metaheuristic algorithms: a critical analysis

Metaheuristic Algorithms:A Critical Analysis

Xin-She YangMiddlesex University London

ICHSA 2017

Xin-She Yang Nature-Inspired Computation ICHSA2017 1 / 54

The Essence of an Algorithm The Essence of an Algorithm

The Essence of an Algorithm

Essence of an Optimization Algorithm

To move to a new, better point x(t+1) from an existing location x

(t).

x1

x2

x(t)

x(t+1)

?

Population-based algorithms use multiple, interacting paths.

Different algorithms

Different strategies/approaches in generating these moves!


The Essence of an Algorithm The Essence of an Algorithm

The Essence of an Algorithm

In essence, an algorithm can be written (mathematically) as

xt+1 = A(x t , α),

For any given xt , the algorithm will generate a new solution x

t+1.

Functional or a Dynamical System?

We can view the above equation as

an iterative functional (with fixed points),

or a dynamical system (with attractors),

or a self-organization system (with organized structures),

or a Markov chain (e.g., Markov chain Monte Carlo).

The behavoir of the system (algorithm) can be controlled by A and α.

For details, please refer to: Xin-She Yang et al., Swarm Intelligence and

Bio-Inspired Computation: Theory and Applications, Elsevier, (2013).Xin-She Yang Nature-Inspired Computation ICHSA2017 3 / 54

The Essence of an Algorithm What’s Wrong with Traditional Algorithms?

What’s Wrong with Traditional Algorithms?

Some issues with traditional optimization algorithms such as theNewton-Raphson Method, pattern search, quadratic programming, etc.

Traditional algorithms are mostly local search, thus cannot guaranteeglobal optimality (except for linear programming and convexoptimization).

Results often depend on the initial starting points (except linear andconvex problems).

Methods tend to be problem-specific (e.g., k-opt, branch and bound).

Cannot deal with highly nonlinear optimization problems efficiently.

Struggle to cope problems with discontinuity.


The Essence of an Algorithm Nature-Inspired Optimization Algorithms

Nature-Inspired Optimization Algorithms

Almost all nature-inspired metaheuristic optimization algorithms (heuristicor metaheuristic algorithms, as a global optimizer) strive to

Increase the probability of finding the global optimality (as a globalsearch tool);

Solve a wider class of problems (treating them as a black-box);

Try to deal with highly nonlinear problems, problems withdiscontinuity, and even NP-hard combinatorial problems;

Draw design inspiration from Nature.


Nature-Inspired Algorithms Nature-Inspired Optimization Algorithms

Nature-Inspired Optimization Algorithms

Genetic algorithms (1960s/1970s), evolutionary strategy (Rechenberg& Swefel 1960s), evolutionary programming (Fogel et al. 1960s).

Simulated annealing (Kirkpatrick et al. 1983), Tabu search (Glover1980s), ant colony optimization (Dorigo 1992), genetic programming(Koza 1992), particle swarm optimization (Kennedy & Eberhart1995), differential evolution (Storn & Price 1996/1997)

harmony search (Geem et al. 2001), , artificial bee colony(Karaboga, 2005).

Firefly algorithm (Yang 2008), cuckoo search (Yang & Deb 2009), batalgorithm (Yang, 2010), flower pollination algorithm (2012), ...

For details, please refer to:Xin-She Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014).


http://www.sciencedirect.com/science/book/9780124167438

Nature-Inspired Algorithms Three different ways of thinking ...

Three different ways of thinking ...

Particle swarm optimization: swarm intelligence

Harmony search: music-inspired

Cuckoo search: swarm intelligence and also scale-free search


Swarm Intelligence Seeking Inspiration in Nature: Swarm Intelligence

Seeking Inspiration in Nature: Swarm Intelligence

Swarm Intelligence

Ants, bees, birds, fish ...Simple rules lead to complex behaviour.

Swarming Starlings (YouTube Video)


https://www.youtube.com/watch?v=QOGCSBh3kmM

Swarm Intelligence PSO

PSO

Particle Swarm Optimization (Kennedy and Eberhart, 1995)

x i

g∗

x j

vt+1i = v

ti + αǫ1(g

∗ − xti ) + βǫ2(x

∗

i − xti ),

xt+1i = x

ti + v

t+1i .

