new optimization algorithm for topology optimization

The Center of Innovative Design Optimization Technology Applied Mechanics and Optimal Design Lab.
1

2
1
2
4
Introduction
Diagonal Quadratic Approximation (DQA)
Numerical Examples
5 Conclusions
3 Filtered Diagonal Quadratic Approximation (FDQA)

3
1
2
4
Introduction
Numerical Examples
5 Conclusions

4
● Considered Problem
0
, ,
minimize ( )
subject to ( ) 0, 1, ,
where , , 1, ,
j
i i L i U
f
f j m
x x x i n
 
   
x
x
x L
L
 Nonlinear and continuously differentiable functions are considered.
 The number of design variables is much larger than that of constraints. (n>>m)
 Computational cost for the analysis is very high.
 To solve this kind of problems, Sequential Approximate Optimization (SAO) with
the dual method has been developed in the last two decades.

5
yes
no
Compute    0 0,j jf fx x
0, , .j m L
2
Construct
a dual subproblem
3
Solve the dual subproblem4
Compute
0, , .j m L
Set k=k+18
END
6
7
Optimization parameter
setting
1
Move to the next point
5
 ( 1)
 x k
jf ( 1)
,k
jf 
x
( 1) ( )*
x xk k
What is the dual subproblem?
Converged?

6
Dual Subproblem
Primal Subproblem
At each k th iteration point
 
 
   
0
, ,
minimize ( )
subject to ( ) 0, 1, ,
where , 1, ,
k
k
j
k k
i i L i U
f
f j m
x x x i n
 
   
x
x
x
%
% L
L
   
   
 ( )
0
1
maximize ( ), ( ) ( )
subject to 0, 1, , .
m
k kk
j j
j
j
L f f
j m



 
 
λ
x λ λ x λ x λ% %
L
( )k
x
 If n >> m the dual method is more
efficient than the primal method.
 The primal variable is explicit
function of the dual variable.
Primal variable
Dual variable

7
 Diagonal Quadratic Approximation (DQA)
( )
( ) ( ) ( ) ( ) ( ) 2
,
1 1
1
( ) ( ) ( ) ( ) 0, , .
2
k
n n
jk k k k k
j i i i j i i
i ii
f
f f x x h x x j m
x 
 
      
 
 x x% L
• DQA is expressed by the first order Taylor’s expansion and the quadratic term.
 What is the Convex Separable Approximation?
1. Separable : off-diagonal Hessian terms of the approximate functions are all zero
2. Convex : diagonal Hessian terms of the approximate functions are non-negative
 To effectively construct the dual subproblem,
convex separable approximation should be used for the approximating functions.
• Good approximation method of true diagonal Hessian terms is very important.

8
( )
( ) ( ) ( ) ( ) ( ) 2
,
1 1
1
( ) ( ) ( ) ( ) 0, , .
2
k
n n
jk k k k k
j i i i j i i
i ii
f
f f x x h x x j m
x 
 
      
 
 x x% L
Previous methods approximating the diagonal hessian terms
 Exponential approximation
 MMA approximation
 CONLIN approximation
 TANA approximation
Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and
Multidisciplinary Optimization, Vol. 41, No. 1, 2010, pp. 39-56.
Groenwold, A. A., Etman, L. F. P., Snyman, J. A., and Rooda, J. E., “Incomplete series expansion for function
approximation”, Structural and Multidisciplinary Optimization, Vol. 34, No. 1, 2007, pp. 21-40.
 
