Dynamic Multimodal Optimization: A Preliminary Study

Dynamic Multimodal Optimization Benchmark Functions Brain Storm Optimization

Dynamic Multimodal Optimization: A PreliminaryStudy

Shi Cheng

School of Computer Science

March 29, 2019

Shi Cheng ([email protected]) Dynamic Multimodal Optimization March 29, 2019 1 / 62


Outline

1 Dynamic Multimodal Optimization

2 Benchmark Functions

3 Brain Storm Optimization



Abstract

A dynamic multimodal optimization problem is defined as anoptimization problem with multiple global optima and thecharacteristics of global optima are changed during the search process.

The benchmark problems have played a fundamental role in verifyingthe algorithm’s search ability. Two cases are used to illustrate theapplication scenario of DMO.



Abstract

A set of benchmark functions on DMO, which contains eightproblems, are proposed to show the difficulty of DMO. The propertiesof the proposed benchmark problems, such as the distribution ofsolutions, the scalability, the number of global/local optima, arediscussed.

Brain storm optimization (BSO) algorithm was used to solve theDMO problem. The effectiveness of the BSO algorithm was validatedon a test function.



Problem

f(x)

0 0.1 0.2 0.3x

x sin 1x

min f (x) = x sin 1x , x ∈ (0, 0.4]



Optimization

f(x)

0 0.1 0.2 0.3x

x sin 1x

min f (x) = x sin 1x , x ∈ (0, 0.4]



Optimization

x1

x2

x3

Ω = {x ∈ Rn}

Solution Space →

f2

f1

Λ = {y ∈ Rm}Objective Space

(a) (b)

Figure: The mapping from solution space to objective space.



Optimization

An optimization problem in Rn, or simply an optimization problem, isa mapping f : Rn → Rm, where Rn is term as solution space (orparameter space, problem space, decision space), and Rm is term asobjective space.

The optimization problem is to find

arg minx∈S

f (x) or maxx∈S

f (x)



Backgrounds: Dynamic Optimization

A dynamic optimization problem could be defined as follows:

min ~f (~x , t) = {f1(~x , t), · · · , fM(~x , t)}s.t. ~g(~x , t) > 0

~h(~x , t) = 0

where ~x is the decision variables; ~f is the objectives to be minimizedwith respect to time t.



Backgrounds: Multimodal Optimization

Many optimization algorithms are designed for locating a single globalsolution. Nevertheless, many real-world problems may have multiplesatisfactory solutions exist.

The multimodal optimization problem is a function with multipleglobal/local optimal values.



Backgrounds: Multimodal Optimization

For multimodal optimization, the objective is to locate multiplepeaks/optima in a single run, and to keep these found optima untilthe end of a run.

An algorithm on solving multimodal optimization problems shouldhave two kinds of abilities: find global/local optima as many aspossible and preserve these found solutions until the end of the search.



Definition: Dynamic Multimodal Problem

Dynamic multimodal optimization problem is a combination ofdynamic optimization problem and multimodal optimization problem.

Unlike the multimodal optimization problems, the uncertainty existsin dynamic multimodal optimization problems.

For example, some difficulties of DMO problems are as follows:

1 The number of global optima is changed during the search;2 The locations of global optima are all/partially changed during the

search.



Application Scenarios: Tracking of multiple targets

(a) Initialization (b) t = ta

(c) t = tb (d) t = tc



Dynamic Tracking of Multiple Targets Problem

Several targets are followed by multiple trackers. It is not fixedbetween the target and a tracker. The position of each tracker ischanged with the changing of targets.

1 Figure (a) shows the initial state of all targets and trackers.2 The trackers are surrounded all targets at time ta in Figure (b).3 Figure (c) shows that all targets are moved to new positions at time tb.4 Figure (d) shows that all targets are surrounded again at time tc .



Application scenarios: Dynamic multipath routing

(a) Initialization (b) t = ta (c) t = tb



Dynamic Multipath Routing Problem

Multiple paths are designed between the source objet and targetobject. The positions of source object and the target object arechanged over time.

1 Figure (a) shows the initial positions for the source object and thetarget object.

2 Figure (b) shows that two paths with equal length are designedbetween two objects at time ta. The positions for two objects aremoved at time tb.

