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This is the Tutorial given at the FedCSIS2011 in Poland.
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Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Nature-Inspired Metaheristics Algorithmsfor Optimization and Computational Intelligence
Xin-She Yang
National Physical Laboratory, UK
@ FedCSIS2011
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are wrong, but some are useful.
- George Box, Statistician
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are inaccurate, but some are useful.
- George Box, Statistician
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are inaccurate, but some are useful.
- George Box, Statistician
All algorithms perform equally well on average over all possiblefunctions.
- No-free-lunch theorems (Wolpert & Macready)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are inaccurate, but some are useful.
- George Box, Statistician
All algorithms perform equally well on average over all possiblefunctions. How so?
- No-free-lunch theorems (Wolpert & Macready)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are inaccurate, but some are useful.
- George Box, Statistician
All algorithms perform equally well on average over all possiblefunctions. Not quite! (more later)
- No-free-lunch theorems (Wolpert & Macready)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Intro
Intro
Computational science is now the third paradigm of science,complementing theory and experiment.
- Ken Wilson (Cornell University), Nobel Laureate.
All models are inaccurate, but some are useful.
- George Box, Statistician
All algorithms perform equally well on average over all possiblefunctions. Not quite! (more later)
- No-free-lunch theorems (Wolpert & Macready)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Overview
Overview
Part I
Introduction
Metaheuristic Algorithms
Monte Carlo and Markov Chains
Algorithm Analysis
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Overview
Overview
Part I
Introduction
Metaheuristic Algorithms
Monte Carlo and Markov Chains
Algorithm Analysis
Part II
Exploration & Exploitation
Dealing with Constraints
Applications
Discussions & Bibliography
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c ,
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
=⇒
min t =
∫ d
0
1
vds =
∫ d
0
√
1 + y′2
√
2g [h − y(x)]dx
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
=⇒
min t =
∫ d
0
1
vds =
∫ d
0
√
1 + y′2
√
2g [h − y(x)]dx
=⇒
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
=⇒
min t =
∫ d
0
1
vds =
∫ d
0
√
1 + y′2
√
2g [h − y(x)]dx
=⇒ =⇒
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
A Perfect Algorithm
A Perfect Algorithm
What is the best relationship among E , m and c?
Initial state: m,E ,c , =⇒ =⇒E =mc2
Steepest Descent
=⇒
min t =
∫ d
0
1
vds =
∫ d
0
√
1 + y′2
√
2g [h − y(x)]dx
=⇒ =⇒
x = A2 (θ − sin θ)
y = h − A2 (1− cos θ)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Computing in Reality
Computing in Reality
A Problem & Problem Solvers⇓
Mathematical/Numerical Models
⇓Computer & Algorithms & Programming
⇓Validation⇓
Results
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
What is an Algorithm?
What is an Algorithm?
Essence of an Optimization Algorithm
To move to a new, better point xi+1 from an existing knownlocation xi .
x1
x2
xi
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
What is an Algorithm?
What is an Algorithm?
Essence of an Optimization Algorithm
To move to a new, better point xi+1 from an existing knownlocation xi .
x1
x2
xi
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
What is an Algorithm?
What is an Algorithm?
Essence of an Optimization Algorithm
To move to a new, better point xi+1 from an existing knownlocation xi .
x1
x2
xi
xi+1
?
Population-based algorithms use multiple, interacting paths.
Different algorithms
Different strategies/approaches in generating these moves!
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Optimization is Like Treasure Hunting
Optimization is Like Treasure Hunting
How to find a treasure, a hidden 1 million dollars?What is your best strategy?
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Optimization Algorithms
Optimization Algorithms
Deterministic
Newton’s method (1669, published in 1711), Newton-Raphson(1690), hill-climbing/steepest descent (Cauchy 1847),least-squares (Gauss 1795),
linear programming (Dantzig 1947), conjugate gradient(Lanczos et al. 1952), interior-point method (Karmarkar1984), etc.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Stochastic/Metaheuristic
Stochastic/Metaheuristic
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 swarmoptimization (Kennedy & Eberhart 1995), differentialevolution (Storn & Price 1996/1997),
harmony search (Geem et al. 2001), honeybee algorithm(Nakrani & Tovey 2004), ..., firefly algorithm (Yang 2008),cuckoo search (Yang & Deb 2009), ...
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent/Hill Climbing
Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very efficient for local search.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Newton’s Method
xn+1 = xn −H−1∇f , H =
∂2f∂x1
2 · · · ∂2f∂x1∂xn
.... . .
...∂2f
∂xn∂x1· · · ∂2f
∂xn2
.
Generation of new moves by gradient.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Newton’s Method
xn+1 = xn −H−1∇f , H =
∂2f∂x1
2 · · · ∂2f∂x1∂xn
.... . .
...∂2f
∂xn∂x1· · · ∂2f
∂xn2
.
Quasi-Newton
If H is replaced by I, we have
xn+1 = xn − αI∇f (xn).
Here α controls the step length.
