a deterministic method are always reproducible. In this paper, we focus on filled function method, which was proposed by Ge in [2]. In [2], the following global optimization problem was considered min f ðxÞ; x2X
q *
This work was supported by the National Natural Science Foundation (No. 10571116). Corresponding author. E-mail address: [email protected] (W. Wang).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.011
ð1Þ
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
55
where X Rn is a closed bounded box containing all global minimizers of f(x) in its interior. It is assumed in this paper that f(x) has only a finite number of local minimizers on X. The basin of f(x) at an isolated local minimizer of f ðxÞ, x*, is defined in [2]. The concept of the filled functions was introduced in [2]. Suppose x* is a current local minimizer of f(x). P ðx; x Þ is said to be a filled function of f(x) at x*, if it satisfies the following properties: (P1) x* is a strictly local maximizer of P ðx; x Þ and the whole basin B* of f(x) at x* becomes a part of a hill of P ðx; x Þ; (P2) P ðx; x Þ has no minimizers or saddle points in any basin of f(x) higher than B*; (P3) If f(x) has a lower basin than B*, then there is a point x 0 in such a basin that minimizes P ðx; x Þ on the line through x and x . Ge specifically proposed the following two-parameter filled function in [2]: ! 2 1 kx x k exp P ðx; x ; r; qÞ ¼ ; r þ f ðxÞ q2
ð2Þ
where the parameters r and q need to be chosen appropriately. In recent years, rapid progress has been made both in theory and in practical applications for the filled function methods (see, e.g., [1,3,8–12]). However, all these open literatures devoted to unconstrained global optimization need the following assumption: f(x) is coercive, i.e., f ðxÞ ! þ1 as kxk ! þ1: But in practice, this coercive assumption on the objective function is not always satisfied. In order to get rid of the coercive assumption on the objective function, make the minimization problem of the proposed filled function be a really unconstrained optimization problem and find the global minimizer which located on the boundary of the box set, in this paper, we give a new definition of the filled function, propose a new filled function and design a new solution algorithm. We consider the following global optimization: min f ðxÞ;
ð3Þ
x2X
where X ¼ fai 6 xi 6 bi ; i ¼ 1; . . . ; ng; 1 < ai < bi < þ1; i ¼ 1; . . . ; n and the function f(x) in (3) is differentiable in Rn. Let gi ðxÞ ¼ ai xi ; and
0
ci ðxÞ ¼ xi bi ; i ¼ 1; . . . ; n;
1 g1 ðxÞ B . C C gðxÞ ¼ B @ .. A; gn ðxÞ
S ¼ fx 2 Rn : gi ðxÞ 6 0; ci ðxÞ 6 0; i ¼ 1; . . . ; ng
0
1 c1 ðxÞ B . C C cðxÞ ¼ B @ .. A: cn ðxÞ
Obviously, (3) is equivalent to the following problem: ðP Þ min f ðxÞ; x2S¼X
ð4Þ
where S ¼ fx 2 Rn : gðxÞ 6 0; cðxÞ 6 0g: In particular, this paper enjoys the following properties: • It does not require that f ðxÞ : Rn ! R is coercive, i.e., f ðxÞ ! þ1 as kxk ! þ1: • It requires f(x) and rf ðxÞ to be Lipschitz continuously differentiable over Rn with Lipschitz constants Lf and Lrf , respectively. • It requires the number of the different local minimal values of problem (P) is finite but the number of minimizers may be infinite. Now, we give a definition of the filled function as the follows: Suppose x* is a current local minimizer of (P). P ðx; x Þ is said to be a filled function of f(x) at x*, if it satisfies the following properties:
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W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
(P1) x* is a strictly local maximizer of P ðx; x Þ on Rn; (P2) rP ðx; x Þ 6¼ 0; when x 2 S 1 or x 2 Rn n S and x 6¼ x ; where S 1 ¼ fx 2 Sjf ðxÞ P f ðx Þg; (P300 ) If x* is a local minimizer and S 2 ¼ fx 2 Sjf ðxÞ < f ðx Þg is not empty, then there exists a point x1 2 S 2 such that x1 is a local minimizer of P ðx; x Þ. Note that (P300 ) in our paper is a revised version of the property (3) in [2,12]. In fact, Property (P300 ) is much stronger than [2,12] since a minimizer of the filled function must be in S 2 ¼ fx 2 S : f ðxÞ < f ðx Þg: Adopting the concept of the filled functions, a global optimization problem can be solved via a two-phase cycle. In Phase 1, we start from a given point and use any local minimization method to find a local minimizer x* of f(x). In Phase 2, we construct a filled function at x* and minimize the filed function in order to identify a point x 0 with f ðx0Þ < f ðx Þ and x0 2 S. If such an x 0 is found, x 0 is certainly in a lower basin than B*. We can then use x 0 as the initial point in Phase 1 again, and hence we can find a better local minimizer x** of f(x) with f ðx Þ < f ðx Þ. This process repeats until the time when minimizing a filled function does not yield a better solution. The current local minimum will be then taken as a global minimizer of f(x). This paper is organized as follows. Following this introduction, a new filled function is proposed and the properties of the new filled function are investigated. In Section 3, a new solution algorithm is suggested. Application of the new filled function algorithm is reported in Section 4 with satisfactory numerical results. Finally, some conclusions are drawn in Section 5. 2. A new filled function and its properties Consider the one parameter filled function for problem (4) 4
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X
3
3
3
f½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ g þ 1g
i¼1
1 þ fmin½0; maxðf ðxÞ f ðx Þ; gi ðxÞ; ci ðxÞ; i ¼ 1; ; nÞg3 ; q
ð5Þ
where x* is a current local minimizer of f(x), k k indicates the Euclidean vector norm. The following theorems show that F ðx; x ; qÞ is a filled function when q > 0 large enough. Theorem 2.1. Assume that x* is a local minimizer of (P), then x* is a strictly local maximizer of F ðx; x ; qÞ: Proof. Obviously, we should prove that for all x 2 N ðx ; r Þ, it holds F ðx; x ; qÞ < F ðx ; x ; qÞ, where r* > 0, N ðx ; r Þ is a neighborhood of x*. There are two cases: Case (1): Since x* is a local minimizer of (P), there exists a neighborhood N ðx ; r Þ of x ; r > 0 such that f ðxÞ P f ðx Þ;
gðx Þ 6 0;
gðxÞ 6 0;
cðx Þ 6 0;
cðxÞ 6 0;
for all x 2 N ðx ; r Þ \ S: For all x 2 N ðx ; r Þ \ S; x 6¼ x ; it holds fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0; we have 4
3
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X 3 ð½maxð0; gi ðxÞ gi ðx ÞÞ i¼1
3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g < 0 ¼ F ðx ; x ; qÞ: Case (2): For all x 2 N ðx ; r Þ \ ðRn n SÞ; there exists an i0 2 1; ; n such that gi0 ðxÞ P 0 or ci0 ðxÞ P 0, thus fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
57
and we have F ðx; x ; qÞ ¼ kx x k4 q logf½maxð0; f ðxÞ f ðx ÞÞ3 þ
n X
ð½maxð0; gi ðxÞ gi ðx ÞÞ3
i¼1 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g < 0 ¼ F ðx ; x ; qÞ: From the above two cases, we get that x* is a strictly local maximizer of F ðx; x ; qÞ:
h
Remark. Obviously, for all x 2 N ðx ; r Þ, r* > 0, we have 4
3
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X 3 3 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g: i¼1
