Metaheuristic Algorithms: A Critical Analysis

Metaheuristic Algorithms:
A Critical Analysis
Xin-She Yang
Middlesex 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(xt
, α),
For any given xt, the algorithm will generate a new solution xt+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).

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 the
Newton-Raphson Method, pattern search, quadratic programming, etc.
Traditional algorithms are mostly local search, thus cannot guarantee
global optimality (except for linear programming and convex
optimization).
Results often depend on the initial starting points (except linear and
convex 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 (heuristic
or metaheuristic algorithms, as a global optimizer) strive to
Increase the probability of ﬁnding the global optimality (as a global
search tool);
Solve a wider class of problems (treating them as a black-box);
Try to deal with highly nonlinear problems, problems with
discontinuity, 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 (Glover
1980s), ant colony optimization (Dorigo 1992), genetic programming
(Koza 1992), particle swarm optimization (Kennedy & Eberhart
1995), 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), bat
algorithm (Yang, 2010), flower pollination algorithm (2012), ...
For details, please refer to:
Xin-She Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014).

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)

Swarm Intelligence PSO
PSO
Particle Swarm Optimization (Kennedy and Eberhart, 1995)
xi
g∗
xj
vt+1
i = vt
i + αǫ1(g∗
− xt
i ) + βǫ2(x∗
i − xt
i ),
xt+1
i = xt
i + vt+1
i .
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:


xi
vi


t+1
=


1 1
−(αǫ1 + βǫ2) 1




xi
vi


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+1
i = xt
i + β0e−γr2
ij (xj − xi ) + α ǫt
i .
⋄ The objective landscape maps to a light-based landscape, and fireflies
swarm into the brightest points/regions.
⋄ There is no g∗, therefore, there is no leader. Also, a highly nonlinear
iterative system, so subdivision into multiswarms is possible.

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, and
each 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 population
and some update equations, FA is not PSO (not even similar to PSO).
Main differences
FA uses a nonlinear attraction mechanism (inverse-quare law plus
exponential decay). PSO mechanism is simply linear (xt
i − g∗).
The population in the FA can subdivide into subgroups and thus can
form multi-swarm automatically (PSO cannot).
The standard FA does not use g∗ (though PSO uses g∗).
xt+1
i = xt
i + β0e−γr2
ij (xj − xi ) + α ǫt
i .
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 let
these host birds raise their chicks, so minimum eﬀorts and maximum
success.
Timing: Cuckoo’s eggs hatch slightly earlier.
Fitness: Similarity of eggs in terms of texture, colour, etc.

Swarm Intelligence Cuckoo Search (Yang and Deb, 2009)
Cuckoo Search (Yang and Deb, 2009)
Cuckoo Behaviour and Idealization
Each cuckoo lays one egg (solution) at a time, and dumps its egg in a
randomly chosen nest.
The best nests with high-quality eggs (solutions) will carry out to the
next generation.
The egg laid by a cuckoo can be discovered by the host bird with a
probability pa and a nest will then be built.

Swarm Intelligence Cuckoo Search
Cuckoo Search
Local random walk:
xt+1
i = xt
i + s ⊗ H(pa − ǫ) ⊗ (xt
j − xt
k).
[xi , xj , xk are 3 different solutions, H(u) is a Heaviside function, ǫ is a
random number drawn from a uniform distribution, and s is the step size.
Global random walk via Lévy flights:
xt+1
i = xt
i + αL(s, λ), L(s, λ) =
λΓ(λ) sin(πλ/2)
π
1
s1+λ
, (s ≫ s0).
Generation of new moves by Lévy 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 Lévy Distribution and Lévy Flights
Lévy Distribution and Lévy Flights
L(s) =
1
π
∞
0
cos(τs)e−α τγ
dτ, (0 < γ ≤ 2),
=
γ
2π
exp −
γ
2(s − µ)
1
(s − µ)3/2
, 0 < µ < s < ∞ (for steps µ > 0).
s
Self-similar, multi-scale, global random walks.

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 to
20 ms. Frequencies of 25 kHz to 100 kHz.
Speed up pulse-emission rate and increase loudness when homing at a
prey.

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+1
i = vt
i + (xt
i − x∗)fi , xt+1
i = xt
i + vt
i .
Variations of Loudness and Pulse Rate
At+1
i ← αAt
i , α ∈ (0, 1], rt+1
i = r0
i [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, feature
selection ...
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
1 L + 19.84d2
1 r,
subject to



g1(x) = −d1 + 0.0193r ≤ 0
g2(x) = −d2 + 0.00954r ≤ 0
g3(x) = −πr2L − 4π
3 r3 + 1296000 ≤ 0
g4(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).

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) = 0.7854x1x2
2 (3.3333x2
3 + 14.9334x3 − 43.0934)
−1.508x1(x2
6 + x2
7 ) + 7.4777(x3
6 + x3
7 ) + 0.7854(x4x2
6 + x5x2
7 ),
subject to
g1 = 27
x1x2
2 x3
− 1 ≤ 0, g2 = 397.5
x1x2
2 x2
3
− 1 ≤ 0,
g3 =
1.93x3
4
x2x3d4
1
− 1 ≤ 0, g4 =
1.93x3
5
x2x3d4
2
− 1 ≤ 0,
g5 = 1
110x3
6
(745x4
hx3
)2 + 16.9 × 106 − 1 ≤ 0,
g6 = 1
85x3
7
(745x5
hx3
)2 + 157.5 × 106 − 1 ≤ 0,
g7 = x2x3
40 − 1 ≤ 0, g8 = 5x2
x1
− 1 ≤ 0,
g9 = x1
12x2
− 1 ≤ 0, g10 = 1.5x6+1.9
x4
− 1 ≤ 0,
g11 = 1.1x7+1.9
x5
− 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. z
must 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 (< 4
kN), .... 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 generation
using 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
Accuracy[%]
61 61 59
61 61 6160 61 61 62 6263 62 62 62 61
Mushrooms
Baseline
Binary
Sigmoid
Hyperbolic 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 efficient
among 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 better
results than K-means methods (Fong, Deb, Yang, Zhang, 2014).

Applications Theoretical Analysis
Theoretical Analysis
Though it still lacks a solid mathematical framework, theoretical studies
start 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 to
re-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 computing
efforts).
However, we lack good theories to understand either self-organization
or 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 a
set 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+1
i (i = 1, ..., n) are generated from xt
i ,
controlled by k parameters and m random numbers.
In principle, the behaviour of an algorithm is controlled by the eigenvalues
of 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
uniform
search
steepest
descent
Tabu Nelder-Mead
CS
PSO/FAEP/ES
SA 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 to
produce 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-dependent
parameters 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 be
directly applied to large-scale/high-dimensional problems (for
d > 1000) ?
Intelligence: Smart algorithms may be a buzz word, but can truly
intelligent 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 and
Metaheuristics, (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, Neural
Computing 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 truss
structures 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: a
model 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 for
multiobjective optimization, Engineering Optimization, Vol. 46, No. 9, 1222-1237 (2014).
M. K. Marchelvam, T. Prabaharan, X. S. Yang, A discrete firefly algorithm for multi-objective
hybrid 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 firefly
algorithm, 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 Bat
Algorithm, 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 multispectral
satellite 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 continuous
optimization problems, Expert Systems with Applications, Vol. 69, 159-175 (2017).

Thank you :) Any generic tool for optimization?
Any generic tool for optimization?

Metaheuristic Algorithms: A Critical Analysis

Related slideshows

More Related Content

Metaheuristic Algorithms: A Critical Analysis