Generation of new moves by gradient.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Steepest Descent Method (Cauchy 1847, Riemann 1863)
Steepest Descent Method (Cauchy 1847, Riemann 1863)
From the Taylor expansion of f (x) about x(n), we have
f (x(n+1)) = f (x(n) + ∆s) ≈ f (x(n) + (∇f (x(n)))T ∆s,
where ∆s = x(n+1) − x(n) is the increment vector.So
f (x(n) + ∆s)− f (x(n)) = (∇f )T∆s < 0.
Therefore, we have∆s = −α∇f (x(n)),
where α > 0 is the step size.In the case of finding maxima, this method is often referred to ashill-climbing.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Conjugate Gradient (CG) Method
Conjugate Gradient (CG) Method
Belong to Krylov subspace iteration methods. The conjugategradient method was pioneered by Magnus Hestenes, EduardStiefel and Cornelius Lanczos in the 1950s. It was named as one ofthe top 10 algorithms of the 20th century.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Conjugate Gradient (CG) Method
Conjugate Gradient (CG) Method
Belong to Krylov subspace iteration methods. The conjugategradient method was pioneered by Magnus Hestenes, EduardStiefel and Cornelius Lanczos in the 1950s. It was named as one ofthe top 10 algorithms of the 20th century.
A linear system with a symmetric positive definite matrix A
Au = b,
is equivalent to minimizing the following function f (u)
f (u) =1
2uTAu− bTu + v,
where v is a vector constant and can be taken to be zero. We caneasily see that ∇f (u) = 0 leads to Au = b.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
CG
CG
The theory behind these iterative methods is closely related to theKrylov subspace Kn spanned by A and b as defined by
Kn(A,b) = {Ib,Ab,A2b, ...,An−1b},
where A0 = I.If we use an iterative procedure to obtain the approximate solutionun to Au = b at nth iteration, the residual is given by
rn = b− Aun,
which is essentially the negative gradient ∇f (un).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
The search direction vector in the conjugate gradient method issubsequently determined by
dn+1 = rn −dT
n Arn
dTn Adn
dn.
The solution often starts with an initial guess u0 at n = 0, andproceeds iteratively. The above steps can compactly be written as
un+1 = un + αndn, rn+1 = rn − αnAdn,
anddn+1 = rn+1 + βndn,
where
αn =rTn rn
dTn Adn
, βn =rTn+1rn+1
rTn rn.
Iterations stop when a prescribed accuracy is reached.Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Gradient-free Methods
Gradient-free Methods
Gradient-base methods
Requires the information of derivatives. Not suitable for problemswith discontinuities.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Gradient-free Methods
Gradient-free Methods
Gradient-base methods
Requires the information of derivatives. Not suitable for problemswith discontinuities.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Gradient-free Methods
Gradient-free Methods
Gradient-base methods
Requires the information of derivatives. Not suitable for problemswith discontinuities.
Gradient-free or derivative-free methods
BFGS, Downhill simplex, Trust-region, SQP ...
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Nelder-Mead Downhill Simplex Method
Nelder-Mead Downhill Simplex Method
The Nelder-Mead method is a downhill simplex algorithm, firstdeveloped by J. A. Nelder and R. Mead in 1965.
A Simplex
In the n-dimensional space, a simplex, which is a generalization ofa triangle on a plane, is a convex hull with n + 1 distinct points.For simplicity, a simplex in the n-dimension space is referred to asn-simplex.
(a) (b) (c)Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Downhill Simplex Method
Downhill Simplex Method
s
xn+1
xr
x s
xr
xn+1
xe
xcxn+1
The first step is to rank and re-order the vertex values
f (x1) ≤ f (x2) ≤ ... ≤ f (xn+1),
at x1, x2, ..., xn+1, respectively. Wikipedia Animation
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Metaheuristic
Metaheuristic
Most are nature-inspired, mimicking certain successful features innature.
Simulated annealing
Genetic algorithms
Ant and bee algorithms
Particle Swarm Optimization
Firefly algorithm and cuckoo search
Harmony search ...
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Simulated Annealling
Simulated Annealling
Metal annealing to increase strength =⇒ simulated annealing.
Probabilistic Move: p ∝ exp[−E/kBT ].
kB=Boltzmann constant (e.g., kB = 1), T=temperature, E=energy.
E ∝ f (x),T = T0αt (cooling schedule) , (0 < α < 1).
T → 0, =⇒p → 0, =⇒ hill climbing.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Simulated Annealling
Simulated Annealling
Metal annealing to increase strength =⇒ simulated annealing.
Probabilistic Move: p ∝ exp[−E/kBT ].
kB=Boltzmann constant (e.g., kB = 1), T=temperature, E=energy.
E ∝ f (x),T = T0αt (cooling schedule) , (0 < α < 1).
T → 0, =⇒p → 0, =⇒ hill climbing.
