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Reducing transformation and global optimization
Applied Mathematics and Computation 218 (2012) 5848–5860
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Reducing transformation and global optimization Djaouida Guettal ⇑, Abdelkader Ziadi Department of Mathematics, University Ferhat Abbas, Setif 19000, Algeria
a r t i c l e
i n f o
Keywords: Global optimization Constrained optimization Reducing transformation a-Dense curves Piyavskii’s algorithm
a b s t r a c t In this paper, we give new results on the Alienor method of dimension reduction. This technique is used to solve multidimensional global optimization problems of type minx2X f(x) where f is a non convex Lipschitz function and X a compact set of Rn ðn P 2Þ defined by Lipschitz constraints. The idea is to construct an a-dense curve h in the feasible set X. The global minimum of f on X is then approximated by the global minimum of f on the curve h. That is, our problem has become a one-dimensional problem which can be solved by the Piyavskii–Shubert method. Examples of these curves and numerical implementations on several test functions are given. Crown Copyright 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction The reducing transformation Alienor has been proposed in the 80s by Cherruault and Guillez [4,5]. It has been essentially developed for solving global optimization problems of coercive functions without constraints or of functions verifying other conditions and with fairly simple constraints. This occurs for examples when the feasible set is a hyper-rectangle [2,3,13]. This method is based on the generation of an a-dense curve in the feasible set which allows to transform the optimization problem in a hyper-rectangle of Rn into an optimization in a compact interval of R [14,20,23]. In our work, we shall consider a more general case. The equality and inequality constraints define a feasible set which is a more complicated compact. Note that existing multidimensional global optimization methods treat either problems without constraints or with simple constraints such as a polytope, sphere or an ellipsoid [1,6–12,19]. 2. The Alienor method First we will give some notation and definitions. We assume that A is any interval of R (in general A = [0, M], with M > 0), and a is a number strictly positive. Definition 1. We say that a curve of Rn defined by:
h:A!X is a-dense in X, if for all x 2 X, $h 2 A such that d(x, h(h)) 6 a where d is the Euclidian distance in Rn . We want to solve the following global optimization problem:
min f ðxÞ; x2X
⇑ Corresponding author. E-mail address: [email protected] (D. Guettal). 0096-3003/$ - see front matter Crown Copyright 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.053
ð1Þ
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5849
where X is the set defined by the following inequalities:
g i ðx1 ; x2 ; . . . ; xn Þ 6 0;
for i ¼ 1; 2; . . . ; m:
In some cases, the feasible set X can be written (after some transformations) in the following form:
X¼
a 6 x1 6 b; ðx1 ; x2 ; . . . ; xn Þ 2 Rn u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ: .. .
9 =
un ðx1 ; x2 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; x2 ; . . . ; xn1 Þ; ; where u2, . . . , un, w2, . . . , wn are Lipschitz functions with constants (resp.) l2, . . . , ln, L2, . . . , Ln with respect to the supremum norm. If it is possible to construct a parametrized curve h(h) = (h1(h), . . . , hn(h)), a-dense in X, for h 2 A, then using the reducing transformation xj = hj(h), j = 1, 2, . . . , n, we approximate the problem (1) by means of the problem:
min f ðhÞ; h2A
ð2Þ
where f⁄(h) = f(h1(h), h2(h), . . . , hn(h)). The difficulty with Alienor method is the construction of the a-dense curve which depends on the structure of the feasible set X. 2.1. Constructing a-dense curves This section is devoted to the construction of a-dense curves. Consider a function h(h) = (h1(h), h2(h), . . . , hn(h)), defined on an interval A of R and whose values belong to the compact X defined above. We are going to study the relationship to must exist between the components h1, h2, . . . , hn in order that the function h generates an a-dense curve in X. Denote by l the Lebesgue measure. The number a is supposed very small according to the interval [a, b]. For n = 2, we have the following theorem: Theorem 1. Let h = (h1, h2) : A ? X be a continuous function and h2, a be strictly positive numbers such that: (i) h1 is surjective. (ii) For any interval I of length h2, there exist h02 ; h002 2 I such that:
(
h2 h02 ¼ u2 ðh1 h02 Þ; h2 ðh002 Þ ¼ w2 h1 h002 :
(iii) For any interval J of A, we have:
lðJÞ < h2 ) lðh1 ðJÞÞ < a: Then, the parametrized curve defined by h(h) = (h1(h), h2(h)) for all h 2 A, is ga-dense in X, with g = max{1, l2, L2}.
Proof. Let (x1, x2) 2 X. Consider the interval [x1 a, x1 + a]. Because h1 is surjective, there exists a closed interval I1 A such that:
h1 ðI1 Þ ¼ ½x1 a; x1 þ a \ ½a; b; and since l(h1(I1)) P a, then l(I1) P h2. Hence, we have jx1 h1(h)j 6 a for all h 2 I1. Moreover there exist h02 ; h002 2 I1 such that:
(
h2 h02 ¼ u2 h1 h02 ; 00 00 h2 h2 ¼ w2 h1 h2 :
Since
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ; and u2, w2 being Lipschitzian of constants l2, L2 (resp.), then:
h2 h02 l2 a 6 x2 6 h2 h002 þ L2 a:
We thus have to study three cases:
8 0 0 > < ð1Þ h2 h2 l2 a 6 x2< h2 h2 ; 0 00 ð2Þ h2 h2 6 x2 6 h2 h2 ; > : ð3Þ h2 h002 < x2 6 h2 h002 þ L2 a:
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(i) If (1) holds, we have:
jx2 h2 h02 j 6 l2 a; and therefore
kx h h02 k ¼ max jx1 h1 h02 j; jx2 h2 h02 j 6 maxfa; l2 ag: (ii) If (2) that satisfied, there exists h0 2 h02 ; h002 such that:
x2 ¼ h2 ðh0 Þ; leading to:
kx hðh0 Þk 6 a: (iii) If the last hypothesis holds, we have:
jx2 h2 h002 j 6 L2 a; and therefore
kx h h002 k 6 maxfa; L2 ag: Summing up, the parametrized curve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in X with g = max{1, l2, L2}.
h
For n P 3, if the Lipschitz constants corresponding to u2, . . . , un, w2, . . . , wn are chosen such that the families (li)26i6n and (Li)26i6n are increasing and a less than the length of the interval hi h0i ; hi h00i for all i = 2, . . . , n, then we have the following result. Theorem 2. Let h = (h1, . . . , hn) : A ? X be a continuous function and h2, . . . , hn, a strictly positive numbers such that: (i) h1 is surjective. (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, . . . , n}, and for all closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Then, the parametrized icurve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in h P Qn1 1 þ n1 i¼2 j¼i maxðlj ; Lj Þ .
X, with g ¼ maxf1; ln ; Ln g
Proof. Let x = (x1, . . . , xn) 2 X. Consider the interval [x1 a, x1 + a], since h1 is surjective, there exists a closed interval I1 A such that:
h1 ðI1 Þ ¼ ½x1 a; x1 þ a \ ½a; b: But l(h1(I1)) P a, so that l(I1) P h2. A consequence of that is jx1 h1(h)j 6 a for all h 2 I1. Moreover, there exist h02 ; h002 2 I1 such that:
(
h2 h02 ¼ u2 h1 h02 ; h2 h002 ¼ w2 h1 h002 :
Because h2 h002 h2 h02 P a, we have
l h2 h02 ; h002
P a;
and hence:
l h02 ; h002 P h3 : On the other hand, we have u2(x1) 6 x2 6 w2(x1), and since:
(
u ðx1 Þ u h1 h0 6 l2 x1 h1 h0 6 l2 a; 2 2 2 2
w ðx1 Þ w h1 h00 6 L2 x1 h1 h00 6 L2 a; 2 2 2 2
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
we deduce:
h2 h02 l2 a 6 x2 6 h2 h002 þ L2 a: We still have three cases to study:
8 0 0 > < ð1Þ h2 h2 l2 a 6 x26 h2 h2 ; ð2Þ h2 h02 6 x2 6 h2 h002 ; > : ð3Þ h2 h002 6 x2 6 h2 h002 þ L2 a: If (1) holds, we take I2 ¼ h02 ; h02 þ h3 . 0 00 If (2) is valid, there exists I2 h2 ; h2 such that:
h2 ðI2 Þ ¼ ½x2 a; x2 þ a \ h2 h02 ; h2 h002 :
If the last condition holds, take I2 ¼ h002 h3 ; h002 . 0 we It is clear that I2 h2 ; h002 and l(I2) P h3 in the three cases. Indeed, for (2) we have:
lðh2 ðI2 ÞÞ P a; and hence l(I2) P h3 because of hypothesis (iii). Moreover, for all h 2 I2, we have: (i) if (1) holds, then
jx2 h2 ðhÞj 6 jx2 h2 h02 j þ jh2 h02 h2 ðhÞj 6 ð1 þ l2 Þa: (ii) If (2) holds, then:
jx2 h2 ðhÞj 6 a: (iii) If the last condition holds, then:
jx2 h2 ðhÞj 6 jx2 h2 h002 j þ jh2 h002 h2 ðhÞj 6 ð1 þ L2 Þa: Consequently, for all h 2 I2, we have:
jx2 h2 ðhÞj 6 ð1 þ maxðl2 ; L2 ÞÞa: We have l(I2) P h3, then there exist h03 ; h003 2 I2 such that:
(
h3 h03 ¼ u3 h1 h03 ; h2 h03 ; 00 00 00 h3 h3 ¼ w3 h1 h3 ; h2 h3 ;
hence
00
h3 h h3 h0 P a; 3 3 consequently
l h3 h03 ; h003
P a;
and therefore:
l h02 ; h002 P h4 : On the other hand, we have u3(x1, x2) 6 x3 6 w3(x1, x2) where u3, w3 are Lipschitz functions, such that:
u ðx1 ; x2 Þ u h1 h0 ; h2 h0 6 l3 max x1 h1 h0 ; x2 h2 h0 6 l3 f1 þ maxðl2 ; L2 Þga; 3 3 3 3 3
3 00 00
00
00
w ðx1 ; x2 Þ w h1 h ; h2 h 6 L3 max x1 h1 h ; x2 h2 h 6 L3 f1 þ maxðl2 ; L2 Þga: 3 3 3 3 3 3
Whence
h3 h03 l3 f1 þ maxðl2 ; L2 Þga 6 x3 6 h3 h003 þ L3 f1 þ maxðl2 ; L2 Þga: Three cases arise:
8 0 0 0 > 2 ; L2 Þga 6 x3 6 h3 h3 ; < ð1 Þ h3 h3 l3 f1 þ maxðl ð20 Þ h3 h03 6 x3 6 h3 h003 ; > 00 : 0 ð3 Þ h3 h3 6 x3 6 h3 h003 þ L3 f1 þ maxðl2 ; L2 Þga: If (10 ) is valid we take I3 ¼ h03 ; h03 þ h4 .