α, β = learning parameters, ǫ1, ǫ2=random numbers.


Swarm Intelligence PSO Demo and Disadvantages

PSO Demo and Disadvantages

The updating equations in PSO form a linear system:

x i

v i

t+1

=

1 1

−(αǫ1 + βǫ2) 1

x i

v i

t

+

0

αǫ1g∗ + βǫ2x

∗

i

.

PSO Demo Premature convergence


Swarm Intelligence Firefly Algorithm

Firefly Algorithm

Firefly Video (YouTube)

Firefly Behaviour and Idealization (Yang, 2008)

Fireflies are unisex and brightness varies with distance.

Less bright ones will be attracted to bright ones.

If no brighter firefly can be seen, a firefly will move randomly.

xt+1i = x

ti + β0e

−γr2ij (x j − x i ) + α ǫti .

⋄ The objective landscape maps to a light-based landscape, and firefliesswarm into the brightest points/regions.

⋄ There is no g∗, therefore, there is no leader. Also, a highly nonlineariterative system, so subdivision into multiswarms is possible.


https://www.youtube.com/watch?v=O69PhjhbWEc

Swarm Intelligence FA Demo and Advantages

FA Demo and Advantages

Paths of fireflies for 3D Rosenbrock function (Husselmann, 2014).


Swarm Intelligence Why is FA so efficient?

Why is FA so efficient?

FA Demo Multimodal optimality

Advantages of Firefly Algorithm over PSO

Automatically subdivide the whole population into subgroups, andeach subgroup swarms around a local mode/optimum.

Control modes/ranges by varying γ.

Control randomization by tuning parameters such as α.

Suitable for multimodal, nonlinear, global optimization problems.


Swarm Intelligence Firefly algorithm is Not PSO

Firefly algorithm is Not PSO

Though both algorithms (in fact, almost all algorithms) use a populationand some update equations, FA is not PSO (not even similar to PSO).

Main differences

FA uses a nonlinear attraction mechanism (inverse-quare law plusexponential decay). PSO mechanism is simply linear (x ti − g∗).

The population in the FA can subdivide into subgroups and thus canform multi-swarm automatically (PSO cannot).

The standard FA does not use g∗ (though PSO uses g∗).xt+1i = x

ti + β0e

−γr2ij (x j − x i) + α ǫti .

FA can find multiple optimal solutions simultaneously (PSO cannot).

FA has a fractal-like search structure (PSO cannot).


Swarm Intelligence Harmony Search

Harmony Search

Zong Woo Geem et al. (2001). Music-inspired metaheuristic.Now almost 14,000 papers [Google scholar] with diverse applications:

Variants for specific applications:

Continuous optimization ...

Discrete variants for scheduling, travelling-salesman problem,combinatorial optimization ...

Water distribution, network optimization ...

Clustering, classifications and feature selection ...

Multiobjective harmony search ...

Hybridization with other algorithms.

A diverse range of applications, for example, see ICHSA2017.


Swarm Intelligence Cuckoo Breeding Behaviour

Cuckoo Breeding Behaviour

Cuckoo Behavior (YouTube Video)

Evolutionary Advantages

Parallism: Dumps eggs in the nests of (multiple) host birds and letthese host birds raise their chicks, so minimum efforts and maximumsuccess.

Timing: Cuckoo’s eggs hatch slightly earlier.

Fitness: Similarity of eggs in terms of texture, colour, etc.


https://www.youtube.com/watch?v=mLuvBz2xFY4

Swarm Intelligence Cuckoo Search (Yang and Deb, 2009)

Cuckoo Search (Yang and Deb, 2009)

Cuckoo Behaviour and IdealizationEach cuckoo lays one egg (solution) at a time, and dumps its egg in arandomly chosen nest.

The best nests with high-quality eggs (solutions) will carry out to thenext generation.

The egg laid by a cuckoo can be discovered by the host bird with aprobability pa and a nest will then be built.


Swarm Intelligence Cuckoo Search

Cuckoo Search

Local random walk:

xt+1i = x

ti + s ⊗ H(pa − ǫ)⊗ (x t

j − xtk).