This is essentially a Markov chain.Generation of new moves by Markov chain.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
An Example
An Example
Xin-She Yang FedCSIS2011
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Genetic Algorithms
Genetic Algorithms
crossover mutation
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Genetic Algorithms
Genetic Algorithms
crossover mutation
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Genetic Algorithms
Genetic Algorithms
crossover mutation
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Generation of new solutions by crossover, mutation and elistism.
Xin-She Yang FedCSIS2011
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Swarm Intelligence
Swarm Intelligence
Ants, bees, birds, fish ...
Simple rules lead to complex behaviour.
Go to Metaheuristic Slides
Xin-She Yang FedCSIS2011
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Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Cuckoo Search
Cuckoo Search
Local random walk:
xt+1i = xt
i + s ⊗ H(pa − ǫ)⊗ (xtj − xt
k).
[xi , xj , xk are 3 different solutions, H(u) is a Heaviside function, ǫis a random number drawn from a uniform distribution, and s isthe step size.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Cuckoo Search
Cuckoo Search
Local random walk:
xt+1i = xt
i + s ⊗ H(pa − ǫ)⊗ (xtj − xt
k).
[xi , xj , xk are 3 different solutions, H(u) is a Heaviside function, ǫis a random number drawn from a uniform distribution, and s isthe step size.
Global random walk via Levy flights:
xt+1i = xt
i + αL(s, λ), L(s, λ) =λΓ(λ) sin(πλ/2)
π
1
s1+λ, (s ≫ s0).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Cuckoo Search
Cuckoo Search
Local random walk:
xt+1i = xt
i + s ⊗ H(pa − ǫ)⊗ (xtj − xt
k).
[xi , xj , xk are 3 different solutions, H(u) is a Heaviside function, ǫis a random number drawn from a uniform distribution, and s isthe step size.
Global random walk via Levy flights:
xt+1i = xt
i + αL(s, λ), L(s, λ) =λΓ(λ) sin(πλ/2)
π
1
s1+λ, (s ≫ s0).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Cuckoo Search
Cuckoo Search
Local random walk:
xt+1i = xt
i + s ⊗ H(pa − ǫ)⊗ (xtj − xt
k).
[xi , xj , xk are 3 different solutions, H(u) is a Heaviside function, ǫis a random number drawn from a uniform distribution, and s isthe step size.
Global random walk via Levy flights:
xt+1i = xt
i + 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.
Xin-She Yang FedCSIS2011
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Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Monte Carlo Methods
Monte Carlo Methods
Almost everyone has used Monte Carlo methods in some way ...
Measure temperatures, choose a product, ...Taste soup, wine ...
Xin-She Yang FedCSIS2011
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Markov Chains
Markov Chains
Random walk – A drunkard’s walk:
ut+1 = µ+ ut + wt ,
where wt is a random variable, and µ is the drift.
For example, wt ∼ N(0, σ2) (Gaussian).
Xin-She Yang FedCSIS2011
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Markov Chains
Markov Chains
Random walk – A drunkard’s walk:
ut+1 = µ+ ut + wt ,
where wt is a random variable, and µ is the drift.
For example, wt ∼ N(0, σ2) (Gaussian).
-10
-5
0
5
10
15
20
25
0 100 200 300 400 500
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Markov Chains
Markov Chains
Random walk – A drunkard’s walk:
ut+1 = µ+ ut + wt ,
where wt is a random variable, and µ is the drift.
For example, wt ∼ N(0, σ2) (Gaussian).
-10
-5
0
5
10
15
20
25
0 100 200 300 400 500-20
-15
-10
-5
0
5
10
-15 -10 -5 0 5 10 15 20
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Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Markov Chains
Markov Chains
Markov chain: the next state only depends on the current stateand the transition probability.
P(i , j) ≡ P(Vt+1 = Sj
∣
∣
∣V0 = Sp, ...,Vt = Si)
= P(Vt+1 = Sj
∣
∣
∣Vt = Sj),
=⇒Pijπ∗i = Pjiπ
∗j , π∗ = stionary probability distribution.
Examples: Brownian motion
ui+1 = µ+ ui + ǫi , ǫi ∼ N(0, σ2).
Xin-She Yang FedCSIS2011
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Markov Chains
Markov Chains
Monopoly (board games)
Monopoly Animation
Xin-She Yang FedCSIS2011
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Markov Chain Monte Carlo
Markov Chain Monte Carlo
Landmarks: Monte Carlo method (1930s, 1945, from 1950s) e.g.,Metropolis Algorithm (1953), Metropolis-Hastings (1970).
Markov Chain Monte Carlo (MCMC) methods – A class ofmethods.
Really took off in 1990s, now applied to a wide range of areas:physics, Bayesian statistics, climate changes, machine learning,finance, economy, medicine, biology, materials and engineering ...
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Convergence Behaviour
Convergence Behaviour
As the MCMC runs, convergence may be reached
When does a chain converge? When to stop the chain ... ?
Are multiple chains better than a single chain?