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If (20 ) is satisfied, there exists I3 h03 ; h003 such that:
h3 ðI3 Þ ¼ ½x3 a; x3 þ a \ h3 h03 ; h3 h003 :
00 00 If the last condition holds, 0 we00 take I3 ¼ h3 h4 ; h3 . Hence, we have I3 h3 ; h3 and l(I3) P h4 in all cases. Moreover, for any h 2 I3 we have: (1) If (10 ) holds, we deduce:
jx3 h3 ðhÞj 6 jx3 h3 h03 j þ jh3 h03 h3 ðhÞj 6 ½1 þ l3 ð1 þ maxðl2 ; L2 ÞÞa: (2) If (20 ) is satisfied, then:
jx3 h3 ðhÞj 6 a: (3) If the last condition holds, we have:
jx3 h3 ðhÞj 6 jx3 h3 h003 j þ jh3 h003 h3 ðhÞj 6 ½1 þ L3 ð1 þ maxðl2 ; L2 ÞÞa: In all cases, we have for every h 2 I3,
jx3 h3 ðhÞj 6 f1 þ maxðl3 ; L3 Þ½1 þ maxðl2 ; L2 Þga; that is to say:
jx3 h3 ðhÞj 6 f1 þ maxðl3 ; L3 Þ þ maxðl3 ; L3 Þ maxðl2 ; L2 Þga: Proceeding like that up to the step (n 1), we obtain for all h 2 In1,
jxn1 hn1 ðhÞj 6 ½1 þ
n1 Y n1 X i¼2
maxðlj ; Lj Þa
j¼i
with l(In1) P hn and In1 In2 I1 A. Therefore, there exist h0n ; h00n 2 In1 such that:
(
hn h0n ¼ un h1 h0n ; . . . ; hn1 h0n ; hn h00n ¼ wn h1 h00n ; . . . ; hn1 h00n :
On the other hand, we have:
un ðx1 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; . . . ; xn1 Þ; and
u ðx1 ; . . . ; xn1 Þ u h1 h0 ; . . . ; hn1 h0 6 ln max x1 h1 h0 ; . . . ; xn1 hn1 h0
n n n n n n " # n1 Y n1 X 6 ln 1 þ maxðlj ; Lj Þ a i¼2
j¼i
w ðx1 ; . . . ; xn1 Þ w h1 h00 ; . . . ; hn1 h00 6 Ln max x1 h1 h00 ; . . . ; xn1 hn1 h00
n n n n n n " # n1 Y n1 X 6 Ln 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
Hence
" # " # n1 Y n1 n1 Y n1 X X 0 00 hn hn ln 1 þ maxðlj ; Lj Þ a 6 xn 6 hn hn þ Ln 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
We have to consider:
8 " # n1 > 0 P n1 Q > 00 > > ð1 Þ hn hn ln 1 þ maxðlj ; Lj Þ a 6 xn 6 hn h0n ; > > > i¼2 j¼i > > < 0 00 ð2 Þ hn hn 6 xn 6 hn h00n ; > > " # > > > n1 00 00 P n1 Q > 00 > > > : ð3 Þ hn hn 6 xn 6 hn hn þ Ln 1 þ i¼2 j¼i maxðlj ; Lj Þ a:
i¼2
j¼i
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5853
If (100 ) holds, we have:
" # n1 Y n1 X
xn hn h0 6 ln 1 þ maxðl ; L Þ a; j j n i¼2
j¼i
and
x h h0 ¼ max x1 h1 h0 ; . . . ; xn1 hn1 h0 ; xn hn h0
n n n n # " # ) ( " n1 Y n1 n1 Y n1 X X 6 max a; 1 þ maxðlj ; Lj Þ a; ln 1 þ maxðlj ; Lj Þ a i¼2
(" 6 max
1þ
j¼i
n1 Y n1 X i¼2
#
i¼2
"
maxðlj ; Lj Þ a; ln 1 þ
j¼i
n1 Y n1 X
j¼i
i¼2
# )
maxðlj ; Lj Þ a :
j¼i
If (200 ) holds, there exists h0 2 In1 such that xn = hn(h0) and we have:
kx hðh0 Þk ¼ maxfjx1 h1 ðh0 Þj; . . . ; jxn1 hn1 ðh0 Þj; jxn hn ðh0 Þjg ( " # ) " # n1 Y n1 n1 Y n1 X X 6 max a; ½1 þ maxðl2 ; L2 Þa; . . . ; 1 þ maxðlj ; Lj Þ a 6 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
i¼2
j¼i
If the last condition holds, we have:
" # n1 Y n1 X
xn hn h00 6 Ln 1 þ maxðl ; L Þ a: j j n i¼2
j¼i
And therefore:
x h h00 ¼ max x1 h1 h00 ; . . . ; xn1 hn1 h00 ; xn hn h00
n n n n ( " # " # ) n1 Y n1 n1 Y n1 X X 6 max a; 1 þ maxðlj ; Lj Þ a; Ln 1 þ maxðlj ; Lj Þ a (" 6 max
1þ
i¼2
j¼i
n1 Y n1 X i¼2
#
"
maxðlj ; Lj Þ a; Ln 1 þ
j¼i
i¼2
j¼i
n1 Y n1 X i¼2
# )
maxðlj ; Lj Þ a :
j¼i
This means thath the parametrized curve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in X, where i P Qn1 g ¼ maxf1; ln ; Ln g 1 þ n1 h i¼2 j¼i maxðlj ; Lj Þ . Remark 1. In Theorem 2, if we suppose that u2, . . . , un, w2, . . . , wn are Lipschitzian functions with the same Lipschitz constant h i n2 ‘, then the curve h(h) = (h1(h), . . . , hn(h)) is c-dense in X with c ¼ maxf1; ‘g 1 þ ‘ ‘11 a. Suppose that u2, . . . , un, w2, . . . , wn are contracting function relatively to the supremum norm. We obtain the following simpler result: Corollary 1. Let h = (h1, . . . , hn) : A ? X be a continuous function and h2, . . . , hn, a strictly positive numbers such that (i) h1 is surjective. (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, . . . , n} and for all closed interval J, we have
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Then, the curve defined by the parametric representation h(h) = (h1(h), . . . , hn(h)) for h 2 A is (n 1)a-dense in X.
Proof. It is a consequence of Theorem 2, because it suffices to take li = Li = 1 for all i = 2, . . . , n. h Corollary 2. Let h = (h1, h2, . . . , hn) be a function from A to X, whose components are Lipschitz functions with constants c1, c2, . . . , cn (resp.). Let h2, h3, . . . , hn, a be strictly positive numbers such that:
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(i) h1 is surjective, (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( 0 hi hi ¼ ui h1 h0i ; . . . ; hi1 h0i hi h00i ¼ wi h1 h00i ; . . . ; hi1 h00i (iii) For all i 2 {2, . . . , n}
ci1 <
a hi
:
Then, the curve h(h) = (h1(h), h2(h), . . . , hn(h)), for h 2 A, is (n 1)a-dense in X.
Proof. It is a consequence of Theorem 2. The conditions (i) and (ii) of the theorem are satisfied. It remains to check condition (iii). Let i 2 {2, 3, . . . , n} and I an interval of A satisfying l(I) < hi. For, h0 , h00 2 I, we have:
jhi1 ðh00 Þ hi1 ðh0 Þj 6 ci1 jh00 h0 j 6 ci1 lðIÞ; involving:
lðhi1 ðIÞÞ 6 ci1 lðIÞ < ci1 hi < a:
Corollary 3. Consider the compact
X ¼ fðx1 ; x2 Þ 2 R2 = a 6 x1 6 b
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þg h i where u2, w2 are contracting functions and let h ¼ ðh1 ; h2 Þ : 0; ap1 ! X be defined by:
ab aþb cosða1 hÞ þ ; 2 2 u ðh1 ðhÞÞ w2 ðh1 ðhÞÞ u ðh1 ðhÞÞ þ w2 ðh1 ðhÞÞ h2 ðhÞ ¼ 2 cosða2 hÞ þ 2 ; 2 2
h1 ðhÞ ¼
where a1, a2, a are strictly positive numbers such that:
a2 P
pðb aÞ a1 : 2a
Then the parametrized curve h(h) = (h1(h), h2(h)) is a-dense in the compact X. Proof. It is a consequence of Theorem 2. Indeed, (i) h1 is surjective by definition. (ii) If we take h2 ¼ ap2 , then for all interval I of length h2, there exist h0 ,h00 2 I such that:
h2 ðh0 Þ ¼ u2 ðh1 ðh0 ÞÞ;
h2 ðh00 Þ ¼ w2 ðh1 ðh00 ÞÞ: cosða2 h0 Þ ¼ 1 ) cosða2 h00 Þ ¼ 1 ( 0 2p k h ¼ a2 ) ; for k 2 N; h00 ¼ ap2 þ 2ap2k
lðIÞ < ap2 , we have l(h1(I)) < a. Since, 8h0 ; h00 2 I :
a b
ab a1 h00 a1 h0 ðb aÞ ðb aÞp cosða1 h00 Þ cosða1 h0 Þ
6 ðb aÞj sin jh1 ðh00 Þ h1 ðh0 Þj ¼
a1 jh00 h0 j < a1 6 a: j6 2 2 2 2 2a2
(iii) for any interval I satisfying
From Theorem 2, the parametrized curve defined by the function h is a-dense in X. h
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Corollary 4. Consider the compact
(
,
a 6 x1 6 b
ðx1 ; x2 ; . . . ; xn Þ 2 Rn
X¼
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ
...