rF ðx; x ; qÞ 2
¼ 4kx x k ðx x Þ 2
3q
ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ þ
Pn
3
½maxð0; f ðxÞ f ðx ÞÞ þ
2
i¼1 P n
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ
i¼1 ð½maxð0; gi ðxÞ gi ðx
ÞÞ3 þ ½maxð0; c
i ðxÞ ci ðx
ÞÞ3 Þ þ 1
;
r2 F ðx;x ; qÞ 2
¼ 4kx x k E 8ðx x Þðx x Þ
T
3q Pn 3 3 3 ½maxð0; f ðxÞ f ðx ÞÞ þ i¼1 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 2 ðmaxð0; f ðxÞ f ðx ÞÞÞ r2 f ðxÞ þ 2ðmaxð0; f ðxÞ f ðx ÞÞÞrf ðxÞrT f ðxÞ
þ
n X 2 ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ r2 gi ðxÞ þ 2 maxð0;gi ðxÞ gi ðx ÞÞrgi ðxÞrT gi ðxÞ i¼1
2 þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ r2 ci ðxÞ þ 2 maxð0; ci ðxÞ ci ðx ÞÞrci ðxÞrT ci ðxÞ
þ 9q
fðmaxð0; f ðxÞ f ðx ÞÞÞÞ2 rf ðxÞ þ 3
Pn
ð½maxð0; f ðxÞ f ðx ÞÞ þ
Pi¼1 n
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ2 rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ2 rci ðxÞg
i¼1 ð½maxð0; gi ðxÞ gi ðx
(
ÞÞ3 þ ½maxð0; c
i ðxÞ ci ðx
ÞÞ3 Þ þ 1Þ2
)T n X 2 2 ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ þ ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ ;
2
i¼1
where E is an identity matrix. It is easily to verify that i
2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ T rgi ðxÞ ¼ ð0; . . . ; 1; . . . ; 0Þ ;
2
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ rci ðxÞ ¼ ð0; . . . ; 1; . . . ; 0ÞT ;
r gi ðxÞ ¼ 0;
i
r ci ðxÞ ¼ 0;
58
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
For any d 2 D ¼ fd 2 Rn : kdk ¼ 1g; jðf ðxÞ f ðx ÞÞd T rf ðxÞj 6 Lf kx x kkrf ðxÞk; jðf ðxÞ f ðx ÞÞd T r2 f ðxÞdj 6 Lf kx x kkr2 f ðxÞk; n X ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞd T rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞd T rci ðxÞ ¼ d T ðx x Þ: i¼1
Let M rf ¼ maxx2N ðx ;r Þ krf ðxÞk; Lr2 f ¼ maxx2N ðx ;r Þ kr2 f ðxÞk; 1 : g¼ Pn 2 2 2 ½f ðxÞ f ðx Þ þ i¼1 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 Note that kx x k 6 r ; then, when q > 0 small enough, it holds 2
2
2
d T r2 F ðx; x ; qÞd ¼ 4kx x k 8ðd T ðx x ÞÞ 3qgfðmaxð0; f ðxÞ f ðx ÞÞÞ d T r2 f ðxÞd þ 2ðmaxð0; f ðxÞ f ðx ÞÞÞðd T rf ðxÞÞ2 þ
n X
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ2 d T r2 gi ðxÞd
i¼1 2
T
2
þ 2 maxð0; gi ðxÞ gi ðx ÞÞðd rgi ðxÞÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T r2 ci ðxÞd
2
2
þ 2 maxð0; ci ðxÞ ci ðx ÞÞðd T rci ðxÞÞ g þ 9qg2 fðmaxð0; f ðxÞ f ðx ÞÞÞ d T rf ðxÞ þ
n X
2
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ d T rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T rci ðxÞg
2
i¼1 2
2
6 4kx x k 3qgðmaxð0; f ðxÞ f ðx ÞÞÞ d T r2 f ðxÞd ( n X 2 2 þ 9q ðmaxð0; f ðxÞ f ðx ÞÞÞ d T rf ðxÞ þ ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ d T rgi ðxÞ i¼1
)2 2
þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T rci ðxÞ 6 4kx x k2 þ 3qL2f Lr2 f kx x k2 þ 9q½L2f M rf kx x k2
n X
ðminð0; xi xi ÞÞ2 d i
i¼1
þ
n X
2
2
ðmaxð0; xi xi ÞÞ d i
i¼1 2
2
2
2 2
6 4kx x k þ 3qL2f Lr2 f kx x k þ 9q½L2f M rf kx x k þ kx x k 2
2
2
6 ð4 þ 3qL2f Lr2 f þ 9qðr Þ ½L2f M rf þ 1 Þkx x k < 0: Therefore, r2 F ðx; x ; qÞ is negative definite. Then x* is an isolated local maximizer of F ðx; x ; qÞ and rF ðx; x ; qÞ 6¼ 0 for all x 2 N ðx ; r Þ. Theorem 2.1 reveals that the proposed new filled function satisfies property (P1). Theorem 2.2. Assume that x* is a local minimizer of (P) and x1 2 S 1 n x or x1 2 Rn n S; then for q > 0 small enough, it holds rF ðx1 ; x ; qÞ 6¼ 0: Proof. Firstly, by the remark, it holds rF ðx; x ; qÞ 6¼ 0 for all x 2 N ðx ; r Þ. Secondly, it is easy to get fmin½0; maxðf ðx1 Þ f ðx Þ; gðx1 Þ; cðx1 ÞÞg ¼ 0 for all x1 2 S 1 n N ðx ; r Þ or x1 2 ðRn n SÞ n N ðx ; r Þ. Thus, we have
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
4
F ðx; x ; qÞ ¼ kx x k q log ½maxð0; f ðxÞ f ðx ÞÞ þ
n X
3
3
59
3
ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 ;
i¼1
" 2
2
rF ðx; x ; qÞ ¼ 4kx x k ðx x Þ 3qg ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ
þ
n X
# 2
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ :
i¼1
Therefore, when q > 0 is small enough,
4
2
T
r F ðx; x ; qÞðx x Þ ¼ 4kx x k 3qg ðmaxð0; f ðxÞ f ðx ÞÞÞ ðx x Þ ðrf ðxÞ rf ðx ÞÞ T
þ ðmaxð0; f ðxÞ f ðx ÞÞÞ2 ðx x ÞT rf ðx ÞÞ þ
n X
2
2
½ðxi xi Þðminð0; xi xi ÞÞ þ ðxi xi Þðmaxð0; xi xi ÞÞ
i¼1 4
4
3
3
6 4kx x k þ 3qfL2f Lrf kx x k þ L2f krf ðx Þkkx x k þ kx x k g " !# L2f krf ðx Þk þ 1 4 2 6 kx x k 4 þ 3q Lf Lrf þ < 0: r Thus, for any x 2 S 1 n x or x 2 Rn n S; it holds rF ðx; x ; qÞ 6¼ 0.
h
Theorem 2.2 reveals that the proposed new filled function satisfies property (P2). Theorem 2.3. Assume that x* is a local minimizer of (P) but not a global minimizer of (P), and clintS ¼ clS, then there exists a point x0 2 S 2 such that x0 is a local minimizer of F ðx; x ; qÞ when q > 0 is small enough. Proof. Since x* is a local minimizer of (P), and x* is not a global minimizer of (P), there exists an x1 is another local minimizer of f(x) such that f ðx1 Þ < f ðx Þ; gðx1 Þ 6 0; cðx1 Þ 6 0: Since f ðxÞ; gi ðxÞ and ci ðxÞ; i ¼ 1; . . . ; n are continuous and clintS ¼ clS, there exists an x2 2 Rn such that f ðx2 Þ < f ðx Þ; gðx2 Þ < 0; cðx2 Þ < 0: So, we have (
F ðx2 ; x ; qÞ
¼
kx2
n X 3 3 x k q log f½ðmaxð0; gi ðx2 Þ gi ðx ÞÞ þ 1 þ ½maxð0; ci ðx2 Þ ci ðx ÞÞ g
)
4
i¼1
1 þ fmaxðf ðx2 Þ f ðx Þ; gi ðx2 Þ; ci ðx2 Þ; i ¼ 1; . . . ; nÞg3 : q Obviously, x2 2 intS \ S 2 . And for all x 2 oS; there exists one i0 2 fi ¼ 1; . . . ; ng such that gi0 ðxÞ ¼ 0 or ci0 ðxÞ ¼ 0. Thus fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0 and n X 4 3 3 F ðx; x ; qÞ ¼ kx x k q log ½maxð0; f ðxÞ f ðx Þ þ ð½maxð0; gi ðxÞ gi ðx ÞÞ 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 :
i¼1
60
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
for all x 2 oS: It is easy to get ( F ðx2 ; x ; qÞ
¼
kx2
4
x k q log
n X
3
3
f½ðmaxð0; gi ðx2 Þ gi ðx ÞÞ þ 1þ½maxð0; ci ðx2 Þ ci ðx ÞÞ g
o
i¼1
1 3 þ fmaxðf ðx2 Þ f ðx Þ; gi ðx2 Þ; ci ðx2 Þ; i ¼ 1; ; nÞg < F ðx; x ; qÞ q n X 3 3 4 ¼ kx x k q log ½maxð0; f ðxÞ f ðx Þ þ ð½maxð0; gi ðxÞ gi ðx ÞÞ 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1
i¼1
when q > 0 small enough. It indicates F ðx2 ; x ; qÞ < F ðx; x ; qÞ, for all x 2 oS when q > 0 small enough. Therefore min F ðx; x ; qÞ ¼ min F ðx; x ; qÞ ¼ F ðx0 ; x ; qÞ 6 F ðx2 ; x ; qÞ x2S
x2SnoS
and S n oS is an open set, thus x0 2 S n oS is a local minimizer of F ðx; x ; qÞ when q > 0 small enough. It is easy to see that f ðx0 Þ 6 f ðx2 Þ < f ðx Þ; gðx0 Þ < 0; cðx0 Þ < 0 since F ðx0 ; x ; qÞ 6 F ðx2 ; x ; qÞ when q > 0 small enough. That is to say, x0 2 S 2 : h Theorem 2.3 clearly state that the proposed filled function satisfies property (P300 ) and Theorems 2.1–2.3 state that the proposed filled function satisfies properties (P1)–(P300 ). Theorem 2.4. If x* is a global minimizer of (P), then F ðx; x ; qÞ < 0 for all x 2 S n x . Proof. Since x* is a global minimizer of (P), we have f ðxÞ P f ðx Þ for all x 2 S n x . Thus, by Theorem 2.1, F ðx; x ; qÞ < 0 for all x 2 S n x . h 3. Algorithm AOPF Based on the results of the previous subsection and in order to compare with the results of algorithm [12] (we call it algorithm ATPF), we constitute a new algorithm AOPF as follows. Initialization step 1. 2. 3. 4.