0
100
200
300
400
500
600
0 100 200 300 400 500 600 700 800 900
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Convergence Behaviour
Convergence Behaviour
t=2
t=0
t=−2U
1
2
3
−∞← t
t=−n
converged
Multiple, interacting chains
Multiple agents trace multiple, interacting Markov chains duringthe Monte Carlo process.
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Analysis
Analysis
Classifications of Algorithms
Trajectory-based: hill-climbing, simulated annealing, patternsearch ...
Population-based: genetic algorithms, ant & bee algorithms,artificial immune systems, differential evolutions, PSO, HS,FA, CS, ...
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Analysis
Analysis
Classifications of Algorithms
Trajectory-based: hill-climbing, simulated annealing, patternsearch ...
Population-based: genetic algorithms, ant & bee algorithms,artificial immune systems, differential evolutions, PSO, HS,FA, CS, ...
Xin-She Yang FedCSIS2011
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Analysis
Analysis
Classifications of Algorithms
Trajectory-based: hill-climbing, simulated annealing, patternsearch ...
Population-based: genetic algorithms, ant & bee algorithms,artificial immune systems, differential evolutions, PSO, HS,FA, CS, ...
Ways of Generating New Moves/Solutions
Markov chains with different transition probability.
Trajectory-based =⇒ a single Markov chain;Population-based =⇒ multiple, interacting chains.
Tabu search (with memory) =⇒ self-avoiding Markov chains.Xin-She Yang FedCSIS2011
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Ergodicity
Ergodicity
Markov Chains & Markov Processes
Most theoretical studies uses Markov chains/process as aframework for convergence analysis.
A Markov chain is said be to regular if some positive power k
of the transition matrix P has only positive elements.
A chain is call time-homogeneous if the change of itstransition matrix P is the same after each step, thus thetransition probability after k steps become Pk .
A chain is ergodic or irreducible if it is aperiodic and positiverecurrent – it is possible to reach every state from any state.
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Convergence Behaviour
Convergence Behaviour
As k →∞, we have the stationary probability distribution π
π = πP, =⇒ thus the first eigenvalue is always 1.
Asymptotic convergence to optimality:
limk→∞
θk → θ∗, (with probability one).
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Convergence Behaviour
Convergence Behaviour
As k →∞, we have the stationary probability distribution π
π = πP, =⇒ thus the first eigenvalue is always 1.
Asymptotic convergence to optimality:
limk→∞
θk → θ∗, (with probability one).
The rate of convergence is usually determined by the secondeigenvalue 0 < λ2 < 1.
An algorithm can converge, but may not be necessarily efficient,as the rate of convergence is typically low.
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Convergence of GA
Convergence of GA
Important studies by Aytug et al. (1996)1, Aytug and Koehler(2000)2, Greenhalgh and Marschall (2000)3, Gutjahr (2010),4 etc.5
The number of iterations t(ζ) in GA with a convergenceprobability of ζ can be estimated by
t(ζ) ≤
⌈
ln(1− ζ)
ln
{
1−min[(1− µ)Ln, µLn]
}
⌉
,
where µ=mutation rate, L=string length, and n=population size.
1H. Aytug, S. Bhattacharrya and G. J. Koehler, A Markov chain analysis of genetic algorithms with power of
2 cardinality alphabets, Euro. J. Operational Research, 96, 195-201 (1996).2H. Aytug and G. J. Koehler, New stopping criterion for genetic algorithms, Euro. J. Operational research,
126, 662-674 (2000).3D. Greenhalgh & S. Marshal, Convergence criteria for genetic algorithms, SIAM J. Computing, 30, 269-282
(2000).4W. J. Gutjahr, Convergence Analysis of Metaheuristics Annals of Information Systems, 10, 159-187 (2010).
5 ´
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Multiobjective Metaheuristics
Multiobjective Metaheuristics
Asymptotic convergence of metaheuristic for multiobjectiveoptimization (Villalobos-Arias et al. 2005)6
The transition matrix P of a metaheuristic algorithm has astationary distribution π such that
|Pkij − πj | ≤ (1− ζ)k−1, ∀i , j , (k = 1, 2, ...),
where ζ is a function of mutation probability µ, string length L
and population size. For example, ζ = 2nLµnL, so µ < 0.5.
6M. Villalobos-Arias, C. A. Coello Coello and O. Hernandez-Lerma, Asymptotic convergence of metaheuristics
for multiobjective optimization problems, Soft Computing, 10, 1001-1005 (2005).
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Multiobjective Metaheuristics
Multiobjective Metaheuristics
Asymptotic convergence of metaheuristic for multiobjectiveoptimization (Villalobos-Arias et al. 2005)6
The transition matrix P of a metaheuristic algorithm has astationary distribution π such that
|Pkij − πj | ≤ (1− ζ)k−1, ∀i , j , (k = 1, 2, ...),
where ζ is a function of mutation probability µ, string length L
and population size. For example, ζ = 2nLµnL, so µ < 0.5.
6M. Villalobos-Arias, C. A. Coello Coello and O. Hernandez-Lerma, Asymptotic convergence of metaheuristics
for multiobjective optimization problems, Soft Computing, 10, 1001-1005 (2005).