 2 ( )
,
( )k
k
i j
i jx
f
h
x



 
x% One point approximation
Two point approximation

9
Previous methods to enhance the convergence property of the SAO
5Fletcher, R., Leyffer, S., and Toint, P. L., "On the global convergence of a filter SQP algorithm," Siam Journal on
Optimization, Vol. 13, No. 1, 2002, pp. 44-59.
4Svanberg, K., "A class of globally convergent optimization methods based on conservative convex separable
approximations," Siam Journal on Optimization, Vol. 12, No. 2, 2001, pp. 555-573.
7Groenwold, A. A., and Etman, L. F. P., "On the conditional acceptance of iterates in SAO algorithms based on convex
separable approximations," Structural and Multidisciplinary Optimization, Vol. 42, No. 2, 2010, pp. 165-178.
NLP filter for the SQP5
GCMMA4
NLP filter for the DQA7
 Filter tests if is acceptable.
 If not acceptable, the inner
iteration is conducted.
 In the inner iteration,
( )*k
x
 ( )
, ,
kk
i j i jh h
• Adjust move limit.
• Enforce conservatism.
User defined parameter

10
1. Propose a DQA with highly accurate diagonal hessian terms
employing a gradient based two-point approximation method.
2. Propose a Nonlinear Programming(NLP) filter appropriate to the
proposed DQA to improve convergence property.
Two-Point Approximation
 f x
 0
f x
  0f x
x  1
f x
  1f x
x
°( )f x
y
n
0, , .j m L
2
Construct
dual subproblem
3
Solve dual subproblem4
Compute
0, , .j m L
5
Compute
0, , .j m L
Set k=k+113
END
Test
Convergence
11
12
setting
1 Move to the next point
Initialize
10
0ρ ρ
 ( )*
x k
jf
 ( 1)
 x k
jf
( 1) ( )
 x xk k
x
 ( 1)
,k
jf 
x
( 1) ( )*
x xk k
NLP
filter
NLP filter

11
1
2
4
Introduction
Numerical Examples
5 Conclusions

12
( )
( ) ( ) ( ) ( ) ( ) 2
,
1 1
1
( ) ( ) ( ) ( ) 0, , .
2
k
n n
jk k k k k
j i i i j i i
i ii
f
f f x x h x x j m
x 
 
      
 
 x x% L
*Kim, J. R., and Choi, D. H., “Enhanced two-point diagonal quadratic approximation methods for design
optimization,” Computer methods in applied mechanics and engineering, Vol. 197, 2008, pp. 846-856
2 ( ) ( )
( )eTDQA k
ij
i j
h
x
f
x


 
x% Ⅱ
( ) 2
( )
( ) ( ) ( ) ( ) 2 1
( 1) 2 ( ) 21 1
1 1
( )
1 1
( ) ( ) ( ) ( )
2 2
( ) ( )
n
k
k
e i i in n
eTDQA k k k i
i i i i i n n
k ki ii
i i i i i i
i i
H y y
f
f f y y G y y
y
H y y H y y


 
 

 
      
    

 
 
x x% Ⅱ
  ip
i i iy x c where
 For the diagonal hessian terms,
we use the second order
derivative of eTDQA.
Determination of these parameters is provided in the previous research.

13
 
    
  
( )
( )
2( 1)( )
2
( ) ( 1)
1
( 1) ( )
if
0 if
i
l l
k
i
k
ii i
pke i
ij i i i in p pk k
l l i l i
l
p f
xx c
Hh G p x c i j
H x c x c
i j
 


   
  
  

 
 
        
          




x
Analytically derive hessian terms of eTDQA
Not convex!
 Separable!
( ) 2
( )
( ) ( ) ( ) ( ) 2 1
( 1) 2 ( ) 21 1
1 1
( )
1 1
( ) ( ) ( ) ( )
2 2
( ) ( )
n
k
k
e i i in n
eTDQA k k k i
i i i i i n n
k ki ii
i i i i i i
i i
H y y
f
f f y y G y y
y
H y y H y y


 
 

 
      
    

 
 
x x% Ⅱ
 where, ip
i i iy x c 

14
● Convexifying Operation of Part 1
      
  
( )
, 1 2
2 ( ( ) ( ) 2( 1)( )
2 ( ) 2
( ) ( 1)
1
( ) ( 1) ( ) i
l l
k
i j
eTDQA k k
pj ki e i
i i i ink p pk ki ii i
l l i l i
l
h P P
f p Hf
G p x c
x xx c
H x c x c
 


 
 
            
           