)
un ðx1 ; x2 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; x2 ; . . . ; xn1 Þ where u2, . . . , un, hw2, . .i. , wn are contracting h ¼ ðh1 ; h2 ; . . . ; hn Þ : 0; ap1 ! X, be defined by:
functions relatively to the supremum norm
and let
the function
ab aþb cosða1 hÞ þ 2 2 u2 ðh1 ðhÞÞ w2 ðh1 ðhÞÞ u ðh1 ðhÞÞ þ w2 ðh1 ðhÞÞ cosða2 hÞ þ 2 h2 ðhÞ ¼ 2 2 ...
h1 ðhÞ ¼
hn ðhÞ ¼
un ðh1 ðhÞ; . . . ; hn1 ðhÞÞ wn ðh1 ðhÞ; . . . ; hn1 ðhÞÞ
cosðan hÞ 2 u ðh1 ðhÞ; . . . ; hn1 ðhÞÞ þ wn ðh1 ðhÞ; . . . ; hn1 ðhÞÞ þ n 2
where a1, . . . , an, a are strictly positive numbers satisfying the relationship:
"
i1 p X ai P 2ðik1Þ ðM k mk Þak 2a k¼1
#
with mk = min uk, Mk = max wk and m1 = a, M1 = b. Then, the curve h(h) = (h1(h), . . . , hn(h)) is (n 1)a-dense in the compact X. Proof. It follows from Theorem 2. Indeed, (1) h1 is surjective by definition. (2) For all i 2 {2, . . . , n} and for all interval I with length hi ¼ api , there exist h0i ¼ 2apik and h00i ¼ api þ 2apik 2 I; k 2 N, such that:
hi ðh0 Þ ¼ ui ðh1 ðh0 Þ; . . . ; hi1 ðh0 ÞÞ; hi ðh00 Þ ¼ wi ðh1 ðh00 Þ; . . . ; hi1 ðh00 ÞÞ:
(3) For all i 2 {2, . . . , n} and for all closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: We proceed by induction on i. For i = 2, the inequality is verified. Assume that the inequality holds for i, i.e. for any closed interval J of length less than ap , we have:
" i1 X
# ðik1Þ M k mk 0 00 00 0 8h ; h 2 J : jhi1 ðh Þ hi1 ðh Þj < 2 ak jh00 h0 j 6 a; 2 k¼1
and let us prove it for i + 1. Consider the closed interval J such that: 0
00
00
0
8h ; h 2 J : jhi ðh Þ hi ðh Þj <
" i X k¼1
2
ðikÞ
ni ðhÞ ¼ ui ðh1 ðhÞ; . . . ; hi1 ðhÞÞ;
ci ðhÞ ¼ wi ðh1 ðhÞ; . . . ; hi1 ðhÞÞ and
hi ðhÞ ¼
p , show that: lðJÞ < aiþ1
# M k mk ak jh00 h0 j 6 a: 2
First we put:
ni ðhÞ ci ðhÞ n ðhÞ þ ci ðhÞ cosðai hÞ þ i : 2 2
i
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D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
For h0 , h00 2 J, we have:
n ðh00 Þ ci ðh00 Þ n ðh00 Þ þ ci ðh00 Þ ni ðh0 Þ ci ðh0 Þ n ðh0 Þ þ ci ðh0 Þ
cosðai h00 Þ þ i cosðai h0 Þ i jhi ðh00 Þ hi ðh0 Þj ¼
i
2 2 2 2
00 00 0 0
ni ðh00 Þ ci ðh00 Þ n ðh Þ þ ci ðh Þ ni ðh Þ ci ðh Þ ¼
cosðai h00 Þ þ i cosðai h0 Þ 2 2 2
n ðh0 Þ þ ci ðh0 Þ ni ðh00 Þ ci ðh00 Þ n ðh00 Þ ci ðh00 Þ i þ cosðai h0 Þ þ i cosðai h0 Þ
2 2 2
n ðh00 Þ ci ðh00 Þ
j cosðai h00 Þ cosðai h0 Þj þ jni ðh00 Þ ni ðh0 Þj þ jc ðh00 Þ c ðh0 Þj 6
i i i
2
M i mi 6 ai jh00 h0 j þ 2 maxfjh1 ðh00 Þ h1 ðh0 Þj; . . . ; jhi1 ðh00 Þ hi1 ðh0 Þjg 2 (since then ui and wi are contracting relatively to the supremum norm).
" "
#
# i1 i X X M i mi p ðik1Þ M k mk ðikÞ M k mk 00 0 00 0 < ai jh h j þ 2 ak jh h j < 2 ak < a: 2 2 2 aiþ1 k¼1 k¼1
Then, the curve defined by the parametric representation h(h) = (h1(h), h2(h), . . . , hn(h)), for h 2 A, is (n 1)a-dense in X. h 2.2. Example of a-dense curves
8 9 , 1 6 x1 6 1 > > < = x21 þ x22 6 1 : Let X ¼ ðx1 ; x2 ; x3 Þ 2 R3 > > : ; x21 þ x22 þ x23 6 1 h i The parametrized curve hðhÞ h 2 0; ap1 defined by:
8 h1 ðhÞ ¼ ð1 aÞ cosða1 hÞ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < h2 ðhÞ ¼ ð1 aÞ cosða2 hÞ 1 ð1 aÞ2 cos2 ða1 hÞ; > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : h ðhÞ ¼ cosða hÞ ð1 ð1 aÞ2 cos2 ða hÞÞð1 ð1 aÞ2 cos2 ða hÞÞ; 3 3 1 2 where a1, a2, a3, a are strictly positive numbers such that:
p a2 ¼ ð1 aÞa1 ; a " # p ð1 aÞ2 a3 ¼ ð1 aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 þ a2 a að2 aÞ
n o aÞ ð1aÞ2 p ffiffiffiffiffiffiffiffiffiffi ffi 1 þ is ca-dense in the compact X, where c ¼ max 1; að1 . ð2aÞ
að2aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Indeed, it follows from Theorem 2. It suffices to take h2 ¼ ap2 ; h3 ¼ ap3 ; u2 ðx1 Þ ¼ ð1 aÞ 1 x21 ; w2 ðx1 Þ ¼ ð1 aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x21 ; u3 ðx1 Þ ¼ 1 x21 x22 ; w3 ðx1 Þ ¼ 1 x21 x22 and (1 a) 6 x1 6 (1 a), then: u2, w2 are Lipschitz functions 2
aÞ ffi ffiffiffiffiffiffiffiffiffiffi with constant pð1 , and u3, w3 are Lipschitzian with constant að2aÞ
ð1aÞ að2aÞ.
