Nature-inspired metaheuristic algorithms for optimization and computional intelligence

Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks

Nature-Inspired Metaheristics Algorithms
for 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


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 possible
functions.
- No-free-lunch theorems (Wolpert & Macready)



Intro

Intro



functions. How so?



Intro

Intro



functions. 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

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 =mc 2



A Perfect Algorithm

A Perfect Algorithm

Steepest Descent



A Perfect Algorithm

A Perfect Algorithm

Steepest Descent
=⇒ d d
1 1 + y ′2
min t = ds = dx
0 v 0 2g [h − y (x)]



A Perfect Algorithm

A Perfect Algorithm

Steepest Descent
=⇒ d d
1 1 + y ′2
min t = ds = dx
0 v 0 2g [h − y (x)]

=⇒


A Perfect Algorithm

A Perfect Algorithm

Steepest Descent
=⇒ d d
1 1 + y ′2
min t = ds = dx
0 v 0 2g [h − y (x)]

=⇒ =⇒


A Perfect Algorithm

A Perfect Algorithm

Steepest Descent
=⇒ d d
1 1 + y ′2
min t = ds = dx
0 v 0 2g [h − y (x)]

A

 x= 2 (θ − sin θ)
=⇒ =⇒
y = h − A (1 − cos θ)

2


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 known
location xi .
xi

x2
x1





Essence of an Optimization Algorithm
To move to a new, better point xi +1 from an existing known
location xi .
xi

?
x2
x1 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 (Karmarkar
1984), 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 swarm
optimization (Kennedy & Eberhart 1995), differential
evolution (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

Steepest Descent/Hill Climbing
Gradient-Based Methods
Use gradient/derivative information – very eﬃcient for local search.



Newton’s Method  
∂2f ∂2f
∂x1 2
··· ∂x1 ∂xn
xn+1 = xn − H−1 ∇f ,

H= .
. .. .
.

.
 . . . 
∂2f ∂2f
∂xn ∂x1 ··· ∂xn 2

Generation of new moves by gradient.



Newton’s Method  
∂2f ∂2f
∂x1 2
··· ∂x1 ∂xn
xn+1 = xn − H−1 ∇f ,

H= .
. .. .
.

.
 . . . 
∂2f ∂2f
∂xn ∂x1 ··· ∂xn 2

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 as
hill-climbing.



Conjugate Gradient (CG) Method

Belong to Krylov subspace iteration methods. The conjugate
gradient method was pioneered by Magnus Hestenes, Eduard
Stiefel and Cornelius Lanczos in the 1950s. It was named as one of
the top 10 algorithms of the 20th century.




Belong to Krylov subspace iteration methods. The conjugate
gradient method was pioneered by Magnus Hestenes, Eduard
Stiefel and Cornelius Lanczos in the 1950s. It was named as one of
the 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)
1
f (u) = uT Au − bT u + v,
2
where v is a vector constant and can be taken to be zero. We can
easily see that ∇f (u) = 0 leads to Au = b.


CG

CG

The theory behind these iterative methods is closely related to the
Krylov subspace Kn spanned by A and b as deﬁned by

Kn (A, b) = {Ib, Ab, A2 b, ..., An−1 b},

where A0 = I.
If we use an iterative procedure to obtain the approximate solution
un 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 is
subsequently determined by

dT Arn
n
dn+1 = rn − dn .
dT Adn
n

The solution often starts with an initial guess u0 at n = 0, and
proceeds iteratively. The above steps can compactly be written as

un+1 = un + αn dn , rn+1 = rn − αn Adn ,

and
dn+1 = rn+1 + βn dn ,
where
rT rn
n rT rn+1
n+1
αn = T
, βn = .
dn Adn rT r n
n
Iterations stop when a prescribed accuracy is reached.


Gradient-free Methods


Gradient-base methods
Requires the information of derivatives. Not suitable for problems
with discontinuities.