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Multiobjective Metaheuristics
Multiobjective Metaheuristics
Asymptotic convergence of metaheuristic for multiobjectiveoptimization (Villalobos-Arias et al. 2005)6
The transition matrix P of a metaheuristic algorithm has astationary distribution π such that
|Pkij − πj | ≤ (1− ζ)k−1, ∀i , j , (k = 1, 2, ...),
where ζ is a function of mutation probability µ, string length L
and population size. For example, ζ = 2nLµnL, so µ < 0.5.
Note: An algorithm satisfying this condition may not converge (formultiobjective optimization)However, an algorithm with elitism, obeying the above condition,does converge!.
6M. Villalobos-Arias, C. A. Coello Coello and O. Hernandez-Lerma, Asymptotic convergence of metaheuristics
for multiobjective optimization problems, Soft Computing, 10, 1001-1005 (2005).
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Other results
Other results
Limited results on convergence analysis exist, concerning (finitestates/domains)
ant colony optimization
generalized hill-climbers and simulated annealing,
best-so-far convergence of cross-entropy optimization,
nested partition method, Tabu search, and
of course, combinatorial optimization.
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Other results
Other results
Limited results on convergence analysis exist, concerning (finitestates/domains)
ant colony optimization
generalized hill-climbers and simulated annealing,
best-so-far convergence of cross-entropy optimization,
nested partition method, Tabu search, and
of course, combinatorial optimization.
However, more challenging tasks for infinite states/domains andcontinuous problems.
Many, many open problems needs satisfactory answers.
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Converged?
Converged?
Converged, often the ‘best-so-far’ convergence, not necessarily atthe global optimality
In theory, a Markov chain can converge, but the number ofiterations tends to be large.
In practice, a finite (hopefully, small) number of generations, if thealgorithm converges, it may not reach the global optimum.
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Converged?
Converged?
Converged, often the ‘best-so-far’ convergence, not necessarily atthe global optimality
In theory, a Markov chain can converge, but the number ofiterations tends to be large.
In practice, a finite (hopefully, small) number of generations, if thealgorithm converges, it may not reach the global optimum.
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Converged?
Converged?
Converged, often the ‘best-so-far’ convergence, not necessarily atthe global optimality
In theory, a Markov chain can converge, but the number ofiterations tends to be large.
In practice, a finite (hopefully, small) number of generations, if thealgorithm converges, it may not reach the global optimum.
How to avoid premature convergence
Equip an algorithm with the ability to escape a local optimum
Increase diversity of the solutions
Enough randomization at the right stage
....(unknown, new) ....
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Coffee Break (15 Minutes)
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All and NFL
All and NFL
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?
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Exploration and Exploitation
Exploration and Exploitation
Characteristics of Metaheuristics
Exploration and Exploitation, or Diversification and Intensification.
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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.
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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/stochasticcomponents. E.g., hill-climbing with random restart.
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Summary
Summary
Exploitation
Exp
lora
tion
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Summary
Summary
Exploitation
Exp
lora
tion
uniformsearch
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Summary
Summary
Exploitation
Exp
lora
tion
uniformsearch
steepestdescent
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Summary
Summary
Exploitation
Exp
lora
tion
uniformsearch
steepestdescent
Tabu Nelder-Mead
CS
PSO/FAEP/ESSA Ant/Bee
Genetic algorithms
Newton-Raphson
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Summary
Summary
Exploitation
Exp
lora
tion
uniformsearch
steepestdescent
Tabu Nelder-Mead
CS
PSO/FAEP/ESSA Ant/Bee
Genetic algorithms
Newton-Raphson
Best?
Free lunch?
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No-Free-Lunch (NFL) Theorems
No-Free-Lunch (NFL) Theorems
Algorithm Performance
Any algorithm is as good/bad as random search, when averagedover all possible problems/functions.
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No-Free-Lunch (NFL) Theorems
No-Free-Lunch (NFL) Theorems
Algorithm Performance
Any algorithm is as good/bad as random search, when averagedover all possible problems/functions.
Finite domains
No universally efficient algorithm!
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No-Free-Lunch (NFL) Theorems
No-Free-Lunch (NFL) Theorems
Algorithm Performance
Any algorithm is as good/bad as random search, when averagedover all possible problems/functions.
Finite domains
No universally efficient algorithm!
Any free taster or dessert?
Yes and no. (more later)
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NFL Theorems (Wolpert and Macready 1997)
NFL Theorems (Wolpert and Macready 1997)
Search space is finite (though quite large), thus the space ofpossible “cost” values is also finite. Objective functionf : X 7→ Y, with F = YX (space of all possible problems).Assumptions: finite domain, closed under permutation (c.u.p).
For m iterations, m distinct visited points form a time-ordered
set dm ={(
dxm(1), dy
m(1))
, ...,(
dxm(m), dy
m(m))}
.