Moreover,
(i) h1 is surjective by definition. (ii) For all i 2 {2,3} and for any interval I with length hi, there exist h0i ¼ 2apik ; h00i ¼ api þ 2apik 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, 3}, and for any closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Indeed, "h0 , h00 2 J:
h00 þ h0 h00 h0
sin a1 jh1 ðh00 Þ h1 ðh0 Þj ¼ j ð1 aÞ cosða1 h00 Þ þ ð1 aÞ cosða1 h0 Þj 6
2ð1 aÞ sin a1
2 2 < ð1 aÞa1 h2 ¼ a:
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5857
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8h0 ; h00 2 J : jh2 ðh00 Þ h2 ðh0 Þj ¼ ð1 aÞ
cosða2 h00 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ð1 aÞ
cosða2 h00 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 ð1 aÞ
cosða2 h00 Þ cosða2 h0 Þj þ j 1 ð1 aÞ2 cos2 ða1 h00 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ
! ð1 aÞ3 ð1 aÞ2 00 0 00 0 6 ð1 aÞa2 jh h j þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 jh h j < ð1 aÞ a2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 h3 ¼ a: að2 aÞ að2 aÞ
n o aÞ ð1aÞ2 p ffiffiffiffiffiffiffiffiffiffi ffi 1 þ Hence by Theorem 2, the curve h(h) is ca-dense in X, where c ¼ max 1; að1 . ð2aÞ að2aÞ
3. Applications 3.1. The test functions
Problem 1 (cos 2). Consider the following global optimization problem:
min f ðxÞ ¼ x21 þ x22 x2X
1 ðcosð5px1 Þ þ cosð5px2 ÞÞ; 10
whose X is the set of x ¼ ðx1 ; x2 Þ 2 R2 such that:
1 6 x1 6 1; 1 1 1 1 x1 6 x2 6 x 1 þ : 2 2 2 2 This function has several local minima in the feasible set. The global minimum is f⁄ = 0.2, is attained at x⁄ = (0, 0). Let us associate the a-dense curve h(h) as follows:
(
h1 ðhÞ ¼ cosða1 hÞ; h2 ðhÞ ¼ 12 ðcosða1 hÞ þ cosða2 hÞÞ
with a2 P 2ap a1 (for any value of a1). Our problem can be reduced to the following one-dimensional problem
min f ðhÞ
h2½0;ap 1
h i with f (h) = f(h(h)), for h 2 0; ap1 . Note that the number a is calculated according to the required accuracy e with which our problem is solved numerically. See Section 3.2. ⁄
Problem 2 (Branin function). Consider the global optimization problem
1 min f ðxÞ ¼ 4x21 2:1x41 þ x61 þ x1 x2 4x22 þ 4x42 ; x2X 3 where X is the set of x ¼ ðx1 ; x2 Þ 2 R2 such that:
1 6 x1 6 1; cosðx1 6Þ
1 6 x2 6 lnðx1 þ 2Þ: 3
This function is symmetric about the origin and has more than 03 of local minima in the feasible set. The global minimum f⁄ = 1.031628, is attained at x1 ¼ ð0:08983; 0:7126Þ and x2 = (0.08983, 0.7126). The associated a-dense curve h(h) = (h1(h), h2(h)) is obtained as follows:
h1 ðhÞ ¼ cosða1 hÞ;
1 1 1 1 cosðcosða1 hÞ þ 6Þ þ lnð2 cosða1 hÞÞ þ cosða2 hÞ þ lnð2 cosða1 hÞÞ cosðcosða1 hÞ þ 6Þ ; h2 ðhÞ ¼ 2 3 2 3
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h
i
where h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 3 (Rastrigin function). Consider the following global optimization problem
min f ðxÞ ¼ x21 þ x22 cosð18x1 Þ cosð18x2 Þ; x2X
where X R2 is the set defined by the following constraints:
1 6 x1 6 1; lnðx1 þ 2Þ þ 1=3 6 x2 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ 4:
This function has about 50 local minima in the feasible set. The global minimum f⁄ = 2, is attained at x⁄ = (0, 0). We construct the curve h(h) = (h1(h), h2(h)) in X as follows:
h1 ðhÞ ¼ cosða1 hÞ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 lnð2 cosða1 hÞÞ þ cos2 ða1 hÞ þ 4 cosða2 hÞ þ cos2 ða1 hÞ þ 4 lnð2 cosða1 hÞÞ þ h2 ðhÞ ¼ 2 3 2 3 h i with h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 4 (ExpCos). Consider the following global optimization problem:
min f ðxÞ ¼ exp x2X
2
x1 þ x22 cosð3px1 Þ cosð3px2 Þ; 2
where X is the set given by the following constraints:
1 6 x1 6 1; 1 1 1 1 x21 þ x1 1 6 x2 6 x21 þ x1 þ 1: 4 2 4 2 This function has several local minima in the feasible set. The global minimum f⁄ = 2.4624, is attained at x⁄ = (1, 1.2475). We construct h(h) in X as follows:
h1 ðhÞ ¼ cosða1 hÞ; 1 1 h2 ðhÞ ¼ ðcos2 ða1 hÞ þ 4Þ cosða2 hÞ cosða1 hÞ; 4 2
h i where h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 5 (Cos6). Consider the following global optimization problem:
min f ðxÞ ¼ x2X
X6 1 2 cosð5 x p x Þ ; i i i¼1 10
X being the ball with center 0 and radius 12 of R6 . We construct the curve h(h) in the following compact:
8 9 ð1 aÞ 6 x1 6 ð12 aÞ > > > > , 2 > > q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi < = 1 1 Pk1 2 1 1 Pk1 2 6 6 x for k ¼ 2; . . . ; 5 a x 6 a x k S ¼ ðx1 ; . . . ; x6 Þ 2 R : i¼1 i i¼1 i 2 2 2 2 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > P5 2 P5 2 : ; 1 1 2 i¼1 xi 6 x6 6 2 i¼1 xi This function has several local minima in the feasible set. The global minimum is f⁄ = 0.6, is attained at x⁄ = (0, 0, 0, 0, 0, 0).
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5859
Keeping the same notation than in Corollary 4 and taking:
8 < pa 12 a a1 for i ¼ 2; pffiffiffiP ai ¼ p2i2 1 k : a 2 a a1 þ 2 i1 for i ¼ 3; . . . ; 6; k¼2 2 ak we find that the curve h(h) = (h1(h), . . . , h6(h)) is 5a-dense curve in the compact X given above. 3.2. Application of the mixed method Alienor–Piyavskii Let us consider a multidimensional problem of minimization:
min f ðxÞ;
ð1Þ
x2X
where f is a Lipschitz function and X is the compact set of Corollary 4. To solve the problem (1) with an accuracy e > 0, we construct a parameterized curve h(h) = (h1(h), . . . , hn(h))a-dense in X for h 2 [0, M], where M is the upper bound of the domain of definition of the function h allowing us to a-densify the above feasible set X. The minimization problem (1) is then approximated by the one-dimensional problem:
min f ðhÞ;
ð2Þ
h2A
where f⁄(h) = f(h1(h),h2(h), . . . ,hn(h)) with h 2 [0, M]. Denote by l1 and l2, respectively, the Lipschitz constants of f and h. The mixed Alienor Piyavskii–Shubert method [16,24] is a combination of two steps: the reducing transformation step and the Piyavskii–Shubert algorithm step. (a) First step. According to Corollary 4, we define the function
h ¼ ðh1 ; h2 ; . . . ; hn Þ : 0;
p !X a1
with
a¼
1 e : ðn 1Þ 2l1
The parametrized curve h(h) = (h1(h), h2(h), . . . , hn(h)) is 2le1 -dense in X. (b) Second step. Now, we apply the algorithm of Piyavskii–Shubert to the function f⁄(h) = f(h(h)), which is a Lipschitz function with constant L = l1l2, in order to compute the global minimum of f⁄(h) [15,18]. 1. Initialization
h1 ¼
p 2a1
;
F e ¼ fe
he ¼ h1 ; fe ¼ f ðhe Þ;
Lp ; 2a1
F 1 ðhÞ ¼ f ðh1 Þ Ljh h1 j:
2. For k = 1, 2, . . . if fe F e 6 2e then stop, otherwise, put hkþ1 ¼ arg min h
h2 0;ap
i F k ðhÞ.
1
If f ðhkþ1 Þ < fe , then put fe ¼ f ðhkþ1 Þ; he ¼ hkþ1 . Put Fk+1(h) = maxj=1,. . .,k+1{f⁄(hj) Ljh hjj}, and F e ¼ min F kþ1 ð½0; ap1 Þ. Put k = k + 1. Go to 2. 3.3. Numerical experiments In this subsection, we present numerical results of Problems 1–5 given above. To illustrate the behavior of the algorithm, we perform the following numerical tests with the accuracy e = 0.1 and Table 1 shows the test results for Problems 1–5. The calculations have been made on Intel Pentium (R) 4 CPU. Note that we have consider the one-dimensional Piyavskii–Shubert algorithm for the minimization of the function f⁄. The constraints introduced in Problems 1–5 have been treated for the first time here. Therefore, we cannot compare our results with previous works. However, it is clear that our technique depends of the one-dimensional optimization method which is used. In this work, we have applied the Piyavskii–Shubert method since it is the best known and most studied in the
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Table 1 Table of test results for Problems 1–5. Problem number
Global minimizer
Global minimum
Number of function evaluations
1 2 3 4 5
(0.0048, 0.0010) (0.033, 0.724) (0.015, 0.34) (1, 1.246) (0, 0, 0, 0, 0, 0)
0.1996 1.0172 1.8347 2.4207 0.6
1271 4230 410 257 129
literature, but other methods can be used as well. The most important is that this approach allows us to solve a new class of constrained global optimization problems. 