COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
A Continuous Approach to Discrete Structural Optimization
T. Tan*, X. S. Li
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology,
Dalian, 116024 China
Email: [email protected], [email protected]
Abstract In this paper we present a continuous approach to solving discrete optimum design, in which the discrete
variables are represented by the linear combination of components of the discrete set with 0-1 coefficients as new
variables. With this approach, the original problem is converted to a completely equivalent continuous optimization
problem. In the implementation, each binary condition is reduced to an equation by means of so-called binary entropy
function. As such, the discrete structural optimization problem can be solved by standard optimization software. It
should be noted that the proposed approach is both mathematically rigorous and easily implemented in engineering
practice. Numerical computations give promising results.
Key words: discrete optimization, continuous approach, 0-1 programming, binary entropy function
INTRODUCTION
Studies on structural optimum design with continuous variables have become quite mature. In many practical
applications, however, the design variables must be selected from a discrete set. This has presented great difficulties.
Although various methods [1, 2] have been developed to deal with this difficult problem, there is no one that is
completely satisfactory when considering efficiency, robustness and solution quality, especially for large-scale
applications.
Recently, continuous approach has been a new trend in the field of discrete optimization field. Our approach proposed
in this paper is different from early continuous relaxation or simple rounding. It first reformulates the original problem
as a 0-1 programming and then replaces discrete set of 0-1 variables by binary entropy function such that the discrete
optimization problem is reduced to an equivalent continuous model. As such, the original problem becomes solvable
by means of standard software for continuous optimization. It should be noted that our approach has rigorous
mathematical basis. Although Templeman and Yates [ 3-5] gave an identical mathematical form of 0-1
programming in their segmental method for the minimum weight design of trusses, the solution methodology is
completely different.
The method for dealing with 0-1 variables given in this paper is very general and applicable to other usages. Numerical
example illustrates the proposed approach.
MATHEMATICAL MODEL OF DISCRETE OPTIMUM DESIGN
The mathematical model of the discrete optimization is generally defined as ( P1) :
min f ( x )
(1)
s.t. g j ( x ) ≤ 0, j = 1,2,", m
(2)
x∈ X
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where f ( x ) is the objective function to be minimized, g ( x ) represents the set of behavioral constraints, and denotes
the set of discrete design variables defined by
{
}
xi ∈ Si = si1 ," , siRi , i = 1, 2," , n
(3)
in which Si stands for a specified discrete set for xi .
Because of the non-convexity and non-differentiability, ( P1) is difficult to be solved directly by means of standard
mathematical programming techniques. To overcome these difficulties, we replace the constraints (3) by a linear
combination:
Ri
xi = ∑ Sirδ ir , i = 1,", n
(4)
r =1
in terms of a set of new variables δ ir (1 ≤ r ≤ Ri ) where δ ir ∈ {0,1} and
reformulated as an equivalent 0-1 programming ( P 2 ) :
Ri
∑δ
i =1
ir
= 1 . With this representation, ( P1) is
min f (δ )
(5)
s.t. g j (δ ) ≤ 0, j = 1, 2,", m
(6)
Ri
∑δ
r =1
ir
= 1, i = 1, 2,", n ;
(7)
δ ir ∈ {0,1} , i = 1, 2," , n, r = 1, 2," , Ri
(8)
It is interesting to note that this model coincides with the segmental model [4]. However, our approach to dealing with
this 0-1 programming is based on rigorous mathematics.
CONTINUOUS APPROACH TO 0-1 PROGRAMMING BASED ON BINARY ENTROPY
The concept of Shannon's entropy is the central role of information theory sometimes referred as measure
of uncertainty. The entropy of a random variable is defined in terms of its probability distribution and can be shown to
be a good measure of randomness or uncertainty. Shannon entropy is defined by
n
H ( p ) = H ( p1 , p2 ," , pn ) = −∑ pi log pi
(9)
i =1
p log p = 0 .
whereas pi is the possibility of a chance event. Define that 0log 0 = 0 because of the limited lim
p →0
Obviously, if the possibility equals to 0, it doesn’t have the contribution to the information entropy.
If X denotes the random defined in the following
⎧1
X =⎨
⎩0
with probability of p
(10)
with probability of 1 - p
and log is to base 2, then the information entropy according to random variable X is defined by
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H ( X ) = − p log p − (1 − p ) log (1 − p )
(11)
The quantity H ( X ) is called binary entropy. The shape of binary entropy is plotted in Figure 1.
H (X )
1
0.5
1
p
Figure 1 Binary entropy function
From the definition and the shape of binary entropy function, we can carry out some properties of the function
(1) H ( X ) ≥ 0 with the equality iff pi = 1 for some i .
(2) H ( X ) ≤ 1 with equality iff pi = 0.5 for all i .
(3) H ( X ) is a concave function.
Based on the concept of binary entropy function, we can carry out the continuous approach to discrete optimization.