Choose a tolerance > 0, e.g., set :¼ 104 . ^ > 0; e.g., set q ^ :¼ 0:1: Choose a fraction q Choose a lower bound of q such that qL > 0, e.g., set qL :¼ 106 : Set k := 1. Main step
Starting from an initial point x1 2 S; minimize f(x) and obtain the first local minimizer x1 of f(x). Choose a set of initial points fxikþ1 : i ¼ 1; 2; . . . ; mg such that xikþ1 2 S n N ðxk ; rk Þ for some rk > 0. Let q = 1. Set i := 1. If i 6 m, then set x :¼ xikþ1 and go to 6; Otherwise, go to 7. If f ðxÞ < f ðx1 Þ, then use x as an initial point for a local minimization method to find xkþ1 such that f ðxkþ1 Þ < f ðxk Þ, set x1 ¼ xkþ1 ; k :¼ k þ 1 and go to 2; Otherwise, go to Inner Circular Step 10. ^ q. 7. Reduce q by setting q :¼ q (a) If q P qL , then go to 4. (b) Otherwise, the algorithm is incapable of finding a better minimizer starting from the initial points, fxikþ1 g: The algorithm stops and x1 is taken as a global minimizer.
1. 2. 3. 4. 5. 6.
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
61
Inner circular step 10 Let xk0 :¼ x; gk0 ¼ rF ðxk0 ; x1 ; qÞ; k 0 :¼ 1: 20 Let d k0 ¼ gk0 : 30 Get the step size ak0 , and let xk0 þ1 ¼ xk0 þ ak0 d k0 . If xk0 þ1 beyond the boundary of S during minimization, then set i :¼ i þ 1 and go to Main Step 5; Otherwise go to 40. 0 4 If f ðxk0 þ1 Þ < f ðx1 Þ, go to Main Step 6; Otherwise, go to 50 50 Let gk0 þ1 ¼ rF ðxk0 þ1 ; x1 ; qÞ, go to 60. 60 If k 0 > n; let x ¼ xk0 þ1 ; go to 10; Otherwise, go to 70 70 Let d k0 þ1 ¼ gk0 þ1 þ bk0 d k0 : where bk0 ¼
gT0
k þ1
ðgk 0 þ1 gk 0 Þ gT0 gk 0
: Let k 0 :¼ k 0 þ 1, go to 30.
k
The motivation and mechanism behind the algorithm are explained below. A set of m ¼ 4n þ 4 initial points is chosen in Step 2 to minimize the filled function. We set the initial points symmetric about the current local minimizer. For example, when n = 2, the initial points are: xk þ rð1; 0Þ; xk þ rð0; 1Þ; xk rð1; 0Þ; xk rð0; 1Þ; xk þ rð1; 1Þ; xk rð1; 1Þ; xk þ 0:5rð1; 0Þ; xk þ 0:5rð0; 1Þ; xk 0:5rð1; 0Þ; xk 0:5rð0; 1Þ; xk þ 0:5rð1; 1Þ; xk 0:5rð1; 1Þ; where r > 0 is a pre-selected step-size. Step 6 represents a transition from minimizing the filled function F ðx; x1 ; qÞ to a local search for the original objective function f(x) when a point x with f ðxÞ < f ðx1 Þ is found. It can be concluded from Theorem 2.3 that this point x must be in a lower basin of f(x). We can thus use x as the initial point to minimize f(x) in this lower basin and obtain a better local minimizer. Step 30 finds a new x in the direction d k0 such that F ðx; x ; qÞ can reduce to certain extent. Then, one can use the new x to minimize the filled function again. In Step 30, use ak0 ¼ 0:1 as the step size. Recall from Theorem 2.3 that the value of q should be selected small enough. Otherwise, there could be no minimizer of F ðx; x1 ; qÞ in a better basin. Thus, the value of q is decreased successively in Step 7 of the solution process if no better solution is found when minimizing the filled function. If all the initial points were chosen to calculate and q reaches its lower bound qL and no better solution is found, the current local minimizer is taken as a global minimizer In Inner Circular Step, we use PRP Conjugate Gradient Method to get the search direction. In fact, it can also use other local method to get the search direction. 4. Numerical experiment Based on the proposed algorithm AOPF and the algorithm ATPF, we tested the same problems. All the numerical experiments are implemented in Matlab 6.5.1, under Windows XP and Pentium (R) 4 CPU 1400 MHZ. In our programs, we obtain local minimizers of the objective function by PRP Conjugate Gradient Method to get the search direction and the Armijo line search to get the step size, krf ðxÞk 6 108 as the terminate condition. The computational results of two algorithms are summarized in Tables for each example problem. The symbols used in the tables are given as follows: k x0 q r
the the the the
iteration number in finding the kth local minimizer initial point which satisfies x0 2 X parameter used for finding the ðk þ 1Þth local minimizer for algorithm AOPF parameter used for finding the ðk þ 1Þth local minimizer for the algorithm ATPF
62
q xk f ðxk Þ Of Tf Os Ts
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
the the the the the the the
parameter used for finding the ðk þ 1Þth local minimizer for the algorithm ATPF kth local minimizer function value of the kth local minimizer CPU time in seconds to obtain the final results for the algorithm AOPF CPU time in seconds to obtain the final results for the algorithm ATPF CPU time in seconds to for the algorithm AOPF to stop at Step 7 CPU time in seconds to for the algorithm ATPF to stop at Step 30
5. Conclusions In this paper, we construct F ðx; x ; qÞ with one parameter and prove that it is a filled function on box constraint problem. From all the numerical results, we can see that algorithm AOPF and algorithm ATPF have the same results, but algorithm AOPF use less time than algorithm ATPF in most cases. We can also see that from all the Tables, in most cases, we must use more time to judge the current point is a global minimizer than Table 1 Computational results for problem 1 with c = 0.2, x0 ¼ ð0; 2Þ AOPF
ATPF
Of
0.0940
Os
0.2103
Tf
0.0630
Ts
k
q
xk
f ðxk Þ
q
r
xk
29
–
–
11.1662
1
1e1
0.0887
1
1
0
1
1e2
1
–
2
1e1
3
1e1
4
1e2
0 2
2:1681 1:7870 2:7380 0:7884 0:9825 0:0548
3.3590
0 2
f ðxk Þ
2:1681 1:7870 2:7380 0:7884 0:9825 0:0548
29 11.1662 0.0887 0
Table 2 Computational results for problem 2, x0 ¼ ð2; 2Þ AOPF Of k 1
ATPF
0.1090
Os
0.1090
q
xk
f ðxk Þ
q
0
1
–
0:000 0:000
Tf
5.3910
Ts
r
xk
–
0:000 0:000
6.3600
f ðxk Þ 0
Table 3 Computational results for problem 2, x0 ¼ ð2; 1Þ AOPF
ATPF
Of
0.0470
Os
0.1250
Tf
0.0620
Ts
k
q
xk
f ðxk Þ
q
r
0.2986
1
–
xk
0
2
1e1
1
–
2
1
1:7476 0:8738 0 0
1:7476 0:8738 0 0
5.4850
f ðxk Þ 0.2986
0
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
63
Table 4 Computational results for problem 3, x0 ¼ ð2; 2Þ AOPF Of k
ATPF
0.0630
Os
0.1410
q
xk
f ðxk Þ
q
0.2155
–
–
1.0316
1
1e5
1
–
2
1e5
1:7036 0:7961 0:0898 0:7127
Tf
0.1560
Ts
r
xk
5.7970
1:7036 0:7961
0:0898 0:7127
f ðxk Þ
0.2155
1.0316
Table 5 Computational results for problem 3, x0 ¼ ð2; 1Þ AOPF
ATPF
Of
0.1470
Os
0.203
Tf
0.0470
Ts
k
q
xk
f ðxk Þ
q
r
0.2155
–
xk
1
–
2
1e1
1:7036 0:7961 0:0898 0:7127
–
1.0316
1
1e5
6.5780
1:7036 0:7961 0:0898 0:7127
f ðxk Þ
0.2155
1.0316
Table 6 Computational results for problem 4, x0 ¼ ð1; 0Þ AOPF Of k
ATPF
0.06210
Os
0.2850
q
xk
f ðxk Þ
q
1
–
–
0
1
1
1
–
2
1
1:000 0:000 0:000 0:000
Tf
0.0470
Ts
r
xk
9.438
1:000 0:000 0:000 0:000
f ðxk Þ 1 0
Table 7 Computational results for problem 5, x0 ¼ ð1; 1Þ AOPF Of k
ATPF
0.0930
Os
0.469
q
xk
f ðxk Þ
q
30
–
–
3
1
1e5
1
–
2
1e5
0:6 0:4 0 1
Tf
0.3130
Ts
r
xk
0:6000 0:4000 0 1
6.438
f ðxk Þ 30 3
to find a global minimizer. Recently, global optimality conditions have been derived in [13] for quadratic binary integer programming problems. However, A global optimality conditions for continuous variables are still an open problem, in general. The criterion of a global minimizer will provide solid stopping conditions for a continuous filled function method.