[x i , x j , xk are 3 different solutions, H(u) is a Heaviside function, ǫ is arandom number drawn from a uniform distribution, and s is the step size.

Global random walk via Levy flights:

xt+1i = x

ti + αL(s, λ), L(s, λ) =

λΓ(λ) sin(πλ/2)

π

1

s1+λ, (s ≫ s0).

Generation of new moves by Levy flights, random walk and elitism.


Swarm Intelligence

The normal distribution

p(x ;µ, σ2) =1

σ√2π

exp[−(x − µ)2

2σ2], −∞ < x <∞.

Isotropic Random Walks

SN =

N∑

i=1

Xi = X1 + ...+ XN = SN−1 + XN ,

s


Swarm Intelligence Levy Distribution and Levy Flights

Levy Distribution and Levy Flights

L(s) =1

π

∫

∞

0cos(τs)e−α τγdτ, (0 < γ ≤ 2),

=

√

γ

2πexp

[

−γ

2(s − µ)

] 1

(s − µ)3/2, 0 < µ < s <∞ (for steps µ > 0).

s

Self-similar, multi-scale, global random walks.Xin-She Yang Nature-Inspired Computation ICHSA2017 20 / 54

Swarm Intelligence Bat Algorithm (Yang, 2010)

Bat Algorithm (Yang, 2010)

BBC Video

Microbats use echolocation for hunting

Ultrasonic short pulses as loud as 110dB with a short period of 5 to20 ms. Frequencies of 25 kHz to 100 kHz.

Speed up pulse-emission rate and increase loudness when homing at aprey.


Swarm Intelligence Bat Algorithm

Bat Algorithm

Acoustics of bat echolocation

λ =v

f∼ 2 mm to 14 mm.

Rules used in the bat algorithm:

fi = fmin + (fmax − fmin)β, β ∈ [0, 1],

vt+1i = v ti + (x t

i − x∗)fi , xt+1i = x

ti + v

ti .

Variations of Loudness and Pulse Rate

At+1i ← αAt

i , α ∈ (0, 1], r t+1i = r0i [1− exp(−γt)].


Swarm Intelligence Other Algorithms

Other Algorithms

Genetic algorithms

Differential evolution

Artificial immune system

Flower pollination algorithm

Memetic algorithm

...

Reviews

Yang, X. S., Nature-Inspired Optimization Algorithms, Elsevier,(2014).


Applications Applications

Applications

Design optimization: structural engineering, product design ...

Scheduling, routing and planning: discrete, combinatorial problems ...

Image processing and data Mining: image segmentation, featureselection ...

Applications in almost all areas (e.g., finance, economics, engineering,industry, ...)

10 Examples in Science and Engineering


Applications Pressure Vessel Design Optimization (Example 1)

Pressure Vessel Design Optimization (Example 1)

r

d1

r

L d2


Applications Pressure Vessel Design

Pressure Vessel Design

This is a well-known test problem for optimization (e.g., see Cagnina et al.

2008) and it can be written as

minimize f (x) = 0.6224d1rL+ 1.7781d2r2 + 3.1661d2

1L+ 19.84d21 r ,

subject to

g1(x) = −d1 + 0.0193r ≤ 0g2(x) = −d2 + 0.00954r ≤ 0g3(x) = −πr2L− 4π

3 r3 + 1296000 ≤ 0g4(x) = L− 240 ≤ 0.

The simple bounds are

0.0625 ≤ d1, d2 ≤ 99× 0.0625, 10.0 ≤ r , L ≤ 200.0.

The best solution (Yang, 2010; Gandomi and Yang, 2011)

f∗ = 6059.714, x∗ = (0.8125, 0.4375, 42.0984, 176.6366).

Now we have a set of analytical solutions (Yang et al., 2013).Xin-She Yang Nature-Inspired Computation ICHSA2017 26 / 54

Applications Speed Reducer/Gear Box Design (Example 2)

Speed Reducer/Gear Box Design (Example 2)

Mixed-Integer Programming:

Continuous variables and integers.