In the derivation of our approach, it owes to the information carried out by relaxation problem ( P 3)
min f (δ )
(5)
s.t. g j (δ ) ≤ 0, j = 1, 2,", m
(6)
Ri
∑δ
r =1
= 1, i = 1, 2,", n
ir
(7)
0 ≤ δ ir ≤ 1, i = 1, 2,", n, r = 1, 2,", Ri
(12)
If denote δˆ is the solution of the relaxation problem ( P 2 ) and δ of problem ( P1) as random variables, then we can
treat δˆi as the possibility of event δ i = 1 and 1 − δˆi as the possibility of event δ i = 0 , that is
(
)
P (δ i = 1) = δˆi
(13)
P (δ i = 0 ) = 1 − δˆi
(14)
Obviously, by means of this hypothesis about the relationship between with the relaxation solution and the discrete
solution, the information entropy of the system can be utilized to measure the values of variables in the iteration. For
every i the corresponding binary entropy is described as
(
) (
H (δ i ) = −δˆi log δˆi − 1 − δˆi log 1 − δˆi
)
(15)
From the properties of binary entropy function, we know: for the one thing, if δˆi = 0 or δˆi = 1 , that is to say the
random variables are determinate other than stochastic, these doesn’t exist any uncertainty, so H (δ i ) = 0 ; for another
thing, if δˆ = 0.5 , that is to say the random variables are indeterminate, so H (δ i ) gets its maximum.
i
From above analysis , we can draw the following point:
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According to this characteristic, combined the relaxation problem with the constraint of binary entropy function, we
can establish an equivalent optimization problem ( P 4 )
min f (δ )
(5)
s.t. g j (δ ) ≤ 0, j = 1, 2,", m
(6)
Ri
∑δ
r =1
ir
= 1, i = 1, 2,", n ;
(7)
− δ ir log δ ir − (1 − δ ir ) log (1 − δ ir ) = 0, i = 1, 2,", n, r = 1, 2,", Ri
(16)
Thus we come to a smooth nonlinear optimization problem with continuous variables δ .
ALGORITHMIC IMPLEMENTATION OF CONTINUOUS APPROACH
Appling our continuous approach to solve the discrete optimum design problem, it makes a difficult problem become
solvable by using available nonlinear optimization algorithms. In the paper, we adopt the popular augmented
Lagrangian method to solve the continuous problem ( P 4 ) .
The augmented Lagrangian function of ( P 4 ) is shown in the following ( P 5 )
I
I
ci
[hi (δ )]2 +
i =1 2
min Φ (δ ,α , λ ) = f (δ ) + ∑ α i hi (δ ) + ∑
i =1
J
1
∑ 2σ
j =1
j
{
⎡ min ( 0, λ j + σ j g j (δ ) ) ⎤ − λ j2
⎣
⎦
2
}
(17)
where α i , i = 1,", I , and λ j , j = 1," , J are the Lagrange multiplier of equality constraints and inequality constraints
respectively, c j > 0, i = 1," , I and σ j , j = 1," , J are the penalty parameters for equality constraints and inequality
constraints respectively. α i , i = 1,", I , and λ j , j = 1," , J can be updated by the following
α ik +1 = α ik + ci hi (δ k )
{
(18)
}
λ jk +1 = min 0, λ jk + σ j g j (δ k )
(19)
To solve a sequential unconstraint problem ( P 5 ) with certain convergence criterion, the solution of the problem ( P1)
is obtained without difficulties.
NUMERICAL EXAMPLE
1. Example 1
min f ( A ) = A1 + 2 A2 + 4 A3 + 6 A4
s.t. −
−
1
1
3
3
−
− −
≥ −2
A1 A2 A3 A4
1
2
2
1
−
−
−
≥ −1.8
A1 A2 A3 A4
Ai ∈ {1, 2,3,5} , i = 1,",5
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(δ11 , δ12 , δ13 , δ14 ) = ( 0,0,0,1) ; (δ 21 , δ 22 , δ 23 , δ 24 ) = ( 0,0,0,1) ;
(δ 31 , δ 32 , δ 33 , δ 34 ) = ( 0,0,0,1) ; (δ 41 , δ 42 , δ 43 , δ 44 ) = ( 0,0,1,0 ) ;
( A , A , A , A ) = ( 5,5,5,3) ; f ( A ) = 53
*
1
*
2
*
3
*
4
∗
2. Example 2
Figure 2 shows a three-bar truss subjected to two distinct loadings F1 and F2 respectively. F1 = F2 = 20 , E = 1 ,
σ = 20 , σ = 15 , S = {0.6, 0.8,1.0,1.2,1.4, 2.0, 3.0, 4.0, 5.0, 6.0} . The truss is symmetrical, A1 = A3 . The
displacement constraints are
(1) U1 ≤ 10.0 ; (2) U1 ≤ 2.0
L
L
2
4
3
2
○
1
○
3
○
L
1
F1
F2
U1
Figure 2: Optimum problem of tree truss bars
The optimum solutions carried out by continuous approach are
∗
∗
(1) ( A1 , A2 ) = ( 0.8,1.0 ) ， W * = 3.2628 ；
∗
∗
(2) ( A1 , A2 ) = ( 2.0, 6.0 ) ， W * = 11.657 ；
Because there are only two design variables the exhaustion method is used in each iteration step to reexamine the
problem. The two cases both converge to globally optimum solutions.