64
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
Table 8 Computational results for problem 6, x0 ¼ ð1; 0Þ AOPF Of k
ATPF
0.2190
Os
0.750
q
xk
f ðxk Þ
1
–
2
1e5
3
1e6
1:0202 0:4389 5:4786 6:0835 6:7083 6:0835
Tf q
0.2340
Ts
r
xk
78.9790
–
–
123.5768
1
1e5
186.7309
1
1e6
2.6560
1:0202 0:4389 5:4786 6:0835 6:7083 6:0835
f ðxk Þ
78.9790
123.5768 186.7309
Table 9 Computational results for problem 8 with n = 2, x0 ¼ ð4; 4Þ AOPF Of k
ATPF
0.2510
Os
0.4380
q
xk
f ðxk Þ
q
3.1098
–
–
0
1
1e4
1
–
2
1e4
1:9899 1:9898 1:0000 1:0000
Tf
0.1880
Ts
r
xk
4.1100
1:9899 1:9898 1:0000 1:0000
f ðxk Þ 3.1098
0
Table 10 Computational results for problem 8 with n = 3, x0 ¼ ð3; 3; 3Þ AOPF
ATPF
Of
0.1780
Os
0.3740
Tf
0.1560
Ts
k
q
xk 0
f ðxk Þ
q
r
xk 0
1
–
1.0367
–
–
2
1e4
0
1
1e5
1
1:9900 @ 1:0000 A 1:0000 0 1 1:0000 @ 1:0000 A 1:0000
10.6090 1
1:9900 @ 1:0000 A 1:0000 0 1 1:0000 @ 1:0000 A 1:0000
f ðxk Þ 1.0367
0
Acknowledgement We thank anonymous referees for their careful reading of the paper, and for their constructive comments, that improved the paper. Appendix Problem Problem Problem Problem Problem Problem Problem
1: 2: 3: 4: 5: 6: 8:
Two-dimensional function in [13] (see Table 1). Three-hump back camel function in [2] (see Tables 2 and 3). Six-hump back camel function in [2]) (see Tables 4 and 5). Treccani function in [2]) (see Table 6). Goldstein and Price function in [2] (see Table 7). Two-dimensional Shubert function in [2] (see Table 8). n-Dimensional function in [2] (see Tables 9–12).
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
65
Table 11 Computational results for problem 8 with n = 7, x0 ¼ ð6; 6; 6; 6; 6; 6; 6Þ AOPF Of k
ATPF
1.4220
Os
1.6750
q
xk
f ðxk Þ
0
1
–
Tf q
1.2340
Ts
r
xk
1
0:9961 B 2:0137 C B C B 2:9958 C B C B 2:9982 C B C B 2:9960 C B C @ 3:0010 A 2:9953
1
39.9687
–
–
1
30.6917
1
1e1
1
28.4985
1
1e4
5
1
0
6
1
B B B B B B B B @
1 0:9726 1:0000 C C 1:0000 C C 1:0000 C C 1:0156 C C 1:0577 A 0:5879
1 1:0000 B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C @ 1:0000 A 0:9999
22.7122
1
1e4
1
1 1:0001 B 1:0001 C B C B 0:9997 C B C B 0:6376 C B C B 2:9823 C B C @ 2:9964 A 2:9985
28.4985
11.3654
1
1e4
1 1:0036 B 1:0000 C B C B 1:0000 C B C B 1:0031 C B C B 0:2284 C B C @ 2:9343 A 1:9997
22.7122
0
0
1.1805
1
1e4
0
7
1 1:0017 B 0:9999 C B C B 0:5130 C B C B 2:8376 C B C B 2:9973 C B C @ 2:9971 A 2:9978
30.6917
0
0
1 1:0036 B 1:0000 C B C B 1:0000 C B C B 1:0031 C B C B 0:2284 C B C @ 2:9343 A 1:9997
1 1:0454 B 0:9818 C B C B 0:9856 C B C B 2:9890 C B C B 2:9876 C B C @ 3:0073 A 2:9956
39.9687
0
0
4
1 1:0001 B 1:0001 C B C B 0:9997 C B C B 0:6376 C B C B 2:9823 C B C @ 2:9964 A 2:9985
0:9961 B 2:0137 C B C B 2:9958 C B C B 2:9982 C B C B 2:9960 C B C @ 3:0010 A 2:9953 0
0
3
1 1:0017 B 0:9999 C B C B 0:5130 C B C B 2:8376 C B C B 2:9973 C B C @ 2:9971 A 2:9978
f ðxk Þ 1
0
0
2
1 1:0454 B 0:9818 C B C B 0:9856 C B C B 2:9890 C B C B 2:9876 C B C @ 3:0073 A 2:9956
1.2340
B B B B B B B B @
1 0:9726 1:0000 C C 1:0000 C C 1:0000 C C 1:0156 C C 1:0577 A 0:5879
1.1805
0
0
1
1e4
1 1:0000 B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C @ 1:0000 A 0:9999
11.3654
0
66
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
Table 12 Computational results for problem 8 with n = 10, x0 ¼ ð6; 6; 6; 6; 6; 6; 6; 6; 6; 6Þ AOPF Of k
ATPF
0.1400
Os
1.0940
q
xk
f ðxk Þ
1
–
2
1e3
0
1
0:0099 B 1:0002 C B C B 1:0029 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0025 C B C @ 1:0023 A 1:0030 0 1 1:0000 B 1:0000 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C @ 1:0001 A 1:0001
Tf q
0.2820
Ts
r
xk
0.3110
–
–
0
1
1e3
0
23.9690 1
0:0099 B 1:0002 C B C B 1:0029 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0025 C B C @ 1:0023 A 1:0030 0 1 1:0000 B 1:0000 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C @ 1:0001 A 1:0001
f ðxk Þ
0.3110
0
References [1] B.C. Cetin, J. Barhen, J.W. Burdick, Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, Journal of Optimization and Applications 77 (1993) 97–126. [2] R.P. Ge, A filled function method for finding a global minimizer of a funciton of several variables, Mathematical Programming 46 (1990) 191–204. [3] R.P. Ge, Y.F. Qin, A class of filled functions for finding global minimizers of a function of several variables, Joural of Optimization Theory and Applications 54 (1987) 241–252. [4] R. Horst, P.M. Pardalos, N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht, 1995. [5] R. Horst, H. Tuy, Global Optimization: Deterministic Approaches, second ed., Springer, Heidelberg, 1993. [6] A.H.G.R. Kan, G.T. TIMMER, A Stochastic Approach to Global optimization, in: P.T. Boggs, R.H. Byrd, R.B. Schnabel (Eds.), Numerical Optimization, SIAM, Philadelphia, Pennsylvania, 1985, pp. 245–262. [7] P.M. Pardalos, H.E. Romeijn, H. Tuy, Recent development and trends in global otpinmization, Journal of Computational and Applied Mathematics 124 (2000) 209–228. [8] X.L. Wang, G.B. Zhou, A new filled function for unconstrained global optimization, Applied Mathematics and Computation 174 (2006) 419–429. [9] Z. Xu, H.X. Huang, P.M. Pardalos, C.X. Xu, Filled functions for unconstrained global optimization, Journal of Global Optinization 20 (2001) 49–65. [10] Y.J. Yang, Y.L. Shang, A new filled function method for unconstrained global optimization, Applied Mathematics and Computation 173 (2006) 501–512. [11] Y. Yao, Dynamic tunneling algorithm for global optimization, IEEE Transactions on Systems, Man and cybernetics 19 (1989) 1222– 1230. [12] L.S. Zhang, C.K. NG, D. Li, W.W. Tian, A new filled function method for global optimization, Journal of Global Optimization 28 (2004) 17–43. [13] Q. Zheng, D. Zhuang, Integral golbal minimization: algorithms, implementations and numerical tests, Jounal of Global Optimization 7 (1995) 421–454.