Applications

f (x1, x2, x3, x4, x5, x6, x7) = 0.7854x1x22 (3.3333x

23 + 14.9334x3 − 43.0934)

−1.508x1(x26 + x27 ) + 7.4777(x36 + x37 ) + 0.7854(x4x26 + x5x

27 ),

subject to

g1 =27

x1x22 x3− 1 ≤ 0, g2 =

397.5x1x

22 x

23− 1 ≤ 0,

g3 =1.93x34x2x3d

41− 1 ≤ 0, g4 =

1.93x35x2x3d

42− 1 ≤ 0,

g5 =1

110x36

√

(745x4hx3

)2 + 16.9 × 106 − 1 ≤ 0,

g6 =1

85x37

√

(745x5hx3

)2 + 157.5 × 106 − 1 ≤ 0,

g7 =x2x340 − 1 ≤ 0, g8 =

5x2x1− 1 ≤ 0,

g9 =x1

12x2− 1 ≤ 0, g10 = 1.5x6+1.9

x4− 1 ≤ 0,

g11 = 1.1x7+1.9x5

− 1 ≤ 0.

Simple bounds are 2.6 ≤ x1 ≤ 3.6, 0.7 ≤ h ≤ 0.8, 17 ≤ x3 ≤ 28,7.3 ≤ x4 ≤ 8.3, 7.8 ≤ x5 ≤ 8.3, 2.9 ≤ x6 ≤ 3.9, and 5.0 ≤ x7 ≤ 5.5. zmust be integers.


Applications Dome Design (Example 3)

Dome Design (Example 3)

120-bar dome: Divided into 7 groups, 120 design elements, about 200 constraints

(Gandomi and Yang 2011; Yang et al. 2012).


Applications Tower Design (Example 4)

Tower Design (Example 4)

26-storey tower: 942 design elements, 244 nodal links, 59 groups/types, > 4000

nonlinear constraints (Yang et al. 2011; Gandomi & Yang 2012).


Applications Car Door Design (Example 5)

Car Door Design (Example 5)

Design better, safer and energy-efficient cars

Minimize weight, low crash deflection (< 32 mm), lower impact force (< 4kN), .... Even a side barrier has 11 design variables!


Applications Scheduling(Example 6)

Scheduling(Example 6)

MK Marichelvam, T Prabaharan, XS Yang, Improved cuckoo search algorithm for hybrid flow

shop scheduling problems to minimize makespan, Applied Soft Computing, vol. 19, no. 1,pp.

93-101 (2014).


Applications Soft Testing and Path Generation

Soft Testing and Path Generation

Srivastava, Mallikarjun, Yang (2013). Optimal test sequence generationusing firefly algorithm, Swarm and Evolutionary Computation, 8(2013)44-53.


Applications Travel Salesman Problem (Example 7)

Travel Salesman Problem (Example 7)


Applications Cuckoo Search for TSP

Cuckoo Search for TSP

Aziz Ouaarab, Belad Ahiod, Xin-She Yang, Discrete cuckoo search algorithm for the travelling

salesman problem, Neural Computing and Applications, (2013).

http://link.springer.com/article/10.1007/s00521-013-1402-2.


Applications Data Mining: Feature Selection (Example 8)

Data Mining: Feature Selection (Example 8)

Bat algorithm and cuckoo search are very competitive (Papa et al, 2013; 2014).

Also effective at dimension reduction (in terms of number of key features).

Dataset # samples #features #classes

Diabetes 768 8 2

DNA 2000 180 3

Mushrooms 8124 112 2

OPF BA CS FA ICS PSO0

10

20

30

40

50

60

70

80

Accura

cy [%

]

61 61 5961 61 6160 61 61 62 6263 62 62 62 61

Mushrooms

BaselineBinarySigmoidHyperbolic Tangent


Applications Image Processing (Example 9)

Image Processing (Example 9)

Satellite images and flooding estimates (Senthilnath & Yang, 2014)


Applications Satellite images

Satellite images

Three methods are compared, and FA obtained the best results.

Algorithms Optimal Worst Mean

GA 0.8044 0.7528 0.7824

PSO 0.8057 0.8009 0.8051

FA 0.8126 0.8019 0.8076

Also, Senthilnath et al. compared 14 different algorithms for classifications,and they found that the firefly algorithm seems to be most efficientamong 14 different classification methods.