4. Conclusion The results described in this paper have enlarged the class of functions used to generate a-dense curves in an arbitrary compact of Rn [17,21,22]. They provide new possibilities for the reducing transformation Alienor in the case of constrained global optimization [9,10]. We have given sufficient conditions on the components of the parametrization to derive new families of a-dense curves having simpler analytical expressions. This allows the improvement of the calculation time in the Alienor method and so helps to solve global optimization problems arising in industrial applications. References [1] M.M. Ali, Z. Kajee-Bagdadi, Improved versions of the LEDE algorithm for constrained global optimization, Applied Mathematics and Computation 215 (2) (2009) 431–440. [2] H. Ammar, Y. Cherruault, Approximation of a several variables function by a one variable function and applications to global optimization, Mathematical and Computer Modelling 18 (2) (1993) 17–21. [3] O. Bendiab, Y. Cherruault, Divanu: a new method for global optimization in dimension n, International Journal of Bio-Medical Computing 38 (1995) 177–180. [4] Y. Cherruault, Optimisation, Méthodes locales et globales, Presses Universitaire de France, 1999. [5] Y. Cherruault, A. Guillez, Une méthode pour la recherche du minimum global d’une fonctionnelle, C.R.A.S, Paris, Série I 296 (1983) 175–178. [6] Yu. G. Evtushenko, V.U. Malkova, A.A. Stanevichyus, Parallelization of the global extremum searching process, Automation and Remote Contro 68 (5) (2007) 787–798. [7] Yu.G. Evtushenko, V.U. Malkova, A.A. Stanevichyus, Parallel global optimization of functions of several variables, Computational Mathematics and Mathematical Physics 49 (2) (2009) 246–260. [8] H. Jiao, Y. Guo, P. Shen, Global optimization of generalized linear fractional programming with nonlinear constraints, Applied Mathematics and Computation 183 (2) (2006) 717–728. [9] R. Horst, A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization, Journal of Optimization Theory and Applications 51 (2) (1986) 271–291. [10] R. Horst, H. Tuy, Global Optimization, Deterministic Approach, Springer-Verlag, Berlin, 1993. [11] Z.H. Huang, X.H. Miao, P. Wang, A revised cut-peak function method for box constrained continuous global optimization, Applied Mathematics and Computation 194 (1) (2007) 224–233. [12] I.E. Lagaris, I.G. Tsoulos, Stopping rules for box-constrained stochastic global optimization, Applied Mathematics and Computation 197 (2) (2008) 622– 632. [13] G. Mora, Y. Cherruault, Characterization and generation of a-dense curves, Computers & Mathematics with Applications 33 (9) (1997) 83–91. [14] G. Mora, Y. Cherruault, A. Ziadi, Functional equations generating space-densifying curves, Computers & Mathematics with Applications 39 (2000) 45– 55. [15] S.A. Piyavskii, An algorithm for finding the absolute minimum for a function, Theory of Optimal Solutions 2 (1967) 13–24. [16] M. Rahal, A. Ziadi, A new extension of Piyavskii’s method to Hölder functions of several variables, Applied Mathematics and Computation 197 (2008) 478–488. [17] H. Sagan, Space-filling Curves, Springer-Verlag, New York, 1994. [18] B.O. Shubert, A sequential method seeking the global maximum of a function, SIAM, Journal of Numerical Analysis 9 (3) (1972) 379–388. [19] A. Törn, A. Zilinska, Global Optimization, Springer-Verlag, New york, 1988. [20] A. Ziadi, Y. Cherruault, Generation of a-dense Curves in a cube of Rn , Kybernetes 27 (4) (1998) 416–425. [21] A. Ziadi, Y. Cherruault, Generation of a-dense curves and application to global optimization, Kybernetes 29 (1) (2000) 71–82. [22] A. Ziadi, Y. Cherruault, G. Mora, The existence of a-dense curves with minimal length in a metric space, Kybernetes 29 (2) (2000) 219–230. [23] A. Ziadi, Y. Cherruault, G. Mora, Global optimization, a new variant of the alienor method, Computers & Mathematics with Applications 41 (2001) 63– 71. [24] A. Ziadi, D. Guettal, Y. Cherruault, Global optimization: Alienor mixed method with Piyavskii–Shubert technique, Kybernetes 34 (7/8) (2005) 1049– 1058.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Reducing transformation and global optimization Djaouida Guettal ⇑, Abdelkader Ziadi Department of Mathematics, University Ferhat Abbas, Setif 19000, Algeria
a r t i c l e
i n f o
Keywords: Global optimization Constrained optimization Reducing transformation a-Dense curves Piyavskii’s algorithm
a b s t r a c t In this paper, we give new results on the Alienor method of dimension reduction. This technique is used to solve multidimensional global optimization problems of type minx2X f(x) where f is a non convex Lipschitz function and X a compact set of Rn ðn P 2Þ defined by Lipschitz constraints. The idea is to construct an a-dense curve h in the feasible set X. The global minimum of f on X is then approximated by the global minimum of f on the curve h. That is, our problem has become a one-dimensional problem which can be solved by the Piyavskii–Shubert method. Examples of these curves and numerical implementations on several test functions are given. Crown Copyright 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction The reducing transformation Alienor has been proposed in the 80s by Cherruault and Guillez [4,5]. It has been essentially developed for solving global optimization problems of coercive functions without constraints or of functions verifying other conditions and with fairly simple constraints. This occurs for examples when the feasible set is a hyper-rectangle [2,3,13]. This method is based on the generation of an a-dense curve in the feasible set which allows to transform the optimization problem in a hyper-rectangle of Rn into an optimization in a compact interval of R [14,20,23]. In our work, we shall consider a more general case. The equality and inequality constraints define a feasible set which is a more complicated compact. Note that existing multidimensional global optimization methods treat either problems without constraints or with simple constraints such as a polytope, sphere or an ellipsoid [1,6–12,19]. 2. The Alienor method First we will give some notation and definitions. We assume that A is any interval of R (in general A = [0, M], with M > 0), and a is a number strictly positive. Definition 1. We say that a curve of Rn defined by:
h:A!X is a-dense in X, if for all x 2 X, $h 2 A such that d(x, h(h)) 6 a where d is the Euclidian distance in Rn . We want to solve the following global optimization problem:
min f ðxÞ; x2X
⇑ Corresponding author. E-mail address: [email protected] (D. Guettal). 0096-3003/$ - see front matter Crown Copyright 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.053
ð1Þ
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5849
where X is the set defined by the following inequalities:
g i ðx1 ; x2 ; . . . ; xn Þ 6 0;
for i ¼ 1; 2; . . . ; m:
In some cases, the feasible set X can be written (after some transformations) in the following form:
X¼
a 6 x1 6 b; ðx1 ; x2 ; . . . ; xn Þ 2 Rn u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ: .. .
9 =
un ðx1 ; x2 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; x2 ; . . . ; xn1 Þ; ; where u2, . . . , un, w2, . . . , wn are Lipschitz functions with constants (resp.) l2, . . . , ln, L2, . . . , Ln with respect to the supremum norm. If it is possible to construct a parametrized curve h(h) = (h1(h), . . . , hn(h)), a-dense in X, for h 2 A, then using the reducing transformation xj = hj(h), j = 1, 2, . . . , n, we approximate the problem (1) by means of the problem:
min f ðhÞ; h2A
ð2Þ
where f⁄(h) = f(h1(h), h2(h), . . . , hn(h)). The difficulty with Alienor method is the construction of the a-dense curve which depends on the structure of the feasible set X. 2.1. Constructing a-dense curves This section is devoted to the construction of a-dense curves. Consider a function h(h) = (h1(h), h2(h), . . . , hn(h)), defined on an interval A of R and whose values belong to the compact X defined above. We are going to study the relationship to must exist between the components h1, h2, . . . , hn in order that the function h generates an a-dense curve in X. Denote by l the Lebesgue measure. The number a is supposed very small according to the interval [a, b]. For n = 2, we have the following theorem: Theorem 1. Let h = (h1, h2) : A ? X be a continuous function and h2, a be strictly positive numbers such that: (i) h1 is surjective. (ii) For any interval I of length h2, there exist h02 ; h002 2 I such that:
(
h2 h02 ¼ u2 ðh1 h02 Þ; h2 ðh002 Þ ¼ w2 h1 h002 :
(iii) For any interval J of A, we have:
lðJÞ < h2 ) lðh1 ðJÞÞ < a: Then, the parametrized curve defined by h(h) = (h1(h), h2(h)) for all h 2 A, is ga-dense in X, with g = max{1, l2, L2}.
Proof. Let (x1, x2) 2 X. Consider the interval [x1 a, x1 + a]. Because h1 is surjective, there exists a closed interval I1 A such that:
h1 ðI1 Þ ¼ ½x1 a; x1 þ a \ ½a; b; and since l(h1(I1)) P a, then l(I1) P h2. Hence, we have jx1 h1(h)j 6 a for all h 2 I1. Moreover there exist h02 ; h002 2 I1 such that:
(
h2 h02 ¼ u2 h1 h02 ; 00 00 h2 h2 ¼ w2 h1 h2 :
Since
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ; and u2, w2 being Lipschitzian of constants l2, L2 (resp.), then:
h2 h02 l2 a 6 x2 6 h2 h002 þ L2 a:
We thus have to study three cases:
8 0 0 > < ð1Þ h2 h2 l2 a 6 x2< h2 h2 ; 0 00 ð2Þ h2 h2 6 x2 6 h2 h2 ; > : ð3Þ h2 h002 < x2 6 h2 h002 þ L2 a:
5850
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
(i) If (1) holds, we have:
jx2 h2 h02 j 6 l2 a; and therefore
kx h h02 k ¼ max jx1 h1 h02 j; jx2 h2 h02 j 6 maxfa; l2 ag: (ii) If (2) that satisfied, there exists h0 2 h02 ; h002 such that:
x2 ¼ h2 ðh0 Þ; leading to:
kx hðh0 Þk 6 a: (iii) If the last hypothesis holds, we have:
jx2 h2 h002 j 6 L2 a; and therefore
kx h h002 k 6 maxfa; L2 ag: Summing up, the parametrized curve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in X with g = max{1, l2, L2}.
h
For n P 3, if the Lipschitz constants corresponding to u2, . . . , un, w2, . . . , wn are chosen such that the families (li)26i6n and (Li)26i6n are increasing and a less than the length of the interval hi h0i ; hi h00i for all i = 2, . . . , n, then we have the following result. Theorem 2. Let h = (h1, . . . , hn) : A ? X be a continuous function and h2, . . . , hn, a strictly positive numbers such that: (i) h1 is surjective. (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, . . . , n}, and for all closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Then, the parametrized icurve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in h P Qn1 1 þ n1 i¼2 j¼i maxðlj ; Lj Þ .