Applied Mathematics and Computation 194 (2007) 54–66 www.elsevier.com/locate/amc
A filled function method with one parameter for box constrained global optimization q Weixiang Wang a
a,d,* ,
Youlin Shang
b,c
, Liansheng Zhang
a
Department of Mathematics, Shanghai University, 99 Shangda Road, Baoshan Shanghai 200444, China b Department of Mathematics, Henan University of Science and Technology, Luoyang 471003, China c Department of Applied Mathematics, Tongji University, Shanghai 200092, China d Department of Mathematics, Shanghai Second Polytechnic University, Shanghai 201209, China
Abstract In this paper, a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed. First, a new definition of the filled function for box constrained minimization problem is given and under mild assumptions, this new auxiliary function is really a filled function. Then a new solution algorithm is proposed according to the theoretical analysis. And some numerical results demonstrate the efficiency of this method for box constrained global optimization. 2007 Elsevier Inc. All rights reserved. Keywords: Local minimizer; Global minimizer; Box constrained minimization; Filled function method
1. Introduction In many real world optimization problems it is essential or desirable to determine the global minimum of the objective function. In the past thirty years, methods that aim to find the global minimum of a function have attracted a lot of interest. Numerous algorithms have been studied and can be found in the surveys by Horst and Pardalos [7], Ge [2] as well as in the introductory textbook by Horst et al. [4]. These global optimization methods can be classified into two groups: stochastic (see e.g., [6]) and deterministic methods (see, e.g., [2,4,5]). Stochastic methods employ random factors, for example points are chosen randomly. Thus every run of such an algorithm could yield a different result. However, results obtained by a deterministic method are always reproducible. In this paper, we focus on filled function method, which was proposed by Ge in [2]. In [2], the following global optimization problem was considered min f ðxÞ; x2X
q *
This work was supported by the National Natural Science Foundation (No. 10571116). Corresponding author. E-mail address: [email protected] (W. Wang).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.011
ð1Þ
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
55
where X Rn is a closed bounded box containing all global minimizers of f(x) in its interior. It is assumed in this paper that f(x) has only a finite number of local minimizers on X. The basin of f(x) at an isolated local minimizer of f ðxÞ, x*, is defined in [2]. The concept of the filled functions was introduced in [2]. Suppose x* is a current local minimizer of f(x). P ðx; x Þ is said to be a filled function of f(x) at x*, if it satisfies the following properties: (P1) x* is a strictly local maximizer of P ðx; x Þ and the whole basin B* of f(x) at x* becomes a part of a hill of P ðx; x Þ; (P2) P ðx; x Þ has no minimizers or saddle points in any basin of f(x) higher than B*; (P3) If f(x) has a lower basin than B*, then there is a point x 0 in such a basin that minimizes P ðx; x Þ on the line through x and x . Ge specifically proposed the following two-parameter filled function in [2]: ! 2 1 kx x k exp P ðx; x ; r; qÞ ¼ ; r þ f ðxÞ q2
ð2Þ
where the parameters r and q need to be chosen appropriately. In recent years, rapid progress has been made both in theory and in practical applications for the filled function methods (see, e.g., [1,3,8–12]). However, all these open literatures devoted to unconstrained global optimization need the following assumption: f(x) is coercive, i.e., f ðxÞ ! þ1 as kxk ! þ1: But in practice, this coercive assumption on the objective function is not always satisfied. In order to get rid of the coercive assumption on the objective function, make the minimization problem of the proposed filled function be a really unconstrained optimization problem and find the global minimizer which located on the boundary of the box set, in this paper, we give a new definition of the filled function, propose a new filled function and design a new solution algorithm. We consider the following global optimization: min f ðxÞ;
ð3Þ
x2X
where X ¼ fai 6 xi 6 bi ; i ¼ 1; . . . ; ng; 1 < ai < bi < þ1; i ¼ 1; . . . ; n and the function f(x) in (3) is differentiable in Rn. Let gi ðxÞ ¼ ai xi ; and
0
ci ðxÞ ¼ xi bi ; i ¼ 1; . . . ; n;
1 g1 ðxÞ B . C C gðxÞ ¼ B @ .. A; gn ðxÞ
S ¼ fx 2 Rn : gi ðxÞ 6 0; ci ðxÞ 6 0; i ¼ 1; . . . ; ng
0
1 c1 ðxÞ B . C C cðxÞ ¼ B @ .. A: cn ðxÞ
Obviously, (3) is equivalent to the following problem: ðP Þ min f ðxÞ; x2S¼X
ð4Þ
where S ¼ fx 2 Rn : gðxÞ 6 0; cðxÞ 6 0g: In particular, this paper enjoys the following properties: • It does not require that f ðxÞ : Rn ! R is coercive, i.e., f ðxÞ ! þ1 as kxk ! þ1: • It requires f(x) and rf ðxÞ to be Lipschitz continuously differentiable over Rn with Lipschitz constants Lf and Lrf , respectively. • It requires the number of the different local minimal values of problem (P) is finite but the number of minimizers may be infinite. Now, we give a definition of the filled function as the follows: Suppose x* is a current local minimizer of (P). P ðx; x Þ is said to be a filled function of f(x) at x*, if it satisfies the following properties:
56
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
(P1) x* is a strictly local maximizer of P ðx; x Þ on Rn; (P2) rP ðx; x Þ 6¼ 0; when x 2 S 1 or x 2 Rn n S and x 6¼ x ; where S 1 ¼ fx 2 Sjf ðxÞ P f ðx Þg; (P300 ) If x* is a local minimizer and S 2 ¼ fx 2 Sjf ðxÞ < f ðx Þg is not empty, then there exists a point x1 2 S 2 such that x1 is a local minimizer of P ðx; x Þ. Note that (P300 ) in our paper is a revised version of the property (3) in [2,12]. In fact, Property (P300 ) is much stronger than [2,12] since a minimizer of the filled function must be in S 2 ¼ fx 2 S : f ðxÞ < f ðx Þg: Adopting the concept of the filled functions, a global optimization problem can be solved via a two-phase cycle. In Phase 1, we start from a given point and use any local minimization method to find a local minimizer x* of f(x). In Phase 2, we construct a filled function at x* and minimize the filed function in order to identify a point x 0 with f ðx0Þ < f ðx Þ and x0 2 S. If such an x 0 is found, x 0 is certainly in a lower basin than B*. We can then use x 0 as the initial point in Phase 1 again, and hence we can find a better local minimizer x** of f(x) with f ðx Þ < f ðx Þ. This process repeats until the time when minimizing a filled function does not yield a better solution. The current local minimum will be then taken as a global minimizer of f(x). This paper is organized as follows. Following this introduction, a new filled function is proposed and the properties of the new filled function are investigated. In Section 3, a new solution algorithm is suggested. Application of the new filled function algorithm is reported in Section 4 with satisfactory numerical results. Finally, some conclusions are drawn in Section 5. 2. A new filled function and its properties Consider the one parameter filled function for problem (4) 4