Gradient-base methods
Requires the information of derivatives. Not suitable for problems
with 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, ﬁrst
developed by J. A. Nelder and R. Mead in 1965.
A Simplex
In the n-dimensional space, a simplex, which is a generalization of
a 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 as
n-simplex.

Xin-She Yang (a) (b) (c) FedCSIS2011


Downhill Simplex Method

Downhill Simplex Method

xe
xr xr
¯
x
s s
xc
xn+1 xn+1 xn+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 in
nature.

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 /kB T ].
kB =Boltzmann constant (e.g., kB = 1), T =temperature, E =energy.

E ∝ f (x), T = T0 αt (cooling schedule) , (0 < α < 1).
T → 0, =⇒p → 0, =⇒ hill climbing.





Metal annealing to increase strength =⇒ simulated annealing.
Probabilistic Move: p ∝ exp[−E /kB T ].
kB =Boltzmann constant (e.g., kB = 1), T =temperature, E =energy.

E ∝ f (x), T = T0 αt (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



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+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ).
i i j 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.



Cuckoo Search

Cuckoo Search

Local random walk:

xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ).
i i j k

the step size.

Global random walk via L´vy ﬂights:
e

λΓ(λ) sin(πλ/2) 1
xt+1 = xt + αL(s, λ),
i i L(s, λ) = , (s ≫ s0 ).
π s 1+λ



Cuckoo Search

Cuckoo Search

Local random walk:

xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ).
i i j k

the step size.

Global random walk via L´vy ﬂights:
e

λΓ(λ) sin(πλ/2) 1
xt+1 = xt + αL(s, λ),
i i L(s, λ) = , (s ≫ s0 ).
π s 1+λ
Generation of new moves by L´vy ﬂights, random walk and elitism.
e


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, σ 2 ) (Gaussian).



Markov Chains

Markov Chains
ut+1 = µ + ut + wt ,

25

20

15

10

5

0

-5

-10
0 100 200 300 400 500



Markov Chains

Markov Chains
ut+1 = µ + ut + wt ,

25 10

20
5

15

0

10

-5

5

-10

0

-15
-5

-10 -20
0 100 200 300 400 500 -15 -10 -5 0 5 10 15 20



Markov Chains

Markov Chains
Markov chain: the next state only depends on the current state
and the transition probability.

P(i , j) ≡ P(Vt+1 = Sj V0 = Sp , ..., Vt = Si )

= P(Vt+1 = Sj Vt = Sj ),

=⇒Pij πi∗ = Pji πj∗ , π ∗ = stionary probability distribution.

Examples: Brownian motion

ui +1 = µ + ui + ǫi , ǫi ∼ N(0, σ 2 ).



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 of
methods.

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
t=−2 converged
U

1
2 t=2
t=−n
3 t=0

Multiple, interacting chains
Multiple agents trace multiple, interacting Markov chains during
the Monte Carlo process.



Analysis

Analysis
Classifications of Algorithms

Trajectory-based: hill-climbing, simulated annealing, pattern
search ...
Population-based: genetic algorithms, ant & bee algorithms,
artificial immune systems, differential evolutions, PSO, HS,
FA, CS, ...



Analysis

Analysis
Classifications of Algorithms

Trajectory-based: hill-climbing, simulated annealing, pattern
search ...
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.


Ergodicity

Ergodicity

Markov Chains & Markov Processes
Most theoretical studies uses Markov chains/process as a
framework 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 its
transition matrix P is the same after each step, thus the
transition probability after k steps become Pk .
A chain is ergodic or irreducible if it is aperiodic and positive
recurrent – 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:

lim θk → θ∗ , (with probability one).
k→∞





As k → ∞, we have the stationary probability distribution π
π = πP, =⇒ thus the ﬁrst eigenvalue is always 1.