  

x x% Ⅱ)
Part 1
 
 
( )
( )
if / 0 1
if / 0 1
k
i i
k
i i
f x p
f x p
    
    
1then 0P 
*S. Park, D. Choi (2011) A new convex separable approximation based on two-point diagonal quadratic approximation
for large-scale structural design optimization, WCSMO 9

15
ip
 ( )
( ) / 0k
if x  x  ( )
( ) / 0k
if x  x
( ) ( 1)k k
i ix x 

( ) ( 1)k k
i ix x 
( ) ( 1)k k
i ix x 
 ( ) ( 1)k k
i ix x 

( )
( 1)
( / )
1
( / )
k
i
k
i
f x
f x 
 

 
( )
( 1)
( / )
0 1
( / )
k
i
k
i
f x
f x 
 
 
 
( )
( 1)
( / )
0
( / )
k
i
k
i
f x
f x 
 

 
● Determination of Exponents Term pi
†
Current point Previous point
   
( 1) ( )
( 1) ( )
1 ln ln
k k
k k
i i i i i
i i
f f
p x c x c
x x


                      
1 1 1 1
-1
-13
3
Eqn.†
Eqn.† Eqn.†
Eqn.†
*S. Park, D. Choi (2011) A new convex separable approximation based on two-point diagonal quadratic approximation
for large-scale structural design optimization, WCSMO 9

16
● Convexifying Operation of part 2
      
  
2 ( ( ) ( ) 2( 1)( )
2 ( ) 2
( ) ( 1)
1
( ) ( 1) ( ) i
l l
eTDQA k k
pj ki e i
i i i ink p pk ki ii i
l l i l i
l
f p Hf
G p x c
x xx c
H x c x c
 


 
            
           
  

x x% Ⅱ)
Part 2
    
  
2( 1)( )
2 2
( ) ( 1)
1
max ,
i
l l
pke i
i i in p pk k
l l l
l
H
P G p x
H x x





  
  
     
   
   
   

After convexifying procedure, all diagonal terms of hessian matrix become positive.

17
1
2
4
Introduction
Numerical Examples
5 Conclusions

18
k: outer iteration number
l: inner iteration number
yes
no
0, , .j m L
2
Construct
a dual subproblem
3
Solve the dual subproblem4
Compute
0, , .j m L
Set k=k+18
END
6
7
setting
1
Move to the next point
5
 ( 1)
 x k
jf ( 1)
,k
jf 
x
( 1) ( )*
x xk k
Converged?

19
k: outer iteration number
l: inner iteration number
yes
no
0, , .j m L
2
Construct
dual subproblem
3
Solve dual subproblem4
Compute
0, , .j m L
5
Compute
0, , .j m L
Set k=k+113
END
11
12
setting
1 Move to the next point
Initialize
10
0ρ ρ
 ( )*
x k
jf
 ( 1)
 x k
jf ( 1)
,k
jf 
x
( 1) ( )*
x xk k
Converged?NLP filter is
adopted
to improve
convergence
property

20
is acceptable
to the current filter?
Slanting
Envelope
Test
Updating filter
( )*
x k
1l l 
Slanting envelope test using NLP Filter
0f f  max 0, , 1, , .jh f j m  Land
 ,h f ( )*k
x
 For the brevity,
 At the kth iteration, the pair is obtained at .
 For the filter, slanting envelop is used to prove convergence.
f
h (1) (1)
,h f
 (2) (2)
,h f
 (3) (3)
,h f
 (4) (4)
,h f
 ,h f
( ) ( )
andj j
f h f h Reh ject    
( ) ( )
orj j
f h f h h Acceptable    
 (5) (5)
,h f
 
( )
( (
)
) )
(
,j j
j j
Domin
f h f and h h
ate f h
Acceptable
   

 (6) (6)
,h f
 ,h f
 ,h f (4) (4)
,h f
yes
no
yes
no
Inner iteration
Slanting envelope
*R. Fletcher, S. Leyffer, P. Toint (2002)
On the global convergence of a filter-SQP algorithm
 Reduce move limit
 Conservative approximate
1l l 
Inner iteration
 Conservative approximation
no
Satisfy sufficient
reduction criterion?