Applications Classifications and Clustering (Example 10)

Classifications and Clustering (Example 10)

K-means clustering with nature-inspired algorithms can produce far betterresults than K-means methods (Fong, Deb, Yang, Zhang, 2014).


Applications Theoretical Analysis

Theoretical Analysis

Though it still lacks a solid mathematical framework, theoretical studiesstart to emerge:

Dynamic system theory

Bayesian statistics

Intermit search theory

Markov chain Monte Carlo

Self-organization theory

Convergence analysis

Exploration and exploitation


Applications Self-Organization

Self-Organization

Self-organizing systems are everywhere,physical, chemical, biological, social, artificial ...


Applications Similarities Between Self-Organization and Optimization

Similarities Between Self-Organization and Optimization

Self-organization:

High degrees of freedom, a sufficient number of states.

Noise, perturbation =⇒ diversity.

Selection mechanism =⇒ structure.

Far-from-equilibrium and large perturbation =⇒ potentially faster tore-organize.

Optimization Algorithms:

A sufficient larger population.

Randomization, stochastic components, exploration=⇒ to escape local optima.

Selection and exploitation =⇒ convergence & optimal solutions.

High-degrees of randomization=⇒ more likely to reach global optimality (but may be slow).


Applications But there are significant differences!

But there are significant differences!

Self-Organization:

Avenues to self-organization may be unclear.

Time may not be important.

Optimization (especially metaheuristics):

How to make an algorithm converge is very important.

Speed of convergence is crucial(to reach truly global optimality with the minimum computingefforts).However, we lack good theories to understand either self-organizationor metaheuristic optimization.


Applications Multi-Agent System (Swarm Intelligence?)

Multi-Agent System (Swarm Intelligence?)

For a multi-agent system or a swarm, an algorithm can be considered as aset of interacting Markov chain or a complex dynamical system

x1

x2...xn

t+1

= A[x1, ..., xn; ǫ1, ..., ǫm; p1(t), ..., pk (t)]

x1

x2...

xn

t

.

A population of solutions xt+1i (i = 1, ..., n) are generated from x

ti ,

controlled by k parameters and m random numbers.

In principle, the behaviour of an algorithm is controlled by the eigenvaluesof A, but in practice, it is almost impossible to figure out the eigenvalues(apart from very simple/rare cases).


Exploration & Exploitation Key Components in All Metaheuristics

Key Components in All Metaheuristics

So many algorithms – what are the common characteristics?

What are the key components?

How to use and balance different components?

What controls the overall behaviour of an algorithm?


Exploration & Exploitation Exploration and Exploitation

Exploration and Exploitation

Characteristics of Metaheuristics

Exploration and Exploitation, or Diversification and Intensification.

Exploitation/Intensification

Intensive local search, exploiting local information.E.g., hill-climbing.

Exploration/Diversification

Exploratory global search, using randomization/stochastic components.E.g., hill-climbing with random restart.


Exploration & Exploitation Summary

Summary

Exploitation

Exploration

uniformsearch

steepestdescent

Tabu Nelder-Mead

CS

PSO/FAEP/ESSA Ant/Bee

Genetic algorithms

Newton-Raphson

Best?

Free lunch?


The Future – Some Open Problems (Swarm Intelligence) The Future – Some Open Problems (Swarm Intelligence)

The Future – Some Open Problems (Swarm Intelligence)

Origin of Swarm Intelligence: What are the main mechanisms toproduce collective intelligence from seemingly simple local rules?Any theory for the rise of swarm intelligence?

Mathematical Analysis: A unified mathematical framework (e.g.,Markov chain theory, dynamical systems) is needed.

Comparison: What are the best/fair performance measures?

Parameter tuning: How to tune the algorithm-dependentparameters so that an algorithm can achieve the best performance?Is there a general framework for this?

Scalability: Can the algorithms that work for small-scale problems bedirectly applied to large-scale/high-dimensional problems (ford > 1000) ?

Intelligence: Smart algorithms may be a buzz word, but can trulyintelligent algorithms be developed?