3 Figure (c) shows that two paths with equal length are designed for twoobjects at new positions.



Proposed Benchmark Generator

To verify the algorithm’s search ability, various benchmark functionswere proposed and compared for different type of optimizationproblems, such as multimodal multi-objective optimization problems,dynamic multiobjective optimization problems.

In the multimodal optimization, for example, 20 multimodal functionsare formulated as maximization problems in [1] and 15 scalablemultimodal functions are formulated as minimization problems in [2].

For simplicity and clarity, we only considered the simple functions,and the rotation is not used here.

1 X. Li, A. Engelbrecht, and M. G. Epitropakis, “Benchmark functions for CEC’2013 special session and competition onniching methods for multimodal function optimization,” Evolutionary Computation and Machine Learning Group, RMITUniversity, Australia, Tech. Rep., 2013.

2 B. Qu, J. Liang, Z. Wang, Q. Chen, and P. Suganthan, “Novel benchmark functions for continuous multimodaloptimization with comparative results,” Swarm and Evolutionary Computation, vol. 26, pp. 2334, 2016.




A dynamic multimodal optimization function is able to be constructedby the Equation (1). The original optima are changed with a dynamicshifted value.

xi = xi ,t + shifti ,t (1)

where xi is a original solution, shifti ,t is a dynamic changed value, andxi ,t is the shifted solution.




The difference between xi and xi ,t is a dynamic changed value shifti ,t .To construct a mapping from xi and xi ,t and two solutions are in thesame search space. An example of shift value is as follows:

shifti ,t =

{− t

T × R if xi ≤ Ubound

− tT × R − R if xi > Ubound

(2)

where t is the number of current iteration, T is the total number ofiteration, R = Ubound − Lbound is the search range, and Ubound andLbound are the upper and lower bound for the search range,respectively.




f(x)

0 1 2xf(x) = 0.6x sin(πx) + 1

f(x)

0 1 2xx =mod((xt − 0.5), ubound)

(a) original function (b) t = ta

f(x)


f(x)


(c) t = tb (d) t = tc



Benchmark functions

The difference between xi and xi ,t is a dynamic changed value shifti ,t .To construct a mapping from xi and xi ,t and two solutions are in thesame search space. An example of shift value is as follows:

shifti ,t =

{− t

T × R if xi ≤ Ubound

− tT × R − R if xi > Ubound

(3)

where t is the number of current iteration, T is the total number ofiteration, R = Ubound − Lbound is the search range, and Ubound andLbound are the upper and lower bound for the search range,respectively.



f1 Two-Peak Trap

f1(x) = 200D +D∑i=1

yi

yi =

−160 + x2

i if xi < 016015 (xi − 15) if 0 ≤ xi ≤ 152005 (15− xi ) if 15 ≤ xi ≤ 20−200 + (xi − 20)2 if xi > 20

where xi = xi ,t + shifti ,t , and xi ∈ [−10, 30].

The original global optima is [20, 20, · · · , 20]D , thus for the f1(xt),the global optima isx∗t = [20− shifti ,t , 20− shifti ,t , · · · , 20− shifti ,t ]

D , and f1(x∗) = 0.



f1 Two-Peak Trap

−10 −5 0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

x

f 1(x)

Two−Peak Trap

x [−10, 30]



f2 Five-Uneven-Peak Trap

f2(x) = 200D +D∑i=1

yi

yi =

−200 + x2i if xi < 0

−80(2.5− xi ) if 0 ≤ xi < 2.5−64(xi − 2.5) if 2.5 ≤ xi < 5−64(7.5− xi ) if 5 ≤ xi < 7.5−28(xi − 7.5) if 7.5 ≤ xi < 12.5−28(17.5− xi ) if 12.5 ≤ xi < 17.5−32(xi − 17.5) if 15 ≤ xi < 22.5−32(27.5− xi ) if 22.5 ≤ xi < 27.5−80(xi − 27.5) if 27.5 ≤ xi ≤ 30−200 + (xi − 30)2 if xi > 30

where xi = xi ,t + shifti ,t , and xi ∈ [−5, 35].