21
is acceptable
to the current filter
and 0f q q    
Updating filter
( )*k
x
 Reduce the move limit
y
n
n
y
1l l 
Inner iteration
Solve the dual subproblem
ix
The move limit is decreased by one half in this study.
 ,
,
k l
i Lx  ,
,
k l
i Ux , 1
,
k l
i Lx
  , 1
,
k l
i Ux
 ,k l
ix
Move limit
( ) ( )
, , 0, , .k k
i j j i jh h j m   L
The approximate functions can be easily conservative by
increasing hessian terms.
 max 1.1,j j  
( ) ( )
( ) ( )k k
j jf fx x%j is determined to match
where
( , ) ( )* ( )*
( ) ( )k l k k
j jf fx x%
The approximate function is conservative if


22
is acceptable
to the current filter?
( )*k
x
0.1 
   ( ) ( )*
0 0
k k
q f f  x x% %
   ( ) ( )*
0 0
k k
f f f  x x
0 andq f q    
yes
no
yes
no
1l l 
Inner iteration
 Conservative approximate
Updating filter
no
yes
Updating filter
1l l 
Inner iteration
Next
iteration
Satisfy sufficient
reduction criterion?
Test if the reduction of real objective function
is smaller than we expected.
Let,
If
reduction of objective function is not
sufficient  go to inner iteration

Otherwise, update filter and go to the next
iteration

23
1
2
4
Introduction
Numerical Examples
5 Conclusions

24
No. Option Problem n m
1
a
Vanderplaats' cantilever beam problem7
20 21
b 200 201
2 Svanberg's 5-variate cantilever beam problem9 5 1
3
a MBB beam topology optimization problem (p=1, 75 by 25)9 1875 1
b MBB beam topology optimization problem (p=3, 75 by 25) 1875 1
9Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and
 Performance of the proposed algorithm is compared with those of previous studies.7,9
 Initial condition and convergence criteria are same as those of previous studies.7,9
 Convergence criterion is .
 For the 3b problem, performance of the proposed algorithm is compared with the MMA and
GCMMA.
( 1) ( )k k
x
 x x

25
0
1
2 1
Minimize ( , )
subject to ( , ) 1 0 1,..., ,
( , ) 20 0 1,..., ,
( , ) 1 0,
1.0 100,
5.0 100.
p
i i i
i
i
j
p j i i
p
p
i
i
f b hl
f j p
f h b j p
y
f
y
b
h




   
   
  
 
 
x
b h
b h
b h
b h


Vanderplaats’ cantilever beam problem No. Method
1a
p=10
SAO-A 64244.83 1.68E-06
SAO-B 64244.83 1.11E-06
SAO-C 64244.83 3.21E-06
SAO-D 64244.83 2.41E-05
DQA 64244.83 5.35E-06
FDQA 64244.83 5.35E-06
*
k *
l
*
0f  max jf
*
k
*
l
: The number of outer iterations
: The number of inner iterations
 DQA and FDQA obtain appropriate
optimum point.
 DQA and FDQA show better efficiency
compared to other methods.
1b
p=100
SAO-A 63678.1 1.78E-06
SAO-B 63678.1 4.90E-06
SAO-C 63678.1 2.98E-07
SAO-D 63678.1 4.06E-05
DQA 63678.1 8.66E-06
FDQA 63678.1 1.66E-06
38 -
101 261
40 94
29 2
10 -
10 0
34 -
457 2535
30 25
29 1
11 -
10 1

26
Svanberg’s 5-variate cantilever beam
 
0 1 1 2 3 4 5
3 3 3
1 1 2 3
3 3
4 5 2
1 2
(0)
Minimize ( ) ( )
subject to ( ) 61/ 37 / 19 /
7 / 1/ 0,
0.001 10 1,2,3,4,5
where 0.0624, 1.0,
5.0, 5.0, 5.0, 5.0, 5.0
i
T
f c x x x x x
f x x x
x x c
x i
c c
    