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X
3
3
3
f½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ g þ 1g
i¼1
1 þ fmin½0; maxðf ðxÞ f ðx Þ; gi ðxÞ; ci ðxÞ; i ¼ 1; ; nÞg3 ; q
ð5Þ
where x* is a current local minimizer of f(x), k k indicates the Euclidean vector norm. The following theorems show that F ðx; x ; qÞ is a filled function when q > 0 large enough. Theorem 2.1. Assume that x* is a local minimizer of (P), then x* is a strictly local maximizer of F ðx; x ; qÞ: Proof. Obviously, we should prove that for all x 2 N ðx ; r Þ, it holds F ðx; x ; qÞ < F ðx ; x ; qÞ, where r* > 0, N ðx ; r Þ is a neighborhood of x*. There are two cases: Case (1): Since x* is a local minimizer of (P), there exists a neighborhood N ðx ; r Þ of x ; r > 0 such that f ðxÞ P f ðx Þ;
gðx Þ 6 0;
gðxÞ 6 0;
cðx Þ 6 0;
cðxÞ 6 0;
for all x 2 N ðx ; r Þ \ S: For all x 2 N ðx ; r Þ \ S; x 6¼ x ; it holds fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0; we have 4
3
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X 3 ð½maxð0; gi ðxÞ gi ðx ÞÞ i¼1
3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g < 0 ¼ F ðx ; x ; qÞ: Case (2): For all x 2 N ðx ; r Þ \ ðRn n SÞ; there exists an i0 2 1; ; n such that gi0 ðxÞ P 0 or ci0 ðxÞ P 0, thus fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
57
and we have F ðx; x ; qÞ ¼ kx x k4 q logf½maxð0; f ðxÞ f ðx ÞÞ3 þ
n X
ð½maxð0; gi ðxÞ gi ðx ÞÞ3
i¼1 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g < 0 ¼ F ðx ; x ; qÞ: From the above two cases, we get that x* is a strictly local maximizer of F ðx; x ; qÞ:
h
Remark. Obviously, for all x 2 N ðx ; r Þ, r* > 0, we have 4
3
F ðx; x ; qÞ ¼ kx x k q logf½maxð0; f ðxÞ f ðx ÞÞ þ
n X 3 3 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1g: i¼1
rF ðx; x ; qÞ 2
¼ 4kx x k ðx x Þ 2
3q
ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ þ
Pn
3
½maxð0; f ðxÞ f ðx ÞÞ þ
2
i¼1 P n
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ
i¼1 ð½maxð0; gi ðxÞ gi ðx
ÞÞ3 þ ½maxð0; c
i ðxÞ ci ðx
ÞÞ3 Þ þ 1
;
r2 F ðx;x ; qÞ 2
¼ 4kx x k E 8ðx x Þðx x Þ
T
3q Pn 3 3 3 ½maxð0; f ðxÞ f ðx ÞÞ þ i¼1 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 2 ðmaxð0; f ðxÞ f ðx ÞÞÞ r2 f ðxÞ þ 2ðmaxð0; f ðxÞ f ðx ÞÞÞrf ðxÞrT f ðxÞ
þ
n X 2 ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ r2 gi ðxÞ þ 2 maxð0;gi ðxÞ gi ðx ÞÞrgi ðxÞrT gi ðxÞ i¼1
2 þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ r2 ci ðxÞ þ 2 maxð0; ci ðxÞ ci ðx ÞÞrci ðxÞrT ci ðxÞ
þ 9q
fðmaxð0; f ðxÞ f ðx ÞÞÞÞ2 rf ðxÞ þ 3
Pn
ð½maxð0; f ðxÞ f ðx ÞÞ þ
Pi¼1 n
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ2 rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ2 rci ðxÞg
i¼1 ð½maxð0; gi ðxÞ gi ðx
(
ÞÞ3 þ ½maxð0; c
i ðxÞ ci ðx
ÞÞ3 Þ þ 1Þ2
)T n X 2 2 ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ þ ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ ;
2
i¼1
where E is an identity matrix. It is easily to verify that i
2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ T rgi ðxÞ ¼ ð0; . . . ; 1; . . . ; 0Þ ;
2
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ rci ðxÞ ¼ ð0; . . . ; 1; . . . ; 0ÞT ;
r gi ðxÞ ¼ 0;
i
r ci ðxÞ ¼ 0;
58
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
For any d 2 D ¼ fd 2 Rn : kdk ¼ 1g; jðf ðxÞ f ðx ÞÞd T rf ðxÞj 6 Lf kx x kkrf ðxÞk; jðf ðxÞ f ðx ÞÞd T r2 f ðxÞdj 6 Lf kx x kkr2 f ðxÞk; n X ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞd T rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞd T rci ðxÞ ¼ d T ðx x Þ: i¼1
Let M rf ¼ maxx2N ðx ;r Þ krf ðxÞk; Lr2 f ¼ maxx2N ðx ;r Þ kr2 f ðxÞk; 1 : g¼ Pn 2 2 2 ½f ðxÞ f ðx Þ þ i¼1 ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 Note that kx x k 6 r ; then, when q > 0 small enough, it holds 2
2
2
d T r2 F ðx; x ; qÞd ¼ 4kx x k 8ðd T ðx x ÞÞ 3qgfðmaxð0; f ðxÞ f ðx ÞÞÞ d T r2 f ðxÞd þ 2ðmaxð0; f ðxÞ f ðx ÞÞÞðd T rf ðxÞÞ2 þ
n X
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ2 d T r2 gi ðxÞd
i¼1 2
T
2
þ 2 maxð0; gi ðxÞ gi ðx ÞÞðd rgi ðxÞÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T r2 ci ðxÞd
2
2
þ 2 maxð0; ci ðxÞ ci ðx ÞÞðd T rci ðxÞÞ g þ 9qg2 fðmaxð0; f ðxÞ f ðx ÞÞÞ d T rf ðxÞ þ
n X
2
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ d T rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T rci ðxÞg
2
i¼1 2
2
6 4kx x k 3qgðmaxð0; f ðxÞ f ðx ÞÞÞ d T r2 f ðxÞd ( n X 2 2 þ 9q ðmaxð0; f ðxÞ f ðx ÞÞÞ d T rf ðxÞ þ ½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ d T rgi ðxÞ i¼1
)2 2
þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ d T rci ðxÞ 6 4kx x k2 þ 3qL2f Lr2 f kx x k2 þ 9q½L2f M rf kx x k2
n X
ðminð0; xi xi ÞÞ2 d i
i¼1
þ
n X
2
2
ðmaxð0; xi xi ÞÞ d i
i¼1 2
2
2
2 2
6 4kx x k þ 3qL2f Lr2 f kx x k þ 9q½L2f M rf kx x k þ kx x k 2
2
2
6 ð4 þ 3qL2f Lr2 f þ 9qðr Þ ½L2f M rf þ 1 Þkx x k < 0: Therefore, r2 F ðx; x ; qÞ is negative definite. Then x* is an isolated local maximizer of F ðx; x ; qÞ and rF ðx; x ; qÞ 6¼ 0 for all x 2 N ðx ; r Þ. Theorem 2.1 reveals that the proposed new filled function satisfies property (P1). Theorem 2.2. Assume that x* is a local minimizer of (P) and x1 2 S 1 n x or x1 2 Rn n S; then for q > 0 small enough, it holds rF ðx1 ; x ; qÞ 6¼ 0: Proof. Firstly, by the remark, it holds rF ðx; x ; qÞ 6¼ 0 for all x 2 N ðx ; r Þ. Secondly, it is easy to get fmin½0; maxðf ðx1 Þ f ðx Þ; gðx1 Þ; cðx1 ÞÞg ¼ 0 for all x1 2 S 1 n N ðx ; r Þ or x1 2 ðRn n SÞ n N ðx ; r Þ. Thus, we have
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
4
F ðx; x ; qÞ ¼ kx x k q log ½maxð0; f ðxÞ f ðx ÞÞ þ
n X
3
3
59
3
ð½maxð0; gi ðxÞ gi ðx ÞÞ þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 ;
i¼1
" 2
2
rF ðx; x ; qÞ ¼ 4kx x k ðx x Þ 3qg ðmaxð0; f ðxÞ f ðx ÞÞÞ rf ðxÞ
þ
n X
# 2
2
½ðmaxð0; gi ðxÞ gi ðx ÞÞÞ rgi ðxÞ þ ðmaxð0; ci ðxÞ ci ðx ÞÞÞ rci ðxÞ :
i¼1
Therefore, when q > 0 is small enough,
4
2
T
r F ðx; x ; qÞðx x Þ ¼ 4kx x k 3qg ðmaxð0; f ðxÞ f ðx ÞÞÞ ðx x Þ ðrf ðxÞ rf ðx ÞÞ T
þ ðmaxð0; f ðxÞ f ðx ÞÞÞ2 ðx x ÞT rf ðx ÞÞ þ
n X
2
2
½ðxi xi Þðminð0; xi xi ÞÞ þ ðxi xi Þðmaxð0; xi xi ÞÞ
i¼1 4
4
3
3
6 4kx x k þ 3qfL2f Lrf kx x k þ L2f krf ðx Þkkx x k þ kx x k g " !# L2f krf ðx Þk þ 1 4 2 6 kx x k 4 þ 3q Lf Lrf þ < 0: r Thus, for any x 2 S 1 n x or x 2 Rn n S; it holds rF ðx; x ; qÞ 6¼ 0.
