Multiobjective Optimization, Lecture Notes- Physics - 2, Study notes for Physics. University of Cambridge
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Multiobjective Optimization, Lecture Notes- Physics - 2, Study notes for Physics. University of Cambridge

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Introduction, Multiobjective archiving, multiobjective SA, multiobjective optimization, multiobjective GA
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5R1MultiobjectiveOptimisation.pdf

5R1 GTP

6. Multiobjective Optimization

6.1 Introduction

Many (perhaps most) real-world design problems are, in fact, multiobjective optimization problems in which the designer seeks to optimize simultaneously several performance attributes of the design and an improvement in one objective is often only gained at the cost of deteriorations in other objectives — trade-offs are necessary.

There are two standard methods for treating multiobjective problems, if a traditional optimiza- tion algorithm which minimizes a single objective is to be employed. One is to construct a composite objective:

, (6.1)

where the are the objectives to be minimized and the are positive-valued weightings. The other is to place constraints on all but one of the objectives, i.e:

, (6.2)

where the are the constraint limits.

Whichever of these approaches is used, the solution of the one objective problem so produced results in the identification of a single point on the trade-off surface, the position of which depends on the designer’s preconceptions (the values of or chosen) as illustrated in Fig- ure 6.1. In order to explore the trade-off surface further a large number of different optimiza- tion runs must be executed each with different weightings or constraints — a potentially time- consuming and computationally expensive exercise if even attempted.

Minimize f ̃ a i f i i

1

!

"

N

!#!"

fi N ai

Minimize f j subject to f i C i i 1 N i j! $%&"'(

Ci

ai Ci

Figure 6.1

: Types of Multiobjective Optimum.

f1

f2

weighted average

optimum

constraint

constrained optimum

trade-off surface

contour a1 f1 a2 f2+

C1

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Of course, eventually a single solution must be chosen, but it is self-evident that the designer will make a better informed decision if the trade-off surface between the conflicting objectives can be inspected before this choice is made. By using suitably adapted stochastic optimization methods it is possible to reveal the trade-off surface of a multiobjective optimization problem in a single run. In the following sections appropriate adaptations to standard Genetic Algo- rithm (GA) and Simulated Annealing (SA) implementations will be discussed.

6.2 Multiobjective Archiving

Adapting any stochastic optimization algorithm to perform multiobjective optimization will inevitably require a common change to the method of archiving. In multiobjective optimiza- tion solutions lying on the trade-off surface (or

Pareto front

as it is also known) are sought.

Any solution on the Pareto front can be identified formally by the fact that it is not

dominated

by any other possible solution. A solution

X

is said to be

dominated

by solution

Y

if

Y

is at least as good on all counts (objectives) and better on at least one, i.e., assuming all

M

objec- tives are to be minimized, if

and . (6.3)

Thus, in multiobjective optimization an archive of the best (i.e. the nondominated) solutions found should be maintained. A suitable archiving scheme, illustrated in Figure 6.2, is as fol- lows:

• All feasible solutions generated are candidates for archiving.

fi Y! " fi# X! " $ i$% 1 M$ &' fi Y! " fi( X! " $ for some i

Figure 6.2: Multiobjective Archiving.

f1

f2

solutions in

new solution will not be archived

current archive

new solution will be archived,

new solution will be archived

dominated solutions will be removed

Case 1

Case 2

Case 3

new solution

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• If a candidate solution dominates any existing members of the archive, those are removed and the new solution is added (Case 1).

• If the new solution is dominated by any existing members of the archive, it is not archived (Case 2).

• If the new solution neither dominates nor is dominated by any members of the archive, it is added to the archive (Case 3).

Using this scheme, it is hoped that, as the search progresses, the archive will converge onto the true trade-off surface between the objectives.