The Future – Some Open Problems (Swarm Intelligence) Intelligent Algorithms?

Intelligent Algorithms?


The Future – Some Open Problems (Swarm Intelligence) Books on Nature-Inspired Algorithms

Books on Nature-Inspired Algorithms

Xin-She Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014).

Xin-She Yang, Cuckoo Search and Firefly Algorithm, Springer, (2014).


The Future – Some Open Problems (Swarm Intelligence) Thank you :) Any questions?

Thank you :) Any questions?

Xin-She Yang et al., Swarm Intelligence and Bio-Inspired Computation:Theory and Applications, Elsevier, (2013).

Xin-She Yang, Artificial Intelligence, Evolutionary Computing andMetaheuristics, (2013).


The Future – Some Open Problems (Swarm Intelligence) References

References

Xin-She Yang, Nature-Inspire Optimization Algorithm, Elsevier, (2014).

Xin-She Yang, Gebrail Bekdas, Sinan Melih Nigdeli, Metaheuristics and Optimization in

Civil Engineering, Springer, (2016).

Xin-She Yang and Suash Deb, Multiobjective cuckoo search for design optimization,Computers & Operations Research, 40(6), 1616–1624 (2013).

X. S. Yang and S. Deb, Cuckoo search: recent advances and applications, NeuralComputing and Applications, vol. 24, no. 1, pp. 169-174 (2014).

X. S. Yang and X. S. He, Bat algorithm: literature review and applications, Int. J.Bio-Inspired Computation, vol. 5, no. 3, pp. 141-149 (2013).

A. H. Gandomi, S. Talatahari and X. S. Yang, S. Deb, Design optimization of trussstructures using cuckoo search algorithm, Structural Design of Tall and Special Buildings,vol. 22, no. 17, pp. 1330-1340 (2013).

A. H. Gandomi and X. S. Yang, Chaotic bat algorithm, Journal of Computational Science,vol. 5, no. 2, pp. 224-232 (2014).

P. R. Srivastava, A. Varshney, P. Nama, X. S. Yang, Software test effort estimation: amodel based on cuckoo search, Int. J. Bio-Inspired Computation, vol. 4, no. 5, pp.278-285 (2012).


The Future – Some Open Problems (Swarm Intelligence)

X. S. Yang, M. Karamanoglu, X.S. He, Flower pollination algorithm: A novel approach formultiobjective optimization, Engineering Optimization, Vol. 46, No. 9, 1222-1237 (2014).

M. K. Marchelvam, T. Prabaharan, X. S. Yang, A discrete firefly algorithm for multi-objectivehybrid flowshop scheduling problems, IEEE Trans. Evolutionary Computation, vol. 18, no. 2,pp. 301-205 (2014).

P. R. Srivastava, B. Millikarjun, X. S. Yang, Optimal test sequence generation using fireflyalgorithm, Swarm and Evolutionary Computation, vol. 8, no. 1, pp. 44-53 (2013).

D. Rodrigues, L. A. M. Pereira, R. Y. M. Nakamura, K. A. P. Costa, X. S. Yang, A. N. Souza,J. P. Papa, A wrapper approach for feature selection and Optimum-Path Forest based on BatAlgorithm, Expert Systems with Applications, vol. 41, no. 5, pp. 2250-2258 (2014).

X. S. Yang, M. Karamanoglu, X. S. He, Multi-objective flower algorithm for optimization,Procedia Computer Science, Vol. 18, pp. 861-868 (2013).

X. S. Yang and J. P. Papa, Bio-inspired Computation and Applications in Image Processing,Elsevier (2016).

J. Senthilnath, S. Kulkarni, J. A. Benediktsson, X. S. Yang, A novel approach for multispectralsatellite image classification based on the bat algorithm, IEEE Geoscience and Remote Sensing

Letters, Vol. 13, No. 4, 599-603 (2016).

A. Chakri, R. Khelif, M. Benouaret, X. S. Yang, New directional bat algorithm for continuousoptimization problems, Expert Systems with Applications, Vol. 69, 159-175 (2017).


http://www.sciencedirect.com/science/book/9780128045367

Thank you :) Any generic tool for optimization?

Any generic tool for optimization?