h
Theorem 2.2 reveals that the proposed new filled function satisfies property (P2). Theorem 2.3. Assume that x* is a local minimizer of (P) but not a global minimizer of (P), and clintS ¼ clS, then there exists a point x0 2 S 2 such that x0 is a local minimizer of F ðx; x ; qÞ when q > 0 is small enough. Proof. Since x* is a local minimizer of (P), and x* is not a global minimizer of (P), there exists an x1 is another local minimizer of f(x) such that f ðx1 Þ < f ðx Þ; gðx1 Þ 6 0; cðx1 Þ 6 0: Since f ðxÞ; gi ðxÞ and ci ðxÞ; i ¼ 1; . . . ; n are continuous and clintS ¼ clS, there exists an x2 2 Rn such that f ðx2 Þ < f ðx Þ; gðx2 Þ < 0; cðx2 Þ < 0: So, we have (
F ðx2 ; x ; qÞ
¼
kx2
n X 3 3 x k q log f½ðmaxð0; gi ðx2 Þ gi ðx ÞÞ þ 1 þ ½maxð0; ci ðx2 Þ ci ðx ÞÞ g
)
4
i¼1
1 þ fmaxðf ðx2 Þ f ðx Þ; gi ðx2 Þ; ci ðx2 Þ; i ¼ 1; . . . ; nÞg3 : q Obviously, x2 2 intS \ S 2 . And for all x 2 oS; there exists one i0 2 fi ¼ 1; . . . ; ng such that gi0 ðxÞ ¼ 0 or ci0 ðxÞ ¼ 0. Thus fmin½0; maxðf ðxÞ f ðx Þ; gðxÞ; cðxÞÞg ¼ 0 and n X 4 3 3 F ðx; x ; qÞ ¼ kx x k q log ½maxð0; f ðxÞ f ðx Þ þ ð½maxð0; gi ðxÞ gi ðx ÞÞ 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1 :
i¼1
60
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
for all x 2 oS: It is easy to get ( F ðx2 ; x ; qÞ
¼
kx2
4
x k q log
n X
3
3
f½ðmaxð0; gi ðx2 Þ gi ðx ÞÞ þ 1þ½maxð0; ci ðx2 Þ ci ðx ÞÞ g
o
i¼1
1 3 þ fmaxðf ðx2 Þ f ðx Þ; gi ðx2 Þ; ci ðx2 Þ; i ¼ 1; ; nÞg < F ðx; x ; qÞ q n X 3 3 4 ¼ kx x k q log ½maxð0; f ðxÞ f ðx Þ þ ð½maxð0; gi ðxÞ gi ðx ÞÞ 3
þ ½maxð0; ci ðxÞ ci ðx ÞÞ Þ þ 1
i¼1
when q > 0 small enough. It indicates F ðx2 ; x ; qÞ < F ðx; x ; qÞ, for all x 2 oS when q > 0 small enough. Therefore min F ðx; x ; qÞ ¼ min F ðx; x ; qÞ ¼ F ðx0 ; x ; qÞ 6 F ðx2 ; x ; qÞ x2S
x2SnoS
and S n oS is an open set, thus x0 2 S n oS is a local minimizer of F ðx; x ; qÞ when q > 0 small enough. It is easy to see that f ðx0 Þ 6 f ðx2 Þ < f ðx Þ; gðx0 Þ < 0; cðx0 Þ < 0 since F ðx0 ; x ; qÞ 6 F ðx2 ; x ; qÞ when q > 0 small enough. That is to say, x0 2 S 2 : h Theorem 2.3 clearly state that the proposed filled function satisfies property (P300 ) and Theorems 2.1–2.3 state that the proposed filled function satisfies properties (P1)–(P300 ). Theorem 2.4. If x* is a global minimizer of (P), then F ðx; x ; qÞ < 0 for all x 2 S n x . Proof. Since x* is a global minimizer of (P), we have f ðxÞ P f ðx Þ for all x 2 S n x . Thus, by Theorem 2.1, F ðx; x ; qÞ < 0 for all x 2 S n x . h 3. Algorithm AOPF Based on the results of the previous subsection and in order to compare with the results of algorithm [12] (we call it algorithm ATPF), we constitute a new algorithm AOPF as follows. Initialization step 1. 2. 3. 4.
Choose a tolerance > 0, e.g., set :¼ 104 . ^ > 0; e.g., set q ^ :¼ 0:1: Choose a fraction q Choose a lower bound of q such that qL > 0, e.g., set qL :¼ 106 : Set k := 1. Main step
Starting from an initial point x1 2 S; minimize f(x) and obtain the first local minimizer x1 of f(x). Choose a set of initial points fxikþ1 : i ¼ 1; 2; . . . ; mg such that xikþ1 2 S n N ðxk ; rk Þ for some rk > 0. Let q = 1. Set i := 1. If i 6 m, then set x :¼ xikþ1 and go to 6; Otherwise, go to 7. If f ðxÞ < f ðx1 Þ, then use x as an initial point for a local minimization method to find xkþ1 such that f ðxkþ1 Þ < f ðxk Þ, set x1 ¼ xkþ1 ; k :¼ k þ 1 and go to 2; Otherwise, go to Inner Circular Step 10. ^ q. 7. Reduce q by setting q :¼ q (a) If q P qL , then go to 4. (b) Otherwise, the algorithm is incapable of finding a better minimizer starting from the initial points, fxikþ1 g: The algorithm stops and x1 is taken as a global minimizer.
1. 2. 3. 4. 5. 6.
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
61
Inner circular step 10 Let xk0 :¼ x; gk0 ¼ rF ðxk0 ; x1 ; qÞ; k 0 :¼ 1: 20 Let d k0 ¼ gk0 : 30 Get the step size ak0 , and let xk0 þ1 ¼ xk0 þ ak0 d k0 . If xk0 þ1 beyond the boundary of S during minimization, then set i :¼ i þ 1 and go to Main Step 5; Otherwise go to 40. 0 4 If f ðxk0 þ1 Þ < f ðx1 Þ, go to Main Step 6; Otherwise, go to 50 50 Let gk0 þ1 ¼ rF ðxk0 þ1 ; x1 ; qÞ, go to 60. 60 If k 0 > n; let x ¼ xk0 þ1 ; go to 10; Otherwise, go to 70 70 Let d k0 þ1 ¼ gk0 þ1 þ bk0 d k0 : where bk0 ¼
gT0
k þ1
ðgk 0 þ1 gk 0 Þ gT0 gk 0
: Let k 0 :¼ k 0 þ 1, go to 30.
k
The motivation and mechanism behind the algorithm are explained below. A set of m ¼ 4n þ 4 initial points is chosen in Step 2 to minimize the filled function. We set the initial points symmetric about the current local minimizer. For example, when n = 2, the initial points are: xk þ rð1; 0Þ; xk þ rð0; 1Þ; xk rð1; 0Þ; xk rð0; 1Þ; xk þ rð1; 1Þ; xk rð1; 1Þ; xk þ 0:5rð1; 0Þ; xk þ 0:5rð0; 1Þ; xk 0:5rð1; 0Þ; xk 0:5rð0; 1Þ; xk þ 0:5rð1; 1Þ; xk 0:5rð1; 1Þ; where r > 0 is a pre-selected step-size. Step 6 represents a transition from minimizing the filled function F ðx; x1 ; qÞ to a local search for the original objective function f(x) when a point x with f ðxÞ < f ðx1 Þ is found. It can be concluded from Theorem 2.3 that this point x must be in a lower basin of f(x). We can thus use x as the initial point to minimize f(x) in this lower basin and obtain a better local minimizer. Step 30 finds a new x in the direction d k0 such that F ðx; x ; qÞ can reduce to certain extent. Then, one can use the new x to minimize the filled function again. In Step 30, use ak0 ¼ 0:1 as the step size. Recall from Theorem 2.3 that the value of q should be selected small enough. Otherwise, there could be no minimizer of F ðx; x1 ; qÞ in a better basin. Thus, the value of q is decreased successively in Step 7 of the solution process if no better solution is found when minimizing the filled function. If all the initial points were chosen to calculate and q reaches its lower bound qL and no better solution is found, the current local minimizer is taken as a global minimizer In Inner Circular Step, we use PRP Conjugate Gradient Method to get the search direction. In fact, it can also use other local method to get the search direction. 4. Numerical experiment Based on the proposed algorithm AOPF and the algorithm ATPF, we tested the same problems. All the numerical experiments are implemented in Matlab 6.5.1, under Windows XP and Pentium (R) 4 CPU 1400 MHZ. In our programs, we obtain local minimizers of the objective function by PRP Conjugate Gradient Method to get the search direction and the Armijo line search to get the step size, krf ðxÞk 6 108 as the terminate condition. The computational results of two algorithms are summarized in Tables for each example problem. The symbols used in the tables are given as follows: k x0 q r
the the the the
iteration number in finding the kth local minimizer initial point which satisfies x0 2 X parameter used for finding the ðk þ 1Þth local minimizer for algorithm AOPF parameter used for finding the ðk þ 1Þth local minimizer for the algorithm ATPF
62
q xk f ðxk Þ Of Tf Os Ts