6.3 M

ULTIOBJECTIVE

SA

SA has traditionally been used for single objective optimization. However, Engrand [1997] has recently proposed a multiobjective variant on the SA algorithm. Although Engrand’s orig- inal algorithm performs adequately, it has a couple of identifiable weaknesses, which are readily overcome. The multiobjective SA algorithm described here in the following sections is therefore the improved version of Engrand’s proposed algorithm (Suppapitnarm

et al

. [2000]).

The structure of the multiobjective SA algorithm is shown in Figure 6.3. The key differences compared to a single objective SA implementation are as follows.

6.3.1 Solution Acceptance

The overall acceptance probability is the product of a series of individual acceptance probabil- ities for each of the

M

objectives to be minimized:

. (6.4)

Note that each objective has its own associated temperature , which obviates the need to scale the objectives carefully with respect to each other, as long as appropriate temperatures can be determined automatically. The individual ‘probabilities’ will be greater than or less than unity depending whether the move from to has decreased or increased that partic- ular objective . Thus, the overall acceptance probability

P

can be regarded as giving an indi- cation as to whether the move is predominantly downhill or uphill .

Any move in which at least one objective is decreased is, however, potentially a move onto the trade-off surface. The acceptance probability of such a move (given by equation (6.4)) depends on the relative changes of all the objectives and the current temperatures, and thus there is no guarantee that the move will be accepted. If it is a move onto the trade-off surface, then it clearly should be accepted (and archived). Therefore, as shown in Figure 6.3, the acceptance logic has an important additional feature:

• Each new solution generated is

first

submitted as a candidate for archiving.

P pi i 1!

M

" exp fi xn 1#$ % fi xn$ %&' (

Ti &) * + ,

i 1!

M

"! !

Ti

pi xn xn 1#

fi P 1- P 1.

48

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• If the solution is archived, then it is automatically accepted.

• If the solution is not archived, then it is accepted with a probability given by equation (6.4).

This ensures that all moves which identify a new solution on the current trade-off surface are accepted and archived.

6.3.2 Annealing Schedules

The formulation of the acceptance probability calculation (equation (6.4)) requires a different temperature for each objective. One way in which this requirement can be met is by imple- menting annealing schedules as follows:

• Initially all the temperatures are set to , so that all feasible moves are accepted. After aTi !

Figure 6.3: Multiobjective SA Algorithm Structure.

Randomly perturb x n x n 1" #"$

Yes

No

Try archiving

Periodically, return to base

Initialise x 0 f X 0" % & T"' '

Evaluate f ( x n 1" # )

Was x n 1" # archived?

Terminate search? StopYes

No

Yes

Noxn 1# ? Accept

Accept x n 1" #

Periodically, reduce T

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predetermined number of trials the temperatures are set to appropriate values using White [1984]’s formulation:

, (6.5)

where is the standard deviation of the variation of observed.

• Periodically thereafter the temperatures are lowered according to the formula:

, (6.6)

where is determined using the formulation of Huang

et al

. [1986]:

. (6.7)

6.3.3 Return To Base (Restarts)

In a traditional single objective SA implementation the “return to base” option retrieves the best solution found and continues the search from there. This only occurs if the search is deemed to have ceased to make satisfactory progress.

In this multiobjective SA implementation when a return to base occurs a solution selected from the archive is retrieved and the search is continued from there. This is done to try to ensure that the entire trade-off surface is explored and therefore returns to base occur much more frequently than in a single objective SA implementation.

Suppapitnarm [1998] investigated various return to base strategies and concluded that a scheme in which returns to base are made to a solution randomly selected from a candidate list consisting of

• the

M

solutions in the archive which have the lowest value of each individual objective, i.e. the solutions at the ends of the trade-off surface, and

• about half-a-dozen solutions selected randomly from the other solutions in the archive

gives consistently good algorithm performance.

6.3.4 Example: Multiobjective Optimisation Of Bicycle Fork Design

Figure 6.5 and Figure 6.6 illustrate the performance of the multiobjective SA algorithm described in optimising the general shape of a pair of bicycle front forks to be manufactured from a predetermined material (steel). Three objective functions were considered:

• minimisation of the material volume, • minimisation of the deflection under frontal impact loading, and • minimisation of the angular deflection under cornering loading.