The performance of an algorithm a iterated m times on a costfunction f is denoted by P(dy
m|f ,m, a).
For any pair of algorithms a and b, the NFL theorem states∑
f
P(dym|f ,m, a) =
∑
f
P(dym|f ,m, b).
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NFL Theorems (Wolpert and Macready 1997)
NFL Theorems (Wolpert and Macready 1997)
Search space is finite (though quite large), thus the space ofpossible “cost” values is also finite. Objective functionf : X 7→ Y, with F = YX (space of all possible problems).Assumptions: finite domain, closed under permutation (c.u.p).
For m iterations, m distinct visited points form a time-ordered
set dm ={(
dxm(1), dy
m(1))
, ...,(
dxm(m), dy
m(m))}
.
The performance of an algorithm a iterated m times on a costfunction f is denoted by P(dy
m|f ,m, a).
For any pair of algorithms a and b, the NFL theorem states∑
f
P(dym|f ,m, a) =
∑
f
P(dym|f ,m, b).
Xin-She Yang FedCSIS2011
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NFL Theorems (Wolpert and Macready 1997)
NFL Theorems (Wolpert and Macready 1997)
Search space is finite (though quite large), thus the space ofpossible “cost” values is also finite. Objective functionf : X 7→ Y, with F = YX (space of all possible problems).Assumptions: finite domain, closed under permutation (c.u.p).
For m iterations, m distinct visited points form a time-ordered
set dm ={(
dxm(1), dy
m(1))
, ...,(
dxm(m), dy
m(m))}
.
The performance of an algorithm a iterated m times on a costfunction f is denoted by P(dy
m|f ,m, a).
For any pair of algorithms a and b, the NFL theorem states∑
f
P(dym|f ,m, a) =
∑
f
P(dym|f ,m, b).
Any algorithm is as good (bad) as a random search!Xin-She Yang FedCSIS2011
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Open Problems
Open Problems
Framework: Need to develop a unified framework foralgorithmic analysis (e.g.,convergence).
Exploration and exploitation: What is the optimal balancebetween these two components? (50-50 or what?)
Performance measure: What are the best performancemeasures ? Statistically? Why ?
Convergence: Convergence analysis of algorithms for infinite,continuous domains require systematic approaches?
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Open Problems
Open Problems
Framework: Need to develop a unified framework foralgorithmic analysis (e.g.,convergence).
Exploration and exploitation: What is the optimal balancebetween these two components? (50-50 or what?)
Performance measure: What are the best performancemeasures ? Statistically? Why ?
Convergence: Convergence analysis of algorithms for infinite,continuous domains require systematic approaches?
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Open Problems
Open Problems
Framework: Need to develop a unified framework foralgorithmic analysis (e.g.,convergence).
Exploration and exploitation: What is the optimal balancebetween these two components? (50-50 or what?)
Performance measure: What are the best performancemeasures ? Statistically? Why ?
Convergence: Convergence analysis of algorithms for infinite,continuous domains require systematic approaches?
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Open Problems
Open Problems
Framework: Need to develop a unified framework foralgorithmic analysis (e.g.,convergence).
Exploration and exploitation: What is the optimal balancebetween these two components? (50-50 or what?)
Performance measure: What are the best performancemeasures ? Statistically? Why ?
Convergence: Convergence analysis of algorithms for infinite,continuous domains require systematic approaches?
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More Open Problems
More Open Problems
Free lunches: Unproved for infinite or continuous domains formultiobjective optimization. (possible free lunches!)What are implications of NFL theorems in practice?If free lunches exist, how to find the best algorithm(s)?
Knowledge: Problem-specific knowledge always helps to findappropriate solutions? How to quantify such knowledge?
Intelligent algorithms: Any practical way to design trulyintelligent, self-evolving algorithms?
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More Open Problems
More Open Problems
Free lunches: Unproved for infinite or continuous domains formultiobjective optimization. (possible free lunches!)What are implications of NFL theorems in practice?If free lunches exist, how to find the best algorithm(s)?
Knowledge: Problem-specific knowledge always helps to findappropriate solutions? How to quantify such knowledge?
Intelligent algorithms: Any practical way to design trulyintelligent, self-evolving algorithms?
Xin-She Yang FedCSIS2011
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More Open Problems
More Open Problems
Free lunches: Unproved for infinite or continuous domains formultiobjective optimization. (possible free lunches!)What are implications of NFL theorems in practice?If free lunches exist, how to find the best algorithm(s)?
Knowledge: Problem-specific knowledge always helps to findappropriate solutions? How to quantify such knowledge?
Intelligent algorithms: Any practical way to design trulyintelligent, self-evolving algorithms?
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Constraints
Constraints
In describing optimization algorithms, we are not concern withconstraints. Algorithms can solve both unconstrained and moreoften constrained problems.
The handling of constraints is an implementation issue, thoughincorrect or inefficient methods of dealing with constraints can slowdown the algorithm efficiency, or even result in wrong solutions.