Asymptotic convergence to optimality:

lim θk → θ∗ , (with probability one).
k→∞

The rate of convergence is usually determined by the second
eigenvalue 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 convergence
probability of ζ can be estimated by
ln(1 − ζ)
t(ζ) ≤ ,
ln 1 − min[(1 − µ)Ln , µLn ]

where µ=mutation rate, L=string length, and n=population size.
1
H. 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).
2
H. Aytug and G. J. Koehler, New stopping criterion for genetic algorithms, Euro. J. Operational research,
126, 662-674 (2000).
3
D. Greenhalgh & S. Marshal, Convergence criteria for genetic algorithms, SIAM J. Computing, 30, 269-282
(2000).
4
Metaheuristics and Gutjahr, Convergence Analysis of Metaheuristics Annals of Information Systems, 10, 159-187 (2010).
W. J. Computational Intelligence


Multiobjective Metaheuristics

Asymptotic convergence of metaheuristic for multiobjective
optimization (Villalobos-Arias et al. 2005)6

The transition matrix P of a metaheuristic algorithm has a
stationary distribution π such that

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

where ζ is a function of mutation probability µ, string length L
and population size. For example, ζ = 2nL µnL , so µ < 0.5.

Xin-She Yang
6 FedCSIS2011
M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics
a



Asymptotic convergence of metaheuristic for multiobjective
optimization (Villalobos-Arias et al. 2005)6

The transition matrix P of a metaheuristic algorithm has a
stationary distribution π such that

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

where ζ is a function of mutation probability µ, string length L
and population size. For example, ζ = 2nL µnL , so µ < 0.5.

Note: An algorithm satisfying this condition may not converge (for
multiobjective optimization)
However, an algorithm with elitism, obeying the above condition,
does converge!.
Xin-She Yang
6 FedCSIS2011
M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics
a


Other results

Other results

Limited results on convergence analysis exist, concerning (ﬁnite
states/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 (ﬁnite
states/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 and
continuous problems.
Many, many open problems needs satisfactory answers.



Converged?

Converged?
Converged, often the ‘best-so-far’ convergence, not necessarily at
the global optimality
In theory, a Markov chain can converge, but the number of
iterations tends to be large.
In practice, a ﬁnite (hopefully, small) number of generations, if the
algorithm converges, it may not reach the global optimum.



Converged?

Converged?
Converged, often the ‘best-so-far’ convergence, not necessarily at
the global optimality
In theory, a Markov chain can converge, but the number of
iterations tends to be large.
In practice, a ﬁnite (hopefully, small) number of generations, if the
algorithm 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) ....


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/stochastic
components. E.g., hill-climbing with random restart.



Summary

Summary Exploration

Exploitation



Summary

Summary
uniform
search
Exploration

Exploitation



Summary

Summary
uniform
search
Exploration

steepest
Exploitation descent



Summary

Summary
uniform
search
CS
Ge
net
Exploration

ic
alg
ori PS
th ms O/
SA EP FA
A nt /E
/Be S
e
Newton-
Raphson
Tabu Nelder-Mead
steepest



Summary

Summary
uniform
search Best?
CS Free lunch?
Ge
net
Exploration

ic
alg
ori PS
th ms O/
SA EP FA
A nt /E
/Be S
e
Newton-
Raphson
Tabu Nelder-Mead
steepest



No-Free-Lunch (NFL) Theorems


Algorithm Performance
Any algorithm is as good/bad as random search, when averaged
over 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 of
possible “cost” values is also finite. Objective function
f : X → Y, with F = Y X (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
x y x y
set dm = dm (1), dm (1) , ..., dm (m), dm (m) .
The performance of an algorithm a iterated m times on a cost
y
function f is denoted by P(dm |f , m, a).

For any pair of algorithms a and b, the NFL theorem states
y y
P(dm |f , m, a) = P(dm |f , m, b).
f f




Search space is finite (though quite large), thus the space of
possible “cost” values is also finite. Objective function
f : X → Y, with F = Y X (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
x y x y
set dm = dm (1), dm (1) , ..., dm (m), dm (m) .
The performance of an algorithm a iterated m times on a cost
y
function f is denoted by P(dm |f , m, a).

For any pair of algorithms a and b, the NFL theorem states
y y
P(dm |f , m, a) = P(dm |f , m, b).
f f
Any algorithm is as good (bad) as a random search!