3. Example 3
Fig. 3 shows geometry of the 25-bar truss structure. Material properties are given as
E = 6.898 ×1010 N / m2 ,
ρ = 2.769 × 103 kg / m3 . The members are divided into eight groups and the group of member linking and the stress
limits are given in Table 1. The structure is subjected to two load cases, as show in Table 2. The displacements at
nodes 1 and 2 in the directions of x and y are constrained to ±8.89 × 10−3 m . For this example, the available discrete
2
values of the variables are: S = {0.51613, 0.64516, 1.9355, 4.5161, 6.4516, 12.903, 19.355, 25.806} ( cm ) .
1
2
2.54m
3
4
6
5
7
2.54m
Z
8
Y
10
X
9
5.08m
5.08m
Figure 3 25-bar space truss
The optimal solution are given in Table 3. The iterations of binary variables δ ir ( i = 1, 2," ,8, r = 1, 2," ,8 )are
shown in Fig. 4 .
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1.0
1.0
Z11
Z12
Z13
Z14
Z15
Z16
Z17
Z18
0.8
0.6
0.4
0.2
0.0
1
3
5
7
9
11
13
Z21
Z22
Z23
Z24
Z25
Z26
Z27
Z28
0.8
0.6
0.4
0.2
0.0
15
1
(a) binary variables relate to x1
3
5
7
9
11
13
15
(b) binary variables relate to x2
1.0
1.0
Z31
Z32
Z33
Z34
Z35
Z36
Z37
Z38
0.8
0.6
0.4
0.2
0.0
1
3
5
7
9
11
13
Z41
Z42
Z43
Z44
Z45
Z46
Z47
Z48
0.8
0.6
0.4
0.2
0.0
15
1
(c) binary variables relate to x3
3
5
7
9
11
13
15
(d) binary variables relate to x4
1.0
1.0
Z51
Z52
Z53
Z54
Z55
Z56
Z57
Z58
0.8
0.6
0.4
0.2
0.0
1
3
5
7
9
11
13
Z61
Z62
Z63
Z64
Z65
Z66
Z67
Z68
0.8
0.6
0.4
0.2
0.0
15
1
(e) binary variables relate to x5
3
5
7
9
11
13
15
(f) binary variables relate to x6
1.0
1.0
Z71
Z72
Z73
Z74
Z75
Z76
Z77
Z78
0.8
0.6
0.4
0.2
0.0
1
3
5
7
9
11
13
0.8
0.6
0.4
0.2
0.0
15
1
(g) binary variables relate to x7
3
5
7
9
11
13
15
Z81
Z82
Z83
Z84
Z85
Z86
Z87
Z88
(h) binary variables relate to x8
Figure 4: The iterations of binary variables
Table 1 Member linking groups and stress limits
Groups
Number of the bar
1
2
3
4
5
6
7
8
1-2
1-4 2-3 1-5 2-6
2-5 2-4 1-3 1-6
3-6 4-5
3-4 4-6
3-10 6-7 4-9 5-8
3-8 4-7 6-9 5-10
3-7 4- 5-9 6-10
Compression
σ ( N / m2 )
-2.420 × 108
-7.994 × 107
-1.194 × 108
-2.420 × 108
-2.420 × 108
-4.662 × 107
-4.662 × 107
-7.664 × 107
stress
limits
Tensile stress limits σ ( N / m2 )
2.758 × 108
2.758 × 108
2.758 × 108
2.758 × 108
2.758 × 108
2.758 × 108
2.758 × 108
2.758 × 108
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Table 2 Load condition for 25-bar truss (KN)
Load case
1
1
1
1
2
2
Joint
1
2
3
6
5
6
X -direction
4.45
0
2.225
2.225
0
0
Y -direction
44.5
44.5
0
0
89
−89
Z -direction
−22.25
−22.25
0
0
−22.25
−22.25
Table 3 Results of 25-bar truss
Optimal solution
W*
2630
x1
x2
0.51613 12.903
x3
19.355
x4
x5
x6
0.51613 0.51613 4.5161
x7
12.903
x8
19.355
CONCLUSION
The continuous approach described in this paper makes it possible to solve discrete optimum design by standard
optimization methods and software. The purpose of this development is to provide engineers a convenient tool to deal
with various applications as well as to arouse researchers with new interests to study discrete optimization. It should be
noted that our approach is centered around the solution of general 0-1 programming problem and gives a general
solution methodology. Therefore, the method given in this paper is not restricted to the structural optimization.
ACKNOWLEDGEMENTS
This research was supported by National Natural Science Foundation of China under Grant (10332010, 10590354).
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