The legs of the forks were assumed to be made of tubular steel with a two part profile, as shown in Figure 6.4, consisting of a straight section and a curved (quadratic profile) section.

Ti !i"

!i fi

Ti# $i Ti"

$i

$i max 0.5 exp 0.7Ti !i

%& ' ( )*"

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The overall height,

H

, and length,

L

, of the forks are fixed, but the height,

h

, and length,

l

, of the curved section may be varied. In addition, the radius of the tubular section at the fork tip,

r

, the tube thickness,

t

, and the degree of tapering of the forks (so that the radius of the tubular

Figure 6.4: Bicycle Front Fork Geometry.

H

h

L

l

Figure 6.5: Archive Evolution.

0.020.010.00 20

40

60

80

100

120

0.020.010.00 20

40

60

80

100

120

Cornering load angular displacement / radians

M at

er ia

l v ol

um e

/ c m

3

After 600 iterations

0.020.010.00 20

40

60

80

100

120

Cornering load angular displacement / radians

M at

er ia

l v ol

um e

/ c m

3

After 100 iterations

0.020.010.00 20

40

60

80

100

120

Cornering load angular displacement / radians

M at

er ia

l v ol

um e

/ c m

3

After 300 iterations

Cornering load angular displacement / radians

M at

er ia

l v ol

um e

/ c m

3

After 1000 iterations

51

5R1 GTP

section increases with distance from the tip) can be varied. Thus, there are five control vari- ables in all. A limit was imposed on the maximum allowed stress in the two loading cases.

Figure 6.5 shows the evolution in two-objective space (material volume versus cornering load angular displacement) of the contents of the archive, demonstrating how the archive converges onto the trade-off surface as the search progresses.

Figure 6.6 shows the final archive contents in the other two-objective projections (material volume versus impact load displacement and cornering load angular displacement versus impact load displacement). The trade-offs between the displacement objectives and the mate- rial volume are clear. The fact that heavier forks displace less under load is not unexpected, but the severity of the ‘elbow’ shaped trade-off surfaces revealed by the optimization is, per- haps, surprising.

6.4 M

ULTIOBJECTIVE

GA

The fact that GAs search from population to population rather than from one individual solu- tion to another makes them very well suited to performing multiobjective optimization. It is easy to conceive of a population being evolved onto the trade-off surface by a suitably config- ured GA. In fact, with an appropriate archiving scheme in place, the only modification required to a single objective GA, in order to perform multiobjective optimization, is in the

selection

scheme. As with single objective GAs, a wide variety of multiobjective selection schemes have been devised. Three of the most widely used (and most easily implemented) will be described here.

6.4.1 Pareto-Based Selection

This selection scheme was first proposed by Goldberg [1989]. It works by ranking the current population as follows:

Figure 6.6: Final Archives In Two-Objective Space.

20151050 20

40

60

80

100

120

Frontal impact displacement / mm

M at

er ia

l v ol

um e

/ c m

3

20151050 0.00

0.01

0.02

Co rn

er in

g lo

ad a

ng ul

ar di

sp la

ce m

en t /

ra di

an s

Frontal impact displacement / mm

52

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• First, the nondominated members of the population are identified. These are assigned rank 1.

• The rank 1 individuals are then removed from consideration and the nondominated mem- bers of the remaining population are identified. These are assigned rank 2.

• The rank 2 individuals are also removed from consideration and the nondominated members of the remaining population are identified. These are assigned rank 3.

• And so on until the entire population has been ranked.

An example of the result of this process is shown in Figure 6.7.

Having ranked all members of the current population, they are assigned selection probabilities based on their rankings in a manner similar to Baker [1985]’s single criterion ranking selection procedure. The probability of a rank

n

member of the current population being selected is given by:

, (6.8)

in which

N

is the population size,

S

is a selection pressure and

, (6.9)

where is the number of solutions in rank

i

. Selection using these probabilities can then be performed using any of the standard GA methods.

pn S N 1 Rn!"# $ Rn 2!"