  
   
  
 

x
x
x
x
Method
T2:R 1.339956 -
T2:E 1.339956 -
T2:MMA 1.339956 -
T2:TANA-3 1.339956 -
GCMMA 1.339956 -
DQA 1.339957 -1.26E-06
FDQA 1.339957 -1.26E-06
*
k *
l
*
0f  max jf
 DQA and FDQA obtain appropriate optimum point.
 DQA and FDQA show good efficiency.
10 8
13 7
20 15
10 4
19 20
8 -
8 0

27
MBB beam topology optimization
0
1
1 0
1
Minimize ( )
subject to ( ) 0,
0.001 1
n
T p T
e e e e
e
n
e
e
e
f x
f x fV
x


 
  

 


x u Ku u k u
x
Ku f
1, 0.3, 1E P  
P
No. Method
3a
R 165.8839 -1.11E-13
T2:R 165.8839 -1.11E-13
E 165.8839 -5.60E-12
T2:E 165.8838 -5.46E-16
MMA 165.8839 5.84E-11
T2:MMA 165.8838 -4.53E-11
T2:R 165.8839 -1.11E-13
T2:E 165.8838 -5.46E-16
T2:MMA 165.8838 -6.06E-16
GCMMA 165.9624 1.90E-08
DQA 165.4939 -1.02E-11
FDQA 165.4936 3.18E-11
*
k *
l
*
0f  max jf
 All of methods obtain similar objective
function.
 For 3a problem, the E method is
better than the proposed FDQA.
 FDQA is more efficient than DQA.
59 -
58 -
33 -
35 -
68 -
51 -
58 0
35 0
43 11
37 103
56 -
39 2

28
MBB beam topology optimization
Method
10-3
DQA 205.6923 6.87E-11
MMA 205.784 -5.22E-05
FDQA 203.9638 -1.78E-07
GCMMA 315.4972 -1.29E-06
*
k
*
l
*
0f  max jfx
 For 3b problem, the proposed FDQA
shows best performance.
 Optimization is not converged except
for the proposed FDQA when εx=10-4.
DQA MMA
Optimized layouts
FDQA GCMMA
The penalization parameter is set to 3.
10-4
DQA - -
MMA - -
FDQA 203.9638 -1.78E-07
GCMMA - -
956 -
301 -
72 74
347 1444
not converged -
not converged -
72 74
not converged -

29
5
1
2
4
Introduction
Numerical Examples
Conclusions

30
 Propose an SAO algorithm with highly accurate hessian terms
by using the eTDQA.
 Propose a filtered SAO algorithm appropriate to the proposed DQA.
 Investigate the efficiency and accuracy of the proposed algorithm
by solving the numerical examples.
 Improve convergence property without worsening the efficiency
through the proposed algorithm.

31

32

33
Previous methods related with the accuracy of approximate method for the SAO
Method Year Keyword Author
Linear or reciprocal
approximation
1986 CONLIN Fluery, C.
1987 MMA Svanberg, K.
Exponential
approximation
1990 TPEA Fadel, G. M., Riley, M. F., Barthelemy, J. M.
Diagonal
quadratic
approximation
1995
Quasi-Newton
update
Duysinx, P. Z., Zhang, W. H., Fluery, C.
2002 Dynamic-Q Snyman, J. A., Hay, A. M.
2007
Incomplete series
expansion
Groenwold, A. A., Etman, L. F. P.,
Snyman, J. A., Rooda, J. E.
20109 Approximated
approximations9 Groenwold, A. A., Etman, L. F. P., Wood, D. W.
9Groenwold, A. A., Etman, L. F. P., and Wood, D. W., "Approximated approximations for SAO," Structural and
Several SAO with the dual methods are compared.

