Nature-inspired metaheuristic algorithms for optimization and computional intelligence

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

Intro

Computational science is now the third paradigm of science,complementing theory and experiment.

- Ken Wilson (Cornell University), Nobel Laureate.




Intro

Intro






Intro

Intro



All models are wrong, but some are useful.

- George Box, Statistician




Intro

Intro



All models are inaccurate, but some are useful.





Intro

Intro





All algorithms perform equally well on average over all possiblefunctions.

- No-free-lunch theorems (Wolpert & Macready)




Intro

Intro





All algorithms perform equally well on average over all possiblefunctions. How so?





Intro

Intro





All algorithms perform equally well on average over all possiblefunctions. Not quite! (more later)





Intro

Intro





All algorithms perform equally well on average over all possiblefunctions. Not quite! (more later)





Overview

Overview

Part I

Introduction

Metaheuristic Algorithms

Monte Carlo and Markov Chains

Algorithm Analysis




Overview

Overview

Part I

Introduction

Metaheuristic Algorithms

Monte Carlo and Markov Chains

Algorithm Analysis

Part II

Exploration & Exploitation

Dealing with Constraints

Applications

Discussions & Bibliography




A Perfect Algorithm

A Perfect Algorithm

What is the best relationship among E , m and c?




A Perfect Algorithm

A Perfect Algorithm





A Perfect Algorithm

A Perfect Algorithm


Initial state: m,E ,c ,




A Perfect Algorithm

A Perfect Algorithm


Initial state: m,E ,c , =⇒




A Perfect Algorithm

A Perfect Algorithm


Initial state: m,E ,c , =⇒




A Perfect Algorithm

A Perfect Algorithm


Initial state: m,E ,c , =⇒ =⇒E =mc2




A Perfect Algorithm

A Perfect Algorithm



Steepest Descent




A Perfect Algorithm

A Perfect Algorithm



Steepest Descent




A Perfect Algorithm

A Perfect Algorithm



Steepest Descent

=⇒

min t =

∫ d

0

1

vds =

∫ d

0

√

1 + y′2

√

2g [h − y(x)]dx




A Perfect Algorithm

A Perfect Algorithm



Steepest Descent

=⇒

min t =

∫ d

0

1

vds =

∫ d

0

√

1 + y′2

√

2g [h − y(x)]dx

=⇒




A Perfect Algorithm

A Perfect Algorithm



Steepest Descent

=⇒

min t =

∫ d

0

1

vds =

∫ d

0

√

1 + y′2

√

2g [h − y(x)]dx

=⇒ =⇒




A Perfect Algorithm

A Perfect Algorithm



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)




Computing in Reality

Computing in Reality

A Problem & Problem Solvers⇓

Mathematical/Numerical Models

⇓Computer & Algorithms & Programming

⇓Validation⇓

Results




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








x1

x2

xi








x1

x2

xi

xi+1

?

Population-based algorithms use multiple, interacting paths.

Different algorithms

Different strategies/approaches in generating these moves!




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?




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.




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), ...




Steepest Descent/Hill Climbing


Gradient-Based Methods

Use gradient/derivative information – very efficient for local search.







































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.




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.




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.




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.






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.




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).




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



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 ...




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



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




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 ...




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.






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.




An Example

An Example




Genetic Algorithms

Genetic Algorithms

crossover mutation




Genetic Algorithms

Genetic Algorithms

crossover mutation




Genetic Algorithms

Genetic Algorithms

crossover mutation










Generation of new solutions by crossover, mutation and elistism.




Swarm Intelligence

Swarm Intelligence

Ants, bees, birds, fish ...

Simple rules lead to complex behaviour.

Go to Metaheuristic Slides




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.




Cuckoo Search

Cuckoo Search

Local random walk:

xt+1i = xt

i + s ⊗ H(pa − ǫ)⊗ (xtj − xt

k).


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).




Cuckoo Search

Cuckoo Search

Local random walk:

xt+1i = xt

i + s ⊗ H(pa − ǫ)⊗ (xtj − xt

k).



xt+1i = xt


π

1

s1+λ, (s ≫ s0).