Open Problems

Open Problems

Framework: Need to develop a uniﬁed framework for
algorithmic analysis (e.g.,convergence).
Exploration and exploitation: What is the optimal balance
between these two components? (50-50 or what?)
Performance measure: What are the best performance
measures ? Statistically? Why ?
Convergence: Convergence analysis of algorithms for inﬁnite,
continuous domains require systematic approaches?



More Open Problems

More Open Problems

Free lunches: Unproved for infinite or continuous domains for
multiobjective 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 find
appropriate solutions? How to quantify such knowledge?
Intelligent algorithms: Any practical way to design truly
intelligent, self-evolving algorithms?



Constraints

Constraints
In describing optimization algorithms, we are not concern with
constraints. Algorithms can solve both unconstrained and more
often constrained problems.

The handling of constraints is an implementation issue, though
incorrect or ineﬃcient methods of dealing with constraints can slow
down 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 chosen
algorithm in implementation.



Aims

Aims


The ease of programming and implementation
Improve (or at least not hinder) the eﬃciency of the chosen
algorithm in implementation.

Scalability
The used approach should be able to deal with small, large and
very 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.
2
≤
y
Optimal x+
−
x+
2y
≤3

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 − x 2 − (y − 2)2 subject to x + 2y = 5.

Deﬁning a combined function Φ using a multiplier λ, we have

Φ = 10 − x 2 − (y − 2)2 + λ(x + 2y − 5).

The optimality conditions are
∂Φ ∂Φ ∂Φ
= 2x +λ = 0, = −2(y −2)+2λ = 0, = x +2y −5,
∂x ∂y ∂λ
whose solutions become
49
x = 1/5, y = 12/5, λ = 2/5, =⇒ fmax = .
5



Barrier Functions

Barrier Functions

As an equality h(x) = 0 can be written as two inequalities h(x) ≤ 0
and −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
0 if u ≤ 0
I−1 [u] =
∞ if u > 0.

Not so easy to deal with numerically. Also discontinuous!



Logarithmic Barrier Functions


A log barrier function

¯− (u) = − 1 log(−u),
I u < 0,
t
where t > 0 is an accuracy parameters (can be very large).





A log barrier function

¯− (u) = − 1 log(−u),
I u < 0,
t
where t > 0 is an accuracy parameters (can be very large).
Then, the above minimization problem becomes
N N
¯− (gi (x)) = f (x) + 1
minimize f (x) + I − log[−gi (x)].
t
i =1 i =1

This is an unconstrained problem and easy to implement!



Penalty Methods

Penalty Methods
For a nonlinear optimization problem with equality and inequality
constraints,
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 constrained
problem is transformed into an unconstrained problem. Now we
deﬁne
M N
Π(x, µi , νj ) = f (x) + µi φ2 (x) +
i νj ψj2 (x),
i =1 j=1

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 for
all i and ν = νj for all j. That is, we can use a simpliﬁed

M N
Π(x, µ, ν) = f (x) + µ Qi [φi (x)]φ2 (x) + ν
i Hj [ψj (x)]ψj2 (x).
i =1 j=1

Here the barrier/indicator-like functions

0 if ψj (x) ≤ 0 0 if φi (x) = 0
Hj = , Qi = .
1 if ψj (x) > 0 1 if φi (x) = 0

In general, for most applications, µ and ν can be taken as 1010 to
1015 . We will use these values in most implementations.



Pressure Vessel Design Optimization

Pressure Vessel Design Optimization

d1 d2

r r

L

Nature-inspired metaheuristic algorithms for optimization and computional intelligence

Nature-inspired metaheuristic algorithms for optimization and computional intelligence

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (20)

Similar a Nature-inspired metaheuristic algorithms for optimization and computional intelligence

Similar a Nature-inspired metaheuristic algorithms for optimization and computional intelligence (20)

Más de Xin-She Yang

Más de Xin-She Yang (20)

Último

Último (20)

Nature-inspired metaheuristic algorithms for optimization and computional intelligence