Methods of handling constraints
Direct methods
Langrange multipliers
Barrier functions
Penalty methods
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Aims
Aims
Either converting a constrained problem to an unconstrained one
or changing the search space into a regular domain
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Aims
Aims
Either converting a constrained problem to an unconstrained one
or changing the search space into a regular domain
The ease of programming and implementation
Improve (or at least not hinder) the efficiency of the chosenalgorithm in implementation.
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Aims
Aims
Either converting a constrained problem to an unconstrained one
or changing the search space into a regular domain
The ease of programming and implementation
Improve (or at least not hinder) the efficiency of the chosenalgorithm in implementation.
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Aims
Aims
Either converting a constrained problem to an unconstrained one
or changing the search space into a regular domain
The ease of programming and implementation
Improve (or at least not hinder) the efficiency of the chosenalgorithm in implementation.
Scalability
The used approach should be able to deal with small, large andvery large scale problems.
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Common Approaches
Common Approaches
Direct method
Simple, but not versatile, difficult in programming.
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Common Approaches
Common Approaches
Direct method
Simple, but not versatile, difficult in programming.
Lagrange multipliers
Main for equality constraints.
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Common Approaches
Common Approaches
Direct method
Simple, but not versatile, difficult in programming.
Lagrange multipliers
Main for equality constraints.
Barrier functions
Very powerful and widely used in convex optimization.
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Common Approaches
Common Approaches
Direct method
Simple, but not versatile, difficult in programming.
Lagrange multipliers
Main for equality constraints.
Barrier functions
Very powerful and widely used in convex optimization.
Penalty methods
Simple and versatile, widely used.
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Common Approaches
Common Approaches
Direct method
Simple, but not versatile, difficult in programming.
Lagrange multipliers
Main for equality constraints.
Barrier functions
Very powerful and widely used in convex optimization.
Penalty methods
Simple and versatile, widely used.
Others
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Direct Methods
Direct Methods
Minimize f (x , y) = (x − 2)2 + 4(y − 3)2
subject to −x + y ≤ 2, x + 2y ≤ 3.
−x+
y≤
2
x + 2y ≤ 3
Optimal
Direct Methods: to generate solutions/points inside the region!(easy for rectangular regions)
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Method of Lagrange Multipliers
Method of Lagrange Multipliers
Maximize f (x , y) = 10− x2 − (y − 2)2 subject to x + 2y = 5.
Defining a combined function Φ using a multiplier λ, we have
Φ = 10− x2 − (y − 2)2 + λ(x + 2y − 5).
The optimality conditions are
∂Φ
∂x= 2x +λ = 0,
∂Φ
∂y= −2(y−2)+2λ = 0,
∂Φ
∂λ= x+2y−5,
whose solutions become
x = 1/5, y = 12/5, λ = 2/5, =⇒ fmax =49
5.
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Barrier Functions
Barrier Functions
As an equality h(x) = 0 can be written as two inequalities h(x) ≤ 0and −h(x) ≤ 0, we only use inequalities.
For a general optimization problem:
minimize f (x), subject to g(xi ) ≤ 0(i = 1, 2, ...,N),
we can define a Indicator or barrier function
I−1[u] =
{
0 if u ≤ 0∞ if u > 0.
Not so easy to deal with numerically. Also discontinuous!
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Logarithmic Barrier Functions
Logarithmic Barrier Functions
A log barrier function
I−(u) = −1
tlog(−u), u < 0,
where t > 0 is an accuracy parameters (can be very large).
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Logarithmic Barrier Functions
Logarithmic Barrier Functions
A log barrier function
I−(u) = −1
tlog(−u), u < 0,
where t > 0 is an accuracy parameters (can be very large).Then, the above minimization problem becomes
minimize f (x) +
N∑
i=1
I−(gi (x)) = f (x) +
N∑
i=1
−1
tlog[−gi (x)].
This is an unconstrained problem and easy to implement!
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Penalty Methods
Penalty Methods
For a nonlinear optimization problem with equality and inequalityconstraints,
minimize
x∈ℜn f (x), x = (x1, ..., xn)T ∈ ℜn,
subject to φi (x) = 0, (i = 1, ...,M),
ψj (x) ≤ 0, (j = 1, ...,N),
the idea is to define a penalty function so that the constrainedproblem is transformed into an unconstrained problem. Now wedefine
Π(x, µi , νj ) = f (x) +M
∑
i=1
µiφ2i (x) +
N∑
j=1
νjψ2j (x),
where µi ≫ 1 and νj ≥ 0 which should be large enough,depending on the solution quality needed.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
In addition, for simplicity of implementation, we can use µ = µi forall i and ν = νj for all j . That is, we can use a simplified
Π(x, µ, ν) = f (x) + µ
M∑
i=1
Qi [φi (x)]φ2i (x) + ν
N∑
j=1
Hj [ψj (x)]ψ2j (x).
Here the barrier/indicator-like functions
Hj =
{
0 if ψj(x) ≤ 01 if ψj(x) > 0
, Qi =
{
0 if φi (x) = 01 if φi (x) 6= 0
.