The original global optima is xi = 0 or 30 for i = 1, 2, · · · ,D, thus forthe f2(xt), the global optima is x∗i ,t = 0− shifti ,t or 30− shifti ,t , andf2(x∗) = 0.



f2 Five-Uneven-Peak Trap

−5 0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

x

f 2(x)

Five−Uneven−Peak Trap

x [−5, 35]



f3 Equal Minima

f3(x) = D +D∑i=1

yi

yi =

x2i if xi < 0− sin6(5πxi ) if 0 ≤ xi ≤ 1(xi − 1)2 if xi > 1

where xi = xi ,t + shifti ,t , and xi ∈ [−0.5, 1.5].

The original global optima is xi = 0.1, 0.3, 0.5, 0.7, or 0.9 fori = 1, 2, · · · ,D, thus for the f3(xt), the global optima is x∗i ,t =0.1− shifti ,t , 0.3− shifti ,t , 0.5− shifti ,t , 0.7− shifti ,t , or 0.9− shifti ,t ,and f3(x∗) = 0.



f3 Equal Minima

−0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

f 3(x)

Equal Minima

x [−0.5, 1.5]



f4 Decreasing Minima

f4(x) = D +D∑i=1

yi

yi =

x2i if xi < 0 or xi > 1

− exp[−2 log(2) · ( xi−0.10.8 )2] · sin6(5πxi )if 0 ≤ xi ≤ 1


The original global optima is [0.1, 0.1, · · · , 0.1]D , thus for the f1(xt),the global optima isx∗ = [0.1− shifti ,t , 0.1− shifti ,t , · · · , 0.1− shifti ,t ]

D , and f4(x∗) = 0.



f4 Decreasing Minima

−0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

f 4(x)

Decreasing Minima

x [−0.5, 1.5]



f5 Uneven Minima

f5(x) =D∑i=1

yi + D

yi =

{x2i if 0 ≤ xi ≤ 1

− sin6(5π(x3/4i − 0.05)) if xi < 0 or xi > 1


The original global optima is xi = 0.07969939, 0.24665545,0.45062669, 0.68142022, or 0.93389520 for i = 1, 2, · · · ,D, thus forthe f3(xt), the global optima is x∗t = xi − shifti ,t , and f5(x∗) = 0.



f5 Uneven Minima

−0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

f 5(x)

Uneven Minima

x [−0.5, 1.5]



f6 Himmelblau’s Function

f6(x) =D−1∑

i=1,3,5,...

[(y2i + yi+1 − 11)2 + (yi + y2

i+1 − 7)2]

yi =

{xi + 3 if i%2 == 1xi + 2 if i%2 == 0

where D must be an even number, xi = xi ,t + shifti ,t , andxi ∈ [−10, 10].

The original global optima is xi = [0, 0], or [−6.779310,−5.283185],or [0.584428,−3.848126], or [−5.805118, 1.131312], fori = 1, 2, · · · , D2 , thus for the f6(xt), the global optima isx∗t = xi − shifti ,t , and f6(x∗) = 0.



f6 Himmelblau’s Function



f7 Six-Hump Camel Back

f7(x) =D−1∑

i=1,3,5,...

{−4[(4− 2.1y2i +

y4i

3)y2

i

+ yiyi+1 + (−4 + 4y2i+1)y2

i+1]}+ 2.063257D

yi =

{xi − 0.089842 if i%2 == 1xi + 0.712656 if i%2 == 0

where D must be an even number, xi = xi ,t + shifti ,t , and xi ∈ [−2, 2].

The original global optima is xi = [0, 0], or[0.17968401,−1.4253124], for i = 1, 2, · · · , D2 , thus for the f7(xt),the global optima is x∗t = xi − shifti ,t , and f7(x∗) = 0.The differencebetween xi and xi ,t is a dynamic changed value shifti ,t .



f7 Six-Hump Camel Back

−2−1

01

2

−2

−1

0

1

2−1000

0

1000

2000

3000

4000

5000

Six−Hump Camel Back



f8 Vincent Function

f8(x) =1

D

D∑i=1

(yi + 1.0)

yi =

(0.25− xi )

2 − sin(10 log(0.25)) if xi < 0.25− sin(10 log(xi )) if 0.25 ≤ xi ≤ 10(xi − 10)2 − sin(10 log(10)) if xi > 10

where xi = xi ,t + shifti ,t , and xi ∈ [−0.5, 11], f8(x∗) = 0.