X, with g ¼ maxf1; ln ; Ln g
Proof. Let x = (x1, . . . , xn) 2 X. Consider the interval [x1 a, x1 + a], since h1 is surjective, there exists a closed interval I1 A such that:
h1 ðI1 Þ ¼ ½x1 a; x1 þ a \ ½a; b: But l(h1(I1)) P a, so that l(I1) P h2. A consequence of that is jx1 h1(h)j 6 a for all h 2 I1. Moreover, there exist h02 ; h002 2 I1 such that:
(
h2 h02 ¼ u2 h1 h02 ; h2 h002 ¼ w2 h1 h002 :
Because h2 h002 h2 h02 P a, we have
l h2 h02 ; h002
P a;
and hence:
l h02 ; h002 P h3 : On the other hand, we have u2(x1) 6 x2 6 w2(x1), and since:
(
u ðx1 Þ u h1 h0 6 l2 x1 h1 h0 6 l2 a; 2 2 2 2
w ðx1 Þ w h1 h00 6 L2 x1 h1 h00 6 L2 a; 2 2 2 2
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
we deduce:
h2 h02 l2 a 6 x2 6 h2 h002 þ L2 a: We still have three cases to study:
8 0 0 > < ð1Þ h2 h2 l2 a 6 x26 h2 h2 ; ð2Þ h2 h02 6 x2 6 h2 h002 ; > : ð3Þ h2 h002 6 x2 6 h2 h002 þ L2 a: If (1) holds, we take I2 ¼ h02 ; h02 þ h3 . 0 00 If (2) is valid, there exists I2 h2 ; h2 such that:
h2 ðI2 Þ ¼ ½x2 a; x2 þ a \ h2 h02 ; h2 h002 :
If the last condition holds, take I2 ¼ h002 h3 ; h002 . 0 we It is clear that I2 h2 ; h002 and l(I2) P h3 in the three cases. Indeed, for (2) we have:
lðh2 ðI2 ÞÞ P a; and hence l(I2) P h3 because of hypothesis (iii). Moreover, for all h 2 I2, we have: (i) if (1) holds, then
jx2 h2 ðhÞj 6 jx2 h2 h02 j þ jh2 h02 h2 ðhÞj 6 ð1 þ l2 Þa: (ii) If (2) holds, then:
jx2 h2 ðhÞj 6 a: (iii) If the last condition holds, then:
jx2 h2 ðhÞj 6 jx2 h2 h002 j þ jh2 h002 h2 ðhÞj 6 ð1 þ L2 Þa: Consequently, for all h 2 I2, we have:
jx2 h2 ðhÞj 6 ð1 þ maxðl2 ; L2 ÞÞa: We have l(I2) P h3, then there exist h03 ; h003 2 I2 such that:
(
h3 h03 ¼ u3 h1 h03 ; h2 h03 ; 00 00 00 h3 h3 ¼ w3 h1 h3 ; h2 h3 ;
hence
00
h3 h h3 h0 P a; 3 3 consequently
l h3 h03 ; h003
P a;
and therefore:
l h02 ; h002 P h4 : On the other hand, we have u3(x1, x2) 6 x3 6 w3(x1, x2) where u3, w3 are Lipschitz functions, such that:
u ðx1 ; x2 Þ u h1 h0 ; h2 h0 6 l3 max x1 h1 h0 ; x2 h2 h0 6 l3 f1 þ maxðl2 ; L2 Þga; 3 3 3 3 3
3 00 00
00
00
w ðx1 ; x2 Þ w h1 h ; h2 h 6 L3 max x1 h1 h ; x2 h2 h 6 L3 f1 þ maxðl2 ; L2 Þga: 3 3 3 3 3 3
Whence
h3 h03 l3 f1 þ maxðl2 ; L2 Þga 6 x3 6 h3 h003 þ L3 f1 þ maxðl2 ; L2 Þga: Three cases arise:
8 0 0 0 > 2 ; L2 Þga 6 x3 6 h3 h3 ; < ð1 Þ h3 h3 l3 f1 þ maxðl ð20 Þ h3 h03 6 x3 6 h3 h003 ; > 00 : 0 ð3 Þ h3 h3 6 x3 6 h3 h003 þ L3 f1 þ maxðl2 ; L2 Þga: If (10 ) is valid we take I3 ¼ h03 ; h03 þ h4 .
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If (20 ) is satisfied, there exists I3 h03 ; h003 such that:
h3 ðI3 Þ ¼ ½x3 a; x3 þ a \ h3 h03 ; h3 h003 :
00 00 If the last condition holds, 0 we00 take I3 ¼ h3 h4 ; h3 . Hence, we have I3 h3 ; h3 and l(I3) P h4 in all cases. Moreover, for any h 2 I3 we have: (1) If (10 ) holds, we deduce:
jx3 h3 ðhÞj 6 jx3 h3 h03 j þ jh3 h03 h3 ðhÞj 6 ½1 þ l3 ð1 þ maxðl2 ; L2 ÞÞa: (2) If (20 ) is satisfied, then:
jx3 h3 ðhÞj 6 a: (3) If the last condition holds, we have:
jx3 h3 ðhÞj 6 jx3 h3 h003 j þ jh3 h003 h3 ðhÞj 6 ½1 þ L3 ð1 þ maxðl2 ; L2 ÞÞa: In all cases, we have for every h 2 I3,
jx3 h3 ðhÞj 6 f1 þ maxðl3 ; L3 Þ½1 þ maxðl2 ; L2 Þga; that is to say:
jx3 h3 ðhÞj 6 f1 þ maxðl3 ; L3 Þ þ maxðl3 ; L3 Þ maxðl2 ; L2 Þga: Proceeding like that up to the step (n 1), we obtain for all h 2 In1,
jxn1 hn1 ðhÞj 6 ½1 þ
n1 Y n1 X i¼2
maxðlj ; Lj Þa
j¼i
with l(In1) P hn and In1 In2 I1 A. Therefore, there exist h0n ; h00n 2 In1 such that:
(
hn h0n ¼ un h1 h0n ; . . . ; hn1 h0n ; hn h00n ¼ wn h1 h00n ; . . . ; hn1 h00n :
On the other hand, we have:
un ðx1 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; . . . ; xn1 Þ; and
u ðx1 ; . . . ; xn1 Þ u h1 h0 ; . . . ; hn1 h0 6 ln max x1 h1 h0 ; . . . ; xn1 hn1 h0
n n n n n n " # n1 Y n1 X 6 ln 1 þ maxðlj ; Lj Þ a i¼2
j¼i
w ðx1 ; . . . ; xn1 Þ w h1 h00 ; . . . ; hn1 h00 6 Ln max x1 h1 h00 ; . . . ; xn1 hn1 h00
n n n n n n " # n1 Y n1 X 6 Ln 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
Hence
" # " # n1 Y n1 n1 Y n1 X X 0 00 hn hn ln 1 þ maxðlj ; Lj Þ a 6 xn 6 hn hn þ Ln 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
We have to consider:
8 " # n1 > 0 P n1 Q > 00 > > ð1 Þ hn hn ln 1 þ maxðlj ; Lj Þ a 6 xn 6 hn h0n ; > > > i¼2 j¼i > > < 0 00 ð2 Þ hn hn 6 xn 6 hn h00n ; > > " # > > > n1 00 00 P n1 Q > 00 > > > : ð3 Þ hn hn 6 xn 6 hn hn þ Ln 1 þ i¼2 j¼i maxðlj ; Lj Þ a:
i¼2
j¼i
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5853
If (100 ) holds, we have:
" # n1 Y n1 X
xn hn h0 6 ln 1 þ maxðl ; L Þ a; j j n i¼2
j¼i
and
x h h0 ¼ max x1 h1 h0 ; . . . ; xn1 hn1 h0 ; xn hn h0
n n n n # " # ) ( " n1 Y n1 n1 Y n1 X X 6 max a; 1 þ maxðlj ; Lj Þ a; ln 1 þ maxðlj ; Lj Þ a i¼2
(" 6 max
1þ
j¼i
n1 Y n1 X i¼2
#
i¼2
"
maxðlj ; Lj Þ a; ln 1 þ
j¼i
n1 Y n1 X
j¼i
i¼2
# )
maxðlj ; Lj Þ a :
j¼i
If (200 ) holds, there exists h0 2 In1 such that xn = hn(h0) and we have:
kx hðh0 Þk ¼ maxfjx1 h1 ðh0 Þj; . . . ; jxn1 hn1 ðh0 Þj; jxn hn ðh0 Þjg ( " # ) " # n1 Y n1 n1 Y n1 X X 6 max a; ½1 þ maxðl2 ; L2 Þa; . . . ; 1 þ maxðlj ; Lj Þ a 6 1 þ maxðlj ; Lj Þ a: i¼2
j¼i
i¼2
j¼i
If the last condition holds, we have:
" # n1 Y n1 X
xn hn h00 6 Ln 1 þ maxðl ; L Þ a: j j n i¼2
j¼i
And therefore:
x h h00 ¼ max x1 h1 h00 ; . . . ; xn1 hn1 h00 ; xn hn h00
n n n n ( " # " # ) n1 Y n1 n1 Y n1 X X 6 max a; 1 þ maxðlj ; Lj Þ a; Ln 1 þ maxðlj ; Lj Þ a (" 6 max
1þ
i¼2
j¼i
n1 Y n1 X i¼2
#
"
maxðlj ; Lj Þ a; Ln 1 þ
j¼i
i¼2
j¼i
n1 Y n1 X i¼2
# )
maxðlj ; Lj Þ a :
j¼i
This means thath the parametrized curve defined by h(h) = (h1(h), . . . , hn(h)) for h 2 A, is ga-dense in X, where i P Qn1 g ¼ maxf1; ln ; Ln g 1 þ n1 h i¼2 j¼i maxðlj ; Lj Þ . Remark 1. In Theorem 2, if we suppose that u2, . . . , un, w2, . . . , wn are Lipschitzian functions with the same Lipschitz constant h i n2 ‘, then the curve h(h) = (h1(h), . . . , hn(h)) is c-dense in X with c ¼ maxf1; ‘g 1 þ ‘ ‘11 a. Suppose that u2, . . . , un, w2, . . . , wn are contracting function relatively to the supremum norm. We obtain the following simpler result: Corollary 1. Let h = (h1, . . . , hn) : A ? X be a continuous function and h2, . . . , hn, a strictly positive numbers such that (i) h1 is surjective. (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, . . . , n} and for all closed interval J, we have
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Then, the curve defined by the parametric representation h(h) = (h1(h), . . . , hn(h)) for h 2 A is (n 1)a-dense in X.
Proof. It is a consequence of Theorem 2, because it suffices to take li = Li = 1 for all i = 2, . . . , n. h Corollary 2. Let h = (h1, h2, . . . , hn) be a function from A to X, whose components are Lipschitz functions with constants c1, c2, . . . , cn (resp.). Let h2, h3, . . . , hn, a be strictly positive numbers such that:
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D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
(i) h1 is surjective, (ii) For all i 2 {2, . . . , n} and for all interval I with length hi, there exist h0i and h00i 2 I such that:
( 0 hi hi ¼ ui h1 h0i ; . . . ; hi1 h0i hi h00i ¼ wi h1 h00i ; . . . ; hi1 h00i (iii) For all i 2 {2, . . . , n}
ci1 <
a hi
:
Then, the curve h(h) = (h1(h), h2(h), . . . , hn(h)), for h 2 A, is (n 1)a-dense in X.