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
the the the the the the the
parameter used for finding the ðk þ 1Þth local minimizer for the algorithm ATPF kth local minimizer function value of the kth local minimizer CPU time in seconds to obtain the final results for the algorithm AOPF CPU time in seconds to obtain the final results for the algorithm ATPF CPU time in seconds to for the algorithm AOPF to stop at Step 7 CPU time in seconds to for the algorithm ATPF to stop at Step 30
5. Conclusions In this paper, we construct F ðx; x ; qÞ with one parameter and prove that it is a filled function on box constraint problem. From all the numerical results, we can see that algorithm AOPF and algorithm ATPF have the same results, but algorithm AOPF use less time than algorithm ATPF in most cases. We can also see that from all the Tables, in most cases, we must use more time to judge the current point is a global minimizer than Table 1 Computational results for problem 1 with c = 0.2, x0 ¼ ð0; 2Þ AOPF
ATPF
Of
0.0940
Os
0.2103
Tf
0.0630
Ts
k
q
xk
f ðxk Þ
q
r
xk
29
–
–
11.1662
1
1e1
0.0887
1
1
0
1
1e2
1
–
2
1e1
3
1e1
4
1e2
0 2
2:1681 1:7870 2:7380 0:7884 0:9825 0:0548
3.3590
0 2
f ðxk Þ
2:1681 1:7870 2:7380 0:7884 0:9825 0:0548
29 11.1662 0.0887 0
Table 2 Computational results for problem 2, x0 ¼ ð2; 2Þ AOPF Of k 1
ATPF
0.1090
Os
0.1090
q
xk
f ðxk Þ
q
0
1
–
0:000 0:000
Tf
5.3910
Ts
r
xk
–
0:000 0:000
6.3600
f ðxk Þ 0
Table 3 Computational results for problem 2, x0 ¼ ð2; 1Þ AOPF
ATPF
Of
0.0470
Os
0.1250
Tf
0.0620
Ts
k
q
xk
f ðxk Þ
q
r
0.2986
1
–
xk
0
2
1e1
1
–
2
1
1:7476 0:8738 0 0
1:7476 0:8738 0 0
5.4850
f ðxk Þ 0.2986
0
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
63
Table 4 Computational results for problem 3, x0 ¼ ð2; 2Þ AOPF Of k
ATPF
0.0630
Os
0.1410
q
xk
f ðxk Þ
q
0.2155
–
–
1.0316
1
1e5
1
–
2
1e5
1:7036 0:7961 0:0898 0:7127
Tf
0.1560
Ts
r
xk
5.7970
1:7036 0:7961
0:0898 0:7127
f ðxk Þ
0.2155
1.0316
Table 5 Computational results for problem 3, x0 ¼ ð2; 1Þ AOPF
ATPF
Of
0.1470
Os
0.203
Tf
0.0470
Ts
k
q
xk
f ðxk Þ
q
r
0.2155
–
xk
1
–
2
1e1
1:7036 0:7961 0:0898 0:7127
–
1.0316
1
1e5
6.5780
1:7036 0:7961 0:0898 0:7127
f ðxk Þ
0.2155
1.0316
Table 6 Computational results for problem 4, x0 ¼ ð1; 0Þ AOPF Of k
ATPF
0.06210
Os
0.2850
q
xk
f ðxk Þ
q
1
–
–
0
1
1
1
–
2
1
1:000 0:000 0:000 0:000
Tf
0.0470
Ts
r
xk
9.438
1:000 0:000 0:000 0:000
f ðxk Þ 1 0
Table 7 Computational results for problem 5, x0 ¼ ð1; 1Þ AOPF Of k
ATPF
0.0930
Os
0.469
q
xk
f ðxk Þ
q
30
–
–
3
1
1e5
1
–
2
1e5
0:6 0:4 0 1
Tf
0.3130
Ts
r
xk
0:6000 0:4000 0 1
6.438
f ðxk Þ 30 3
to find a global minimizer. Recently, global optimality conditions have been derived in [13] for quadratic binary integer programming problems. However, A global optimality conditions for continuous variables are still an open problem, in general. The criterion of a global minimizer will provide solid stopping conditions for a continuous filled function method.
64
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
Table 8 Computational results for problem 6, x0 ¼ ð1; 0Þ AOPF Of k
ATPF
0.2190
Os
0.750
q
xk
f ðxk Þ
1
–
2
1e5
3
1e6
1:0202 0:4389 5:4786 6:0835 6:7083 6:0835
Tf q
0.2340
Ts
r
xk
78.9790
–
–
123.5768
1
1e5
186.7309
1
1e6
2.6560
1:0202 0:4389 5:4786 6:0835 6:7083 6:0835
f ðxk Þ
78.9790
123.5768 186.7309
Table 9 Computational results for problem 8 with n = 2, x0 ¼ ð4; 4Þ AOPF Of k
ATPF
0.2510
Os
0.4380
q
xk
f ðxk Þ
q
3.1098
–
–
0
1
1e4
1
–
2
1e4
1:9899 1:9898 1:0000 1:0000
Tf
0.1880
Ts
r
xk
4.1100
1:9899 1:9898 1:0000 1:0000
f ðxk Þ 3.1098
0
Table 10 Computational results for problem 8 with n = 3, x0 ¼ ð3; 3; 3Þ AOPF
ATPF
Of
0.1780
Os
0.3740
Tf
0.1560
Ts
k
q
xk 0
f ðxk Þ
q
r
xk 0
1
–
1.0367
–
–
2
1e4
0
1
1e5
1
1:9900 @ 1:0000 A 1:0000 0 1 1:0000 @ 1:0000 A 1:0000
10.6090 1
1:9900 @ 1:0000 A 1:0000 0 1 1:0000 @ 1:0000 A 1:0000
f ðxk Þ 1.0367
0
Acknowledgement We thank anonymous referees for their careful reading of the paper, and for their constructive comments, that improved the paper. Appendix Problem Problem Problem Problem Problem Problem Problem
1: 2: 3: 4: 5: 6: 8:
Two-dimensional function in [13] (see Table 1). Three-hump back camel function in [2] (see Tables 2 and 3). Six-hump back camel function in [2]) (see Tables 4 and 5). Treccani function in [2]) (see Table 6). Goldstein and Price function in [2] (see Table 7). Two-dimensional Shubert function in [2] (see Table 8). n-Dimensional function in [2] (see Tables 9–12).
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
65
Table 11 Computational results for problem 8 with n = 7, x0 ¼ ð6; 6; 6; 6; 6; 6; 6Þ AOPF Of k
ATPF
1.4220
Os
1.6750
q
xk
f ðxk Þ
0
1
–
Tf q
1.2340
Ts
r
xk
1
0:9961 B 2:0137 C B C B 2:9958 C B C B 2:9982 C B C B 2:9960 C B C @ 3:0010 A 2:9953
1
39.9687
–
–
1
30.6917
1
1e1
1
28.4985
1
1e4
5
1
0
6
1
B B B B B B B B @
1 0:9726 1:0000 C C 1:0000 C C 1:0000 C C 1:0156 C C 1:0577 A 0:5879
1 1:0000 B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C @ 1:0000 A 0:9999
22.7122
1
1e4
1
1 1:0001 B 1:0001 C B C B 0:9997 C B C B 0:6376 C B C B 2:9823 C B C @ 2:9964 A 2:9985
28.4985
11.3654
1
1e4
1 1:0036 B 1:0000 C B C B 1:0000 C B C B 1:0031 C B C B 0:2284 C B C @ 2:9343 A 1:9997
22.7122
0
0
1.1805
1
1e4
0
7
1 1:0017 B 0:9999 C B C B 0:5130 C B C B 2:8376 C B C B 2:9973 C B C @ 2:9971 A 2:9978
30.6917
0
0
1 1:0036 B 1:0000 C B C B 1:0000 C B C B 1:0031 C B C B 0:2284 C B C @ 2:9343 A 1:9997
1 1:0454 B 0:9818 C B C B 0:9856 C B C B 2:9890 C B C B 2:9876 C B C @ 3:0073 A 2:9956
39.9687
0
0
4
1 1:0001 B 1:0001 C B C B 0:9997 C B C B 0:6376 C B C B 2:9823 C B C @ 2:9964 A 2:9985
0:9961 B 2:0137 C B C B 2:9958 C B C B 2:9982 C B C B 2:9960 C B C @ 3:0010 A 2:9953 0
0
3
1 1:0017 B 0:9999 C B C B 0:5130 C B C B 2:8376 C B C B 2:9973 C B C @ 2:9971 A 2:9978
f ðxk Þ 1
0
0
2
1 1:0454 B 0:9818 C B C B 0:9856 C B C B 2:9890 C B C B 2:9876 C B C @ 3:0073 A 2:9956
1.2340
B B B B B B B B @
1 0:9726 1:0000 C C 1:0000 C C 1:0000 C C 1:0156 C C 1:0577 A 0:5879
1.1805
0
0
1
1e4
1 1:0000 B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C B 1:0000 C B C @ 1:0000 A 0:9999
11.3654
0
66
W. Wang et al. / Applied Mathematics and Computation 194 (2007) 54–66
Table 12 Computational results for problem 8 with n = 10, x0 ¼ ð6; 6; 6; 6; 6; 6; 6; 6; 6; 6Þ AOPF Of k
ATPF
0.1400
Os
1.0940
q
xk
f ðxk Þ
1
–
2
1e3
0
1
0:0099 B 1:0002 C B C B 1:0029 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0025 C B C @ 1:0023 A 1:0030 0 1 1:0000 B 1:0000 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C @ 1:0001 A 1:0001
Tf q
0.2820
Ts
r
xk
0.3110
–
–
0
1
1e3
0
23.9690 1
0:0099 B 1:0002 C B C B 1:0029 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0024 C B C B 1:0025 C B C @ 1:0023 A 1:0030 0 1 1:0000 B 1:0000 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C B 1:0001 C B C @ 1:0001 A 1:0001
f ðxk Þ
0.3110
0
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