The original global optima is xi = 0.333018, 0.624228, 1.170088,2.193280, 4.111207, or 7.706277 for i = 1, 2, · · · ,D, thus for thef8(xt), the global optima is x∗t = xi − shifti ,t , and f8(x∗) = 0.



f8 Vincent Function

−2 0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

x

f 8(x)

Vincent Function

x [−0.5, 11]



Properties of eight benchmark functions

Func. Function Name Global Optimum Local Optima Search Range

f1 Two-Peak Trap 1 2D − 1 [−10, 30]Df2 Five-Uneven-Peak Trap 2D 5D − 2D [−5, 35]Df3 Equal Minima 5D 0 [−0.5, 1.5]Df4 Decreasing Minima 1 5D − 1 [−0.5, 1.5]Df5 Uneven Minima 5D 0 [−0.5, 1.5]Df6 Himmelblau’s Function 4D/2 0 [−10, 10]Df7 Six-Hump Camel Back 2D/2 0 [−2, 2]Df8 Vincent Function 6D 0 [−0.5, 11]D



Performance Criteria

The value and number of found global optima could be used inperformance criteria.

Two criteria are used to measure the number of found global optima.One is the total number of global optima found in all runs. The otherindicator is the quality of the global optima found, i.e., the precisionfor the solutions, over multiple runs [1]. The equations of PRcalculation are given in Eq. (4).

PR =

∑NRrun=1 NPFi

NKP × NR=

NPF

NKP × NR(4)

where NPFi denotes the number of global optima found in the end ofthe i-th run, NKP the number of known global optima [1].

1 X. Li, A. Engelbrecht, and M. G. Epitropakis, “Benchmark functions for CEC’2013 special session and competition onniching methods for multimodal function optimization,” Evolutionary Computation and Machine Learning Group, RMITUniversity, Australia, Tech. Rep., 2013.



Performance Criteria

The position and the number of optima are not changed for a specificproblem in the multimodal optimization.

One calculation in the performance criteria is enough for staticproblems. However, this calculation should be evaluated at leastseveral times for dynamic problems.

Calculating the performance criteria value several times andcomparing the mean value may be a good way to verify analgorithm’s effectiveness in dynamic multimodal optimization.



Brain storm optimization algorithms

1 Randomly generate n potential solutions (individuals);

2 Clustering/Classifying n individuals into m groups;

3 Evaluate the n individuals;

4 Rank individuals in each group and record the best individual asgroup center in each group;

5 Occasionally, randomly replace a group center;

6 Generate new individuals;

7 If n new individuals have been generated, go to step 8; otherwise goto step 6;

8 Terminate if pre-determined maximum number of iterations has beenreached; otherwise go to step 2.




Start

Initialization

Solution Evaluation

Solution Clustering/Classification

Solution SelectionSolution Generation

End?

Stop

No

Yes




Start

Initialization

Converging Operation

Diverging Operation

End?

Stop

Yes

No



Dynamic Multimodal Problem

A dynamic multimodal optimization problem is a combination of thedynamic optimization problem and multimodal optimization problem.

In order to give a simple illustration, an example of a dynamicmultimodal optimization problem is as follows.

f (x) = | sin(t × π × x)− x

t × π |

where x ∈ [−2, 2] and t ∈ [ 1π , π].



Dynamic Multimodal Problem: Example

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

x [−2, 2], t = 1/

f(x)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

x [−2, 2], t = /3

f(x)

(a) t = 1/π (b) t = π/3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

x [−2, 2], t = /2

f(x)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

x [−2, 2], t =

f(x)

(c) t = π/2 (d) t = π



Nonlinear equation system with time variable

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1 [−2, 2], t = 1/

x 2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1 [−2, 2], t = /3

x 2

(a) t = 1/π (b) t = π/3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1 [−2, 2], t = /2

x 2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1 [−2, 2], t =

x 2

(c) t = π/2 (d) t = π




It can be seen that there are 3, 7, 13 equal optima with t = π3 , t = π

2 ,and t = π, respectively.

It gives an example of problem with the equal optima, i.e., the valueof all optima is equal to 0.

The dynamic multimodal optimization problem also could be extendedto problems with unequal optima, which will be much harder to solve.




The dynamic multimodal optimization could be applied to real-worldproblems, for example, nonlinear equation system with time variable{

sin(t × π × x1)− x2 = 0x1 − t × π × x2 = 0

(5)

where x ∈ [0, 1] and t ∈ [ 1π , π].