Proof. It is a consequence of Theorem 2. The conditions (i) and (ii) of the theorem are satisfied. It remains to check condition (iii). Let i 2 {2, 3, . . . , n} and I an interval of A satisfying l(I) < hi. For, h0 , h00 2 I, we have:
jhi1 ðh00 Þ hi1 ðh0 Þj 6 ci1 jh00 h0 j 6 ci1 lðIÞ; involving:
lðhi1 ðIÞÞ 6 ci1 lðIÞ < ci1 hi < a:
Corollary 3. Consider the compact
X ¼ fðx1 ; x2 Þ 2 R2 = a 6 x1 6 b
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þg h i where u2, w2 are contracting functions and let h ¼ ðh1 ; h2 Þ : 0; ap1 ! X be defined by:
ab aþb cosða1 hÞ þ ; 2 2 u ðh1 ðhÞÞ w2 ðh1 ðhÞÞ u ðh1 ðhÞÞ þ w2 ðh1 ðhÞÞ h2 ðhÞ ¼ 2 cosða2 hÞ þ 2 ; 2 2
h1 ðhÞ ¼
where a1, a2, a are strictly positive numbers such that:
a2 P
pðb aÞ a1 : 2a
Then the parametrized curve h(h) = (h1(h), h2(h)) is a-dense in the compact X. Proof. It is a consequence of Theorem 2. Indeed, (i) h1 is surjective by definition. (ii) If we take h2 ¼ ap2 , then for all interval I of length h2, there exist h0 ,h00 2 I such that:
h2 ðh0 Þ ¼ u2 ðh1 ðh0 ÞÞ;
h2 ðh00 Þ ¼ w2 ðh1 ðh00 ÞÞ: cosða2 h0 Þ ¼ 1 ) cosða2 h00 Þ ¼ 1 ( 0 2p k h ¼ a2 ) ; for k 2 N; h00 ¼ ap2 þ 2ap2k
lðIÞ < ap2 , we have l(h1(I)) < a. Since, 8h0 ; h00 2 I :
a b
ab a1 h00 a1 h0 ðb aÞ ðb aÞp cosða1 h00 Þ cosða1 h0 Þ
6 ðb aÞj sin jh1 ðh00 Þ h1 ðh0 Þj ¼
a1 jh00 h0 j < a1 6 a: j6 2 2 2 2 2a2
(iii) for any interval I satisfying
From Theorem 2, the parametrized curve defined by the function h is a-dense in X. h
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D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
Corollary 4. Consider the compact
(
,
a 6 x1 6 b
ðx1 ; x2 ; . . . ; xn Þ 2 Rn
X¼
u2 ðx1 Þ 6 x2 6 w2 ðx1 Þ
...
)
un ðx1 ; x2 ; . . . ; xn1 Þ 6 xn 6 wn ðx1 ; x2 ; . . . ; xn1 Þ where u2, . . . , un, hw2, . .i. , wn are contracting h ¼ ðh1 ; h2 ; . . . ; hn Þ : 0; ap1 ! X, be defined by:
functions relatively to the supremum norm
and let
the function
ab aþb cosða1 hÞ þ 2 2 u2 ðh1 ðhÞÞ w2 ðh1 ðhÞÞ u ðh1 ðhÞÞ þ w2 ðh1 ðhÞÞ cosða2 hÞ þ 2 h2 ðhÞ ¼ 2 2 ...
h1 ðhÞ ¼
hn ðhÞ ¼
un ðh1 ðhÞ; . . . ; hn1 ðhÞÞ wn ðh1 ðhÞ; . . . ; hn1 ðhÞÞ
cosðan hÞ 2 u ðh1 ðhÞ; . . . ; hn1 ðhÞÞ þ wn ðh1 ðhÞ; . . . ; hn1 ðhÞÞ þ n 2
where a1, . . . , an, a are strictly positive numbers satisfying the relationship:
"
i1 p X ai P 2ðik1Þ ðM k mk Þak 2a k¼1
#
with mk = min uk, Mk = max wk and m1 = a, M1 = b. Then, the curve h(h) = (h1(h), . . . , hn(h)) is (n 1)a-dense in the compact X. Proof. It follows from Theorem 2. Indeed, (1) h1 is surjective by definition. (2) For all i 2 {2, . . . , n} and for all interval I with length hi ¼ api , there exist h0i ¼ 2apik and h00i ¼ api þ 2apik 2 I; k 2 N, such that:
hi ðh0 Þ ¼ ui ðh1 ðh0 Þ; . . . ; hi1 ðh0 ÞÞ; hi ðh00 Þ ¼ wi ðh1 ðh00 Þ; . . . ; hi1 ðh00 ÞÞ:
(3) For all i 2 {2, . . . , n} and for all closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: We proceed by induction on i. For i = 2, the inequality is verified. Assume that the inequality holds for i, i.e. for any closed interval J of length less than ap , we have:
" i1 X
# ðik1Þ M k mk 0 00 00 0 8h ; h 2 J : jhi1 ðh Þ hi1 ðh Þj < 2 ak jh00 h0 j 6 a; 2 k¼1
and let us prove it for i + 1. Consider the closed interval J such that: 0
00
00
0
8h ; h 2 J : jhi ðh Þ hi ðh Þj <
" i X k¼1
2
ðikÞ
ni ðhÞ ¼ ui ðh1 ðhÞ; . . . ; hi1 ðhÞÞ;
ci ðhÞ ¼ wi ðh1 ðhÞ; . . . ; hi1 ðhÞÞ and
hi ðhÞ ¼
p , show that: lðJÞ < aiþ1
# M k mk ak jh00 h0 j 6 a: 2
First we put:
ni ðhÞ ci ðhÞ n ðhÞ þ ci ðhÞ cosðai hÞ þ i : 2 2
i
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D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
For h0 , h00 2 J, we have:
n ðh00 Þ ci ðh00 Þ n ðh00 Þ þ ci ðh00 Þ ni ðh0 Þ ci ðh0 Þ n ðh0 Þ þ ci ðh0 Þ
cosðai h00 Þ þ i cosðai h0 Þ i jhi ðh00 Þ hi ðh0 Þj ¼
i
2 2 2 2
00 00 0 0
ni ðh00 Þ ci ðh00 Þ n ðh Þ þ ci ðh Þ ni ðh Þ ci ðh Þ ¼
cosðai h00 Þ þ i cosðai h0 Þ 2 2 2
n ðh0 Þ þ ci ðh0 Þ ni ðh00 Þ ci ðh00 Þ n ðh00 Þ ci ðh00 Þ i þ cosðai h0 Þ þ i cosðai h0 Þ
2 2 2
n ðh00 Þ ci ðh00 Þ
j cosðai h00 Þ cosðai h0 Þj þ jni ðh00 Þ ni ðh0 Þj þ jc ðh00 Þ c ðh0 Þj 6
i i i
2
M i mi 6 ai jh00 h0 j þ 2 maxfjh1 ðh00 Þ h1 ðh0 Þj; . . . ; jhi1 ðh00 Þ hi1 ðh0 Þjg 2 (since then ui and wi are contracting relatively to the supremum norm).
" "
#
# i1 i X X M i mi p ðik1Þ M k mk ðikÞ M k mk 00 0 00 0 < ai jh h j þ 2 ak jh h j < 2 ak < a: 2 2 2 aiþ1 k¼1 k¼1
Then, the curve defined by the parametric representation h(h) = (h1(h), h2(h), . . . , hn(h)), for h 2 A, is (n 1)a-dense in X. h 2.2. Example of a-dense curves
8 9 , 1 6 x1 6 1 > > < = x21 þ x22 6 1 : Let X ¼ ðx1 ; x2 ; x3 Þ 2 R3 > > : ; x21 þ x22 þ x23 6 1 h i The parametrized curve hðhÞ h 2 0; ap1 defined by:
8 h1 ðhÞ ¼ ð1 aÞ cosða1 hÞ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < h2 ðhÞ ¼ ð1 aÞ cosða2 hÞ 1 ð1 aÞ2 cos2 ða1 hÞ; > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : h ðhÞ ¼ cosða hÞ ð1 ð1 aÞ2 cos2 ða hÞÞð1 ð1 aÞ2 cos2 ða hÞÞ; 3 3 1 2 where a1, a2, a3, a are strictly positive numbers such that:
p a2 ¼ ð1 aÞa1 ; a " # p ð1 aÞ2 a3 ¼ ð1 aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 þ a2 a að2 aÞ
n o aÞ ð1aÞ2 p ffiffiffiffiffiffiffiffiffiffi ffi 1 þ is ca-dense in the compact X, where c ¼ max 1; að1 . ð2aÞ
að2aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Indeed, it follows from Theorem 2. It suffices to take h2 ¼ ap2 ; h3 ¼ ap3 ; u2 ðx1 Þ ¼ ð1 aÞ 1 x21 ; w2 ðx1 Þ ¼ ð1 aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x21 ; u3 ðx1 Þ ¼ 1 x21 x22 ; w3 ðx1 Þ ¼ 1 x21 x22 and (1 a) 6 x1 6 (1 a), then: u2, w2 are Lipschitz functions 2
aÞ ffi ffiffiffiffiffiffiffiffiffiffi with constant pð1 , and u3, w3 are Lipschitzian with constant að2aÞ
ð1aÞ að2aÞ.