34
Previous methods related with the convergence property of the SAO
Method Year Author
Trust-region like framework
1998
Alexandrow, N. M., Dennis, J. E., Lewis, R. M.,
Torczon, V.
2000 Gonn, A. R., Gould, N. I. M., Toint, P. L.
Nonlinear acceptance filter for SQP
1998 Fletcher, R., Leyffer, S., Toint, P. L.
2002
Fletcher, R., Gould, N. I. M., Leyffer, S.,
Toint, P. L., Wächter, A.
Globally convergent version of MMA 2002 Svanberg, K.
Filter for the dual SAO 2009
Groenwold, A. A., Wood, D. W., Etman, L. F. P.,
Tosserams, S.
Filtered conservatism7
2010 Groenwold, A. A., Etman, L. F. P.
According to the filter option, SAO-A, SAO-B, SAO-C, and SAO-D are compared.

35
Enhanced Two-point Diagonal Quadratic Approximation (eTDQA)*
( ) 2
( )
( ) ( ) ( ) ( ) 2 1
( 1) 2 ( ) 21 1
1 1
( )
1 1
( ) ( ) ( ) ( )
2 2
( ) ( )
n
k
k
e i i in n
eTDQA k k k i
i i i i i n n
k ki ii
i i i i i i
i i
H y y
f
f f y y G y y
y
H y y H y y


 
 

 
      
    

 
 
x x% Ⅱ
1. Intervening variable with shifting constant ci
1
f
ix
to define an intervening variable when
the design variable value is near zero
or negative
  ip
iii cxy 





0otherwise,
1,if
i
L
iii
c
xcx 
*Kim, J. R., and Choi, D. H., “Enhanced two-point diagonal quadratic
approximation methods for design optimization,” Computer methods in
applied mechanics and engineering, Vol. 197, 2008, pp. 846-856
i i i i ec p G H    Each parameter is calculated sequentially
  ip
i i iy x c 
where

36
3. Gi is determined to match when pi calculation fails, otherwise, Gi is set to 0.( ) °( )( 1) ( 1)k k
f fy y- -
Ñ = Ñ
2. pi is determined to match .
5. e is a correction factor to match .
  ip
i i iy x c 
where
( ) 2
( )
( ) ( ) ( ) ( ) 2 1
( 1) 2 ( ) 21 1
1 1
( )
1 1
( ) ( ) ( ) ( )
2 2
( ) ( )
n
k
k
e i i in n
eTDQA k k k i
i i i i i n n
k ki ii
i i i i i i
i i
H y y
f
f f y y G y y
y
H y y H y y


 
 

 
      
    

 
 
x x% Ⅱ
( ) °( )( 1) ( 1)k k
f fy y- -
=
( ) °( )( 1) ( 1)k k
f fy y- -
Ñ = Ñ
Calculation of parameters
4. .
( )
( )
( )
( )1
if / / 0
1 otherwise
k k
i i i
i
G f x f x
H
-ìï ¶ ¶ ¶ ¶ £ï= í
ïïî
g

37
Two-Point Approximation
 f x
 0
f x
  0f x
x  1
f x
  1f x
x
°( )f xCurrent Design Point
Previous Design Point
Design Information
 Conventional SAO with dual method are usually utilize only the current point
information to construct approximate function.
 Two-point approximation methods use function values and first order derivative
values of two design points.
( 1) ( 1) ( 1) ( 1)
1 2, ,...,
Tk k k k
nx x x   
   x
 
( 1)
( 1)
,
k
jk
j
i
f
f
x


 
 
 
x
( ) ( ) ( ) ( )
1 2, ,...,
Tk k k k
nx x x   x
 
( )
( )
,
k
jk
j
i
f
f
x
 
 
 
x

38
K. Svanberg (2002), A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations
• If the condition described below is satisfied, the approximate function is said to be
conservative
• If the approximate functions are not conservative, convergence property cannot be
guaranteed.*
• If the hessian terms are increased, it is possible to obtain optimum point near to
the current point.
• In the proposed method, the approximate functions can be easily conservative by
increasing hessian terms.
( , ) ( )* ( )*
( ) ( )k l k k
j jf fx x%
Example) Let .   0 1, ' 0 2f f 
Then,         
2 21 1
0 ' 0 0 0 1 2
2 2
f x f f x h x x hx       %
8h 
2h 

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