Parameter Settings

The BSO-OS algorithm has been used in the experimental study.

The detailed parameter settings are as follows:

BSO-OS algorithm: pelitist = 0.1, pone = 0.8, slope k = 500.



Experimental Results

0 100 200 300 400 50010

−4

10−3

10−2

Iteration

Err

or

BSO−OS

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3

3.5

4

Iteration

Num

ber

of o

ptim

a

BSO−OS

(a) search error (b) number of global optima obtained

Figure: Results of BSO-OS algorithm solving a dynamic multimodal optimizationproblem



Discussion

There are many iterations in the swarm intelligence algorithm, whichindicates that massive solutions will be evaluated during the search.For the reason of computing and storage efficiency, there is noalgorithm to store all the evaluated solutions.

Several representative solutions, such as the local best or personalbest solutions in particle swarm optimization algorithm, are used as amemory system for search algorithm.

The distribution of solutions is used in BSO algorithms, and there isno explicit memory system of the previous iterations.

This memory system will be beneficial when algorithm solvesproblems in the static environment. However, for problems in thedynamic environment, this memory may mislead the search direction.



Summary

Dynamic multimodal optimization (DMO) problem

The DMO problem is defined as an optimization problem withmultiple global optima and the characteristics of global optima arechanged during the search process.

The benchmark problems have played an important role in verifyingthe algorithm’s search ability. A set of DMO benchmark functionswere designed and discussed.

In future, different swarm intelligence algorithms could be used forsolving DMO problems.



Thematic Issue: Brain Storm Optimization Algorithms

Shi Cheng and Yuhui Shi. Thematic issue on “Brain StormOptimization Algorithms”.Memetic Computing, 10(4), 351-352.



Book: Brain Storm Optimization Algorithms

Shi Cheng and Yuhui Shi. Brain Storm Optimization Algorithms:Concepts, Principles and Applications.Adaptation, Learning, and Optimization. Springer InternationalPublishing AG, 2019.



Call for papers

International Journal of Swarm Intelligence Research (IJSIR)IGI Global.



References I

Y. Shi, “Brain storm optimization algorithm,” in Advances in SwarmIntelligence (Y. Tan, Y. Shi, Y. Chai, and G. Wang, eds.), vol. 6728 ofLecture Notes in Computer Science, pp. 303–309, SpringerBerlin/Heidelberg, 2011.

Y. Shi, “An optimization algorithm based on brainstorming process,”International Journal of Swarm Intelligence Research (IJSIR), vol. 2,pp. 35–62, October-December 2011.

Y. Shi, “Brain storm optimization algorithm in objective space,” inProceedings of 2015 IEEE Congress on Evolutionary Computation, (CEC2015), (Sendai, Japan), pp. 1227–1234, IEEE, 2015.

S. Cheng, Q. Qin, J. Chen, and Y. Shi, “Brain storm optimization algorithm:A review,” Artificial Intelligence Review, vol. 46, no. 4, pp. 445–458, 2016.



References II

S. Cheng, Y. Sun, J. Chen, Q. Qin, X. Chu, X. Lei, and Y. Shi, “Acomprehensive survey of brain storm optimization algorithms,” inProceedings of 2017 IEEE Congress on Evolutionary Computation (CEC2017), (Donostia, San Sebastian, Spain), pp. 1637–1644, IEEE, 2017.

S. Cheng, J. Chen, Y. Sun, and Y. Shi, “Developmental brain stormoptimization algorithms: From a data-driven perspective,” Journal ofZhengzhou University (Engineering Science), vol. 39, no. 3, pp. 22–28, 2018.

S. Cheng, H. Lu, W. Song, J. Chen, and Y. Shi, “Dynamic multimodaloptimization using brain storm optimization algorithms,” in Bio-inspiredComputing: Theories and Applications (BIC-TA 2018), pp. 236–245,Springer Singapore, 2018.



References III

S. Cheng, Y.-n. Guo, X. Lei, Y. Zhang, J. Liang, and Y. Shi, “Dynamicmultimodal optimization: A preliminary study,” in Proceedings of 2019 IEEECongress on Evolutionary Computation (CEC 2019), (Wellington, NewZealand), IEEE, 2019.



Acknowledgement

Thank you for your time!


Dynamic Multimodal Optimization: A Preliminary Study

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