Moreover,
(i) h1 is surjective by definition. (ii) For all i 2 {2,3} and for any interval I with length hi, there exist h0i ¼ 2apik ; h00i ¼ api þ 2apik 2 I such that:
( hi h0i ¼ ui h1 h0i ; . . . ; hi1 h0i ; 00 00 hi hi ¼ wi h1 hi ; . . . ; hi1 h00i :
(iii) For all i 2 {2, 3}, and for any closed interval J, we have:
lðJÞ < hi ) lðhi1 ðJÞÞ < a: Indeed, "h0 , h00 2 J:
h00 þ h0 h00 h0
sin a1 jh1 ðh00 Þ h1 ðh0 Þj ¼ j ð1 aÞ cosða1 h00 Þ þ ð1 aÞ cosða1 h0 Þj 6
2ð1 aÞ sin a1
2 2 < ð1 aÞa1 h2 ¼ a:
D. Guettal, A. Ziadi / Applied Mathematics and Computation 218 (2012) 5848–5860
5857
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8h0 ; h00 2 J : jh2 ðh00 Þ h2 ðh0 Þj ¼ ð1 aÞ
cosða2 h00 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ð1 aÞ
cosða2 h00 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ þ cosða2 h0 Þ 1 ð1 aÞ2 cos2 ða1 h00 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 ð1 aÞ
cosða2 h00 Þ cosða2 h0 Þj þ j 1 ð1 aÞ2 cos2 ða1 h00 Þ 1 ð1 aÞ2 cos2 ða1 h0 Þ
! ð1 aÞ3 ð1 aÞ2 00 0 00 0 6 ð1 aÞa2 jh h j þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 jh h j < ð1 aÞ a2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 h3 ¼ a: að2 aÞ að2 aÞ
n o aÞ ð1aÞ2 p ffiffiffiffiffiffiffiffiffiffi ffi 1 þ Hence by Theorem 2, the curve h(h) is ca-dense in X, where c ¼ max 1; að1 . ð2aÞ að2aÞ
3. Applications 3.1. The test functions
Problem 1 (cos 2). Consider the following global optimization problem:
min f ðxÞ ¼ x21 þ x22 x2X
1 ðcosð5px1 Þ þ cosð5px2 ÞÞ; 10
whose X is the set of x ¼ ðx1 ; x2 Þ 2 R2 such that:
1 6 x1 6 1; 1 1 1 1 x1 6 x2 6 x 1 þ : 2 2 2 2 This function has several local minima in the feasible set. The global minimum is f⁄ = 0.2, is attained at x⁄ = (0, 0). Let us associate the a-dense curve h(h) as follows:
(
h1 ðhÞ ¼ cosða1 hÞ; h2 ðhÞ ¼ 12 ðcosða1 hÞ þ cosða2 hÞÞ
with a2 P 2ap a1 (for any value of a1). Our problem can be reduced to the following one-dimensional problem
min f ðhÞ
h2½0;ap 1
h i with f (h) = f(h(h)), for h 2 0; ap1 . Note that the number a is calculated according to the required accuracy e with which our problem is solved numerically. See Section 3.2. ⁄
Problem 2 (Branin function). Consider the global optimization problem
1 min f ðxÞ ¼ 4x21 2:1x41 þ x61 þ x1 x2 4x22 þ 4x42 ; x2X 3 where X is the set of x ¼ ðx1 ; x2 Þ 2 R2 such that:
1 6 x1 6 1; cosðx1 6Þ
1 6 x2 6 lnðx1 þ 2Þ: 3
This function is symmetric about the origin and has more than 03 of local minima in the feasible set. The global minimum f⁄ = 1.031628, is attained at x1 ¼ ð0:08983; 0:7126Þ and x2 = (0.08983, 0.7126). The associated a-dense curve h(h) = (h1(h), h2(h)) is obtained as follows:
h1 ðhÞ ¼ cosða1 hÞ;
1 1 1 1 cosðcosða1 hÞ þ 6Þ þ lnð2 cosða1 hÞÞ þ cosða2 hÞ þ lnð2 cosða1 hÞÞ cosðcosða1 hÞ þ 6Þ ; h2 ðhÞ ¼ 2 3 2 3
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h
i
where h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 3 (Rastrigin function). Consider the following global optimization problem
min f ðxÞ ¼ x21 þ x22 cosð18x1 Þ cosð18x2 Þ; x2X
where X R2 is the set defined by the following constraints:
1 6 x1 6 1; lnðx1 þ 2Þ þ 1=3 6 x2 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ 4:
This function has about 50 local minima in the feasible set. The global minimum f⁄ = 2, is attained at x⁄ = (0, 0). We construct the curve h(h) = (h1(h), h2(h)) in X as follows:
h1 ðhÞ ¼ cosða1 hÞ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 lnð2 cosða1 hÞÞ þ cos2 ða1 hÞ þ 4 cosða2 hÞ þ cos2 ða1 hÞ þ 4 lnð2 cosða1 hÞÞ þ h2 ðhÞ ¼ 2 3 2 3 h i with h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 4 (ExpCos). Consider the following global optimization problem:
min f ðxÞ ¼ exp x2X
2
x1 þ x22 cosð3px1 Þ cosð3px2 Þ; 2
where X is the set given by the following constraints:
1 6 x1 6 1; 1 1 1 1 x21 þ x1 1 6 x2 6 x21 þ x1 þ 1: 4 2 4 2 This function has several local minima in the feasible set. The global minimum f⁄ = 2.4624, is attained at x⁄ = (1, 1.2475). We construct h(h) in X as follows:
h1 ðhÞ ¼ cosða1 hÞ; 1 1 h2 ðhÞ ¼ ðcos2 ða1 hÞ þ 4Þ cosða2 hÞ cosða1 hÞ; 4 2
h i where h 2 0; ap1 , and a1, a2 are strictly positive numbers such that:
a2 P
2p
a
a1 :
Problem 5 (Cos6). Consider the following global optimization problem:
min f ðxÞ ¼ x2X
X6 1 2 cosð5 x p x Þ ; i i i¼1 10
X being the ball with center 0 and radius 12 of R6 . We construct the curve h(h) in the following compact:
8 9 ð1 aÞ 6 x1 6 ð12 aÞ > > > > , 2 > > q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi < = 1 1 Pk1 2 1 1 Pk1 2 6 6 x for k ¼ 2; . . . ; 5 a x 6 a x k S ¼ ðx1 ; . . . ; x6 Þ 2 R : i¼1 i i¼1 i 2 2 2 2 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > P5 2 P5 2 : ; 1 1 2 i¼1 xi 6 x6 6 2 i¼1 xi This function has several local minima in the feasible set. The global minimum is f⁄ = 0.6, is attained at x⁄ = (0, 0, 0, 0, 0, 0).
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5859
Keeping the same notation than in Corollary 4 and taking:
8 < pa 12 a a1 for i ¼ 2; pffiffiffiP ai ¼ p2i2 1 k : a 2 a a1 þ 2 i1 for i ¼ 3; . . . ; 6; k¼2 2 ak we find that the curve h(h) = (h1(h), . . . , h6(h)) is 5a-dense curve in the compact X given above. 3.2. Application of the mixed method Alienor–Piyavskii Let us consider a multidimensional problem of minimization:
min f ðxÞ;
ð1Þ
x2X
where f is a Lipschitz function and X is the compact set of Corollary 4. To solve the problem (1) with an accuracy e > 0, we construct a parameterized curve h(h) = (h1(h), . . . , hn(h))a-dense in X for h 2 [0, M], where M is the upper bound of the domain of definition of the function h allowing us to a-densify the above feasible set X. The minimization problem (1) is then approximated by the one-dimensional problem:
min f ðhÞ;
ð2Þ
h2A
where f⁄(h) = f(h1(h),h2(h), . . . ,hn(h)) with h 2 [0, M]. Denote by l1 and l2, respectively, the Lipschitz constants of f and h. The mixed Alienor Piyavskii–Shubert method [16,24] is a combination of two steps: the reducing transformation step and the Piyavskii–Shubert algorithm step. (a) First step. According to Corollary 4, we define the function
h ¼ ðh1 ; h2 ; . . . ; hn Þ : 0;
p !X a1
with
a¼
1 e : ðn 1Þ 2l1
The parametrized curve h(h) = (h1(h), h2(h), . . . , hn(h)) is 2le1 -dense in X. (b) Second step. Now, we apply the algorithm of Piyavskii–Shubert to the function f⁄(h) = f(h(h)), which is a Lipschitz function with constant L = l1l2, in order to compute the global minimum of f⁄(h) [15,18]. 1. Initialization
h1 ¼
p 2a1
;
F e ¼ fe
he ¼ h1 ; fe ¼ f ðhe Þ;
Lp ; 2a1
F 1 ðhÞ ¼ f ðh1 Þ Ljh h1 j:
2. For k = 1, 2, . . . if fe F e 6 2e then stop, otherwise, put hkþ1 ¼ arg min h
h2 0;ap
i F k ðhÞ.
1
If f ðhkþ1 Þ < fe , then put fe ¼ f ðhkþ1 Þ; he ¼ hkþ1 . Put Fk+1(h) = maxj=1,. . .,k+1{f⁄(hj) Ljh hjj}, and F e ¼ min F kþ1 ð½0; ap1 Þ. Put k = k + 1. Go to 2. 3.3. Numerical experiments In this subsection, we present numerical results of Problems 1–5 given above. To illustrate the behavior of the algorithm, we perform the following numerical tests with the accuracy e = 0.1 and Table 1 shows the test results for Problems 1–5. The calculations have been made on Intel Pentium (R) 4 CPU. Note that we have consider the one-dimensional Piyavskii–Shubert algorithm for the minimization of the function f⁄. The constraints introduced in Problems 1–5 have been treated for the first time here. Therefore, we cannot compare our results with previous works. However, it is clear that our technique depends of the one-dimensional optimization method which is used. In this work, we have applied the Piyavskii–Shubert method since it is the best known and most studied in the
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Table 1 Table of test results for Problems 1–5. Problem number
Global minimizer
Global minimum
Number of function evaluations
1 2 3 4 5
(0.0048, 0.0010) (0.033, 0.724) (0.015, 0.34) (1, 1.246) (0, 0, 0, 0, 0, 0)
0.1996 1.0172 1.8347 2.4207 0.6
1271 4230 410 257 129
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