N N 1!# $%

Rn 1 rn 2 ri i 1%

n 1!

&" "%

ri

Figure 6.7: An Example of Pareto-Based Population Ranking.

4 4

4

4

3

3 3

3 2

2

2

22

1 1

1 1

f1

f2

1

53

5R1 GTP

6.4.2 Dominance Count Ranking

Fonseca and Fleming [1993] proposed a slightly different scheme in which an individual’s rank corresponds to the number of members of the current population by which it is dominated (plus 1). Thus, as in Goldberg’s scheme, all the nondominated solutions have rank 1, but dom- inated individuals are penalized according the population density in the corresponding region of the trade-off surface. Figure 6.8 shows the same population as that in Figure 6.7 ranked using dominance counting. The difference in the ranking of some solutions is clearly quite dramatic. As with Pareto-based ranking, selection probabilities can then be assigned to indi- viduals based on their ranking.

6.4.3 Tournament Selection

Tournament selection can be applied in multiobjective GAs, just as in single objective GAs. A subset of solutions is randomly selected and the nondominated solutions in this subset identi- fied. If more solutions are nondominated than are required to be selected (in tournament selec- tion schemes either one or two parents are chosen in each tournament), then the required number are chosen randomly from these nondominated candidates.

6.4.4 Other Considerations

The other key component parts of a GA (crossover and mutation operators) do not need to be changed in order to perform multiobjective optimization.

Many variants and elaborations on the comparatively simple multiobjective GA described here have been developed. These have been well reviewed recently by Deb [2001].

Figure 6.8: An Example of Dominance Count Ranking.

13 9

9

10

5

6 7

3

7

2

1 1

1 1

f1

f2

1

3

3

3

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6.4.5 Algorithm Performance

The case study to be presented in the final lecture of this course will describe the application of a multiobjective GA to a difficult real-world problem, the design of Pressurised Water Reactor reload cores.

6.5 References

Baker, J.E., (1985) “Adaptive Selection Methods for Genetic Algorithms”, 101-111 in

Pro- ceedings of an International Conference on Genetic Algorithms and their Applications

(J.J. Grefenstette, editor), Lawrence Erlbaum Associates, Hillsdale, NJ. Deb, K., (2001)

Multi-objective Optimization Using Evolutionary Algorithms

, John Wiley & Sons Ltd., New York, NY.

Engrand, P., (1997) “A Multi-Objective Approach Based on Simulated Annealing and its Ap- plication to Nuclear Fuel Management”,

Proc. 5th Int. Conf. Nuclear Engineering

, Nice, ICONE5-2523.

Fonseca, C.M., and P.J. Fleming (1993) “Genetic Algorithms for Multi-Objective Optimiza- tion: Formulation, Discussion and Generalization”, 416-423 in

Genetic Algorithms:

Pro- ceedings of the Fifth International Conference

(S. Forrest, editor), Morgan Kaufmann, San Mateo, CA.

Goldberg, D.E., (1989)

Genetic Algorithms in Search, Optimization and Machine Learning

, Addison Wesley, Reading, MA.

Huang, M.D., F. Romeo and A. Sangiovanni-Vincentelli (1986) “An Efficient General Cooling Schedule for Simulated Annealing”,

Proc. IEEE Int. Conf. Computer Aided Design

, 381- 384.

Suppapitnarm, A., (1998)

A Simulated Annealing Algorithm for Multiobjective Design Optimi- zation

, MPhil Thesis, University of Cambridge. Suppapitnarm, A., K.A. Seffen, G.T. Parks and P.J. Clarkson (2000) “A Simulated Annealing

Algorithm for Multiobjective Optimization”

Eng. Opt.

33

, 59-85. White, S.R., (1984) “Concepts of Scale in Simulated Annealing”,

Proc. IEEE Int. Conf. Com- puter Design

, 646-651.

55

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