In general, for most applications, µ and ν can be taken as 1010 to1015. We will use these values in most implementations.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Pressure Vessel Design Optimization
Pressure Vessel Design Optimization
r
d1
r
L
d2
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Formulation
Formulation
minimize f (x) = 0.6224d1rL+1.7781d2r2+3.1661d2
1 L+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 ≤ 0h1(x) = [d1/0.0625] − n = 0h2(x) = [d2/0.0625] − k = 0.
The simple bounds are
0.0625 ≤ d1, d2 ≤ 99× 0.0625, 10.0 ≤ r , L ≤ 200.0.
1 ≤ n, k ≤ 99 are integers.Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Minimze Π(x, λ) = f (x)+λ
2∑
i=1
Qi [hi (x)]h2i (x)+λ
4∑
j=1
Hj [gj (x)]g2j (x),
where λ = 1015.
This becomes an unconstrained optimization problemin a regular domain.
Best solutions found so far in the literature
f∗ = $6059.714
at(0.8125, 0.4375, 42.0984, 176.6366).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Applications
Applications
Design optimization: structural engineering, product design ...
Scheduling, routing and planning: often discrete,combinatorial problems ...
Applications in almost all areas (e.g., finance, economics,engineering, industry, ...)
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Dome Design
Dome Design
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Dome Design
Dome Design
120-bar dome: Divided into 7 groups, 120 design elements, about 200
constraints (Kaveh and Talatahari 2010; Gandomi and Yang 2011).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Tower Design
Tower Design
26-storey tower: 942 design elements, 244 nodal links, 59 groups/types,
> 4000 nonlinear constraints (Kaveh & Talatahari 2010; Gandomi & Yang 2011).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Topology Optimization of Nanoscale Device
Topology Optimization of Nanoscale Device
The topology optimization of a nanoscale heat-conducting system is
shape optimization,7 which can be considered an inverse problem for
shape or distribution of materials.
u
e
150 nm
150 nm
flux
Tx
=1
Tx = 1 − x
Tx = 1 − x
T=
0
Benchmark Design
Two materials with heat diffussivities of K1 and K2, respectively. Forexample, Si and Mg2Si, K1/K2 ≈ 10. The aim is to distribute the twomaterials such that the difference |Ta − Tb| is as large as possible.
7A. Evgrafov, K. Maute, R. G. Yang and M. L. Dunn, Topologyoptimization for nano-scale heat transfer, Int. J. Num. Methods in Engrg., 77(2), 285-300 (2009).Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Initial Congfiguration
Initial Congfiguration
Unit square with two different materials (initial configuration).
z
jTa
Tb
=⇒
K2
K1
Then, use FA to redistribute these two materials so as to maximizethe temperature difference.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Optimal shape and distribution of materials: Si (blue) and Mg2Si (red).
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Optimal topology (left) and temperature distribution (right).
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
References
References
Sambridge, M. And Mosegaard, K., (2002). Monte Carlo methodsin geophysical inverse problems, Reviews of Geophysics, 40, 3-1-29.
Scales, J. A., Smith, M. L., and Treitel, S., IntroductoryGeophysical Inverse Theory, Samizdat Press, (2001).
Yang X. S. (2008). Nature-Inspired Metaheuristic Algorithms,Lunver Press, UK.
Yang, X. S., (2009). Firefly algorithms for multimodal optimization,5th Symposium on Stochastic Algorithms, Foundation andApplications (SAGA 2009) (Eds Watanabe O. and Zeugmann T.),LNCS, 5792, pp. 169-178.
Yang X.-S. and Deb S., (2009). ”Cuckoo search via Lvy flights”.World Congress on Nature & Biologically Inspired Computing(NaBIC 2009). IEEE Publications. pp. 210214. arXiv:1003.1594v1.
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Thanks
Thanks
International Journal of Mathematical Modelling and NumericalOptimization (IJMMNO) http://www.inderscience.com/ijmmno
Books:
Computational Optimization, Methods and Algorithms (Slawomir Kozieland Xin-She Yang), Springer (2011).http://www.springerlink.com/content/978-3-642-20858-4
Engineering Optimization: An Introduction with MetaheuristicAppliactions (Xin-She Yang), John Wiley & Sons, (2010).http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470582464.html
Thank you!
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks
Thanks
Thanks
International Journal of Mathematical Modelling and NumericalOptimization (IJMMNO) http://www.inderscience.com/ijmmno
Books:
Computational Optimization, Methods and Algorithms (Slawomir Kozieland Xin-She Yang), Springer (2011).http://www.springerlink.com/content/978-3-642-20858-4
Engineering Optimization: An Introduction with MetaheuristicAppliactions (Xin-She Yang), John Wiley & Sons, (2010).http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470582464.html
Notes
https://sites.google.com/site/tutorialmetaheuristic/tutorials
Thank you!
Xin-She Yang FedCSIS2011
Metaheuristics and Computational Intelligence
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