Cuckoo Search

Cuckoo Search

Local random walk:

xt+1i = xt

i + s ⊗ H(pa − ǫ)⊗ (xtj − xt

k).



xt+1i = xt


π

1

s1+λ, (s ≫ s0).

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




Monte Carlo Methods

Monte Carlo Methods

Almost everyone has used Monte Carlo methods in some way ...

Measure temperatures, choose a product, ...Taste soup, wine ...




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).




Markov Chains

Markov Chains


ut+1 = µ+ ut + wt ,



-10

-5

0

5

10

15

20

25

0 100 200 300 400 500




Markov Chains

Markov Chains


ut+1 = µ+ ut + wt ,



-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




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).




Markov Chains

Markov Chains

Monopoly (board games)

Monopoly Animation




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 ...




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






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.




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, ...




Analysis

Analysis







Analysis

Analysis




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



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.






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).






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.




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 ´




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).








|Pkij − πj | ≤ (1− ζ)k−1, ∀i , j , (k = 1, 2, ...),












|Pkij − πj | ≤ (1− ζ)k−1, ∀i , j , (k = 1, 2, ...),



Note: An algorithm satisfying this condition may not converge (formultiobjective optimization)However, an algorithm with elitism, obeying the above condition,does converge!.






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.




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.




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.




Converged?

Converged?







Converged?

Converged?




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) ....




Coffee Break (15 Minutes)




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?




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.








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.




Summary

Summary

Exploitation

Exp

lora

tion




Summary

Summary

Exploitation

Exp

lora

tion

uniformsearch




Summary

Summary

Exploitation

Exp

lora

tion

uniformsearch

steepestdescent




Summary

Summary

Exploitation

Exp

lora

tion

uniformsearch

steepestdescent

Tabu Nelder-Mead

CS

PSO/FAEP/ESSA Ant/Bee

Genetic algorithms

Newton-Raphson




Summary

Summary

Exploitation

Exp

lora

tion

uniformsearch

steepestdescent

Tabu Nelder-Mead

CS

PSO/FAEP/ESSA Ant/Bee

Genetic algorithms

Newton-Raphson

Best?

Free lunch?




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!








Finite domains

No universally efficient algorithm!

Any free taster or dessert?

Yes and no. (more later)




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).








set dm ={(

dxm(1), dy

m(1))

, ...,(

dxm(m), dy

m(m))}

.


m|f ,m, a).


f

P(dym|f ,m, a) =

∑

f

P(dym|f ,m, b).








set dm ={(

dxm(1), dy

m(1))

, ...,(

dxm(m), dy

m(m))}

.


m|f ,m, a).


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



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?




Open Problems

Open Problems








Open Problems

Open Problems








Open Problems

Open Problems








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?




More Open Problems

More Open Problems







More Open Problems

More Open Problems







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




Aims

Aims

Either converting a constrained problem to an unconstrained one

or changing the search space into a regular domain




Aims

Aims



The ease of programming and implementation

Improve (or at least not hinder) the efficiency of the chosenalgorithm in implementation.




Aims

Aims








Aims

Aims





Scalability

The used approach should be able to deal with small, large andvery large scale problems.




Common Approaches

Common Approaches

Direct method

Simple, but not versatile, difficult in programming.




Common Approaches

Common Approaches

Direct method


Lagrange multipliers

Main for equality constraints.




Common Approaches

Common Approaches

Direct method




Barrier functions

Very powerful and widely used in convex optimization.




Common Approaches

Common Approaches

Direct method




Barrier functions


Penalty methods

Simple and versatile, widely used.




Common Approaches

Common Approaches

Direct method




Barrier functions


Penalty methods

Simple and versatile, widely used.

Others




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)




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.




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!




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).






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!




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.




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.




Pressure Vessel Design Optimization

Pressure Vessel Design Optimization

r

d1

r

L

d2




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



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).




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, ...)




Dome Design

Dome Design




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).




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).




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



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.




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).




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.




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!




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!



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Nature-inspired metaheuristic algorithms for optimization and computional intelligence

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