An Introduction to Evolutionary Algorithms

Karthik Sindhya, PhD

Postdoctoral Researcher

Industrial Optimization Group

Department of Mathematical Information Technology

Karthik.sindhya@jyu.fi http://users.jyu.fi/~kasindhy/

• Nature Inspired Algorithms

• Differential Evolution algorithm

• Constraint handling

• Applications

Overview

Nature Inspired Algorithms

• Nature provide some of the efficient ways to solve problems

– Algorithms imitating processes in nature/inspired from nature – Nature Inspired Algorithms.

• What type of problems?

– Aircraft wing design

• Wind turbine design

• Bionic car

Nature Inspired Algorithms

BBC

Performance improvement by 40%. They reduce turbulence across the surface, increasing angle of attack and decreasing drag. (Source: Popular Mechanics)

Hexagonal plates - resulting in door paneling one-third lighter than conventional paneling, but just as strong. (Source: Popular Mechanics)

• Bullet train

Nature Inspired Algorithms

NATGEO

Train's nose is designed after the beak of a kingfisher, which dives smoothly into water. (Source: Popular Mechanics)

• Optimization – An act, process, or methodology of making something (as a design, system, or

decision) as fully perfect, functional, or effective as possible. (http://www.merriam-

webster.com/dictionary)

• Nature as an optimizer – Birds: Minimize drag.

– Humpback whale: Maximize maneuverability (enhanced lift devices to control flow over the flipper and maintain lift at high angles of attack).

– Boxfish: Minimize drag and maximize rigidity of exoskeleton.

– Kingfisher: Minimize micro-pressure waves.

• Consider an optimization problem of the form

Nature Inspired Algorithms for Optimization

• Objective and constraint functions can be non-differentiable.

• Constraints nonlinear. • Discrete/Discontinuous search space. • Mixed variables (Integer, Real, Boolean etc.) • Large number of constraints and variables. • Objective functions can be multimodal.

– Multimodal functions have more than one optima, but can either have a single or more than one global optima.

• Computationally expensive objective functions and constraints.

Practical Optimization Problems – Charecteristics!

Practical Optimization Problems – Charecteristics!

Simulation model

Decision vector Objective vector

Optimization algorithm

• Different methods for different types of problems.

• Constraint handling e.g. using panalty method is sensitive to penalty parameters.

• Often get stuck in local optima (lack global perspective).

• Usually need knowledge of first/second order derivatives of objective functions and constraints.

Traditional Optimization Techniques – Problems!

Computational intelligence

Nature inspired algorithms

Fuzzy logic systems

Neural networks

Nature Inspired Algorithms for Optimization

Nature inspired algorithms

Evolutionary algorithms

Genetic algorithm

Differential evolution

Swarm optimization

Particle swarm optimization

Ant colony optimization

.... and many more.

Nature Inspired Algorithms for Optimization

Evolution

Humans

Macintosh

Nokia

Evolutionary Algorithms

Charles Darwin

Offsprings created by reproduction, mutation, etc.

Natural selection - A guided search procedure

Individuals suited to the environment survive, reproduce and pass their genetic traits to offspring

Populations adapt to their environment. Variations accumulate over time to generate new species

• Terminologies 1. Individual - carrier of the genetic information (chromosome). It

is characterized by its state in the search space, its fitness (objective function value).

2. Population - pool of individuals which allows the application of genetic operators.

3. Fitness function - The term “fitness function” is often used as a synonym for objective function.

4. Generation - (natural) time unit of the EA, an iteration step of an evolutionary algorithm.

Evolutionary Algorithms

Population

Individual

Crossover

Mutation

Parents Offspring

Evolutionary Algorithms

Evolutionary Algorithms

Step 1 t:= 0

Step 2 Initialize P(t)

Step 3 Evaluate P(t)

Step 4 While not terminate do P’(t) := variation [P(t)]; evaluate [P’(t)]; P(t+1) := select [P’(t) U P(t)]; t := t + 1; od

Reproduced from “Evolutionary Computation: Comments on the History and Current State” – Bäack et. al

Evolutionary algorithms = Selection + Crossover + Mutation

Evolutionary Algorithms

• Mean approaches optimum

• Variance reduces

Evolutionary Algorithms

Effi

cie

ncy

Problem type

Random scheme

Robust scheme

Robustness = Breadth + Efficiency

(Goldberg, 1989)

• Selection - Roulette wheel, Tournement, steady state, etc.

– Motivation is to preserve the best (make multiple copies) and eliminate the worst

• Crossover – simulated binary crossover, Linear crossover, blend crossover, etc.

– Create new solutions by considering more than one individual

– Global search for new and hopefully better solutions

• Mutation – Polynomial mutation, random mutation, etc. – Keep diversity in the population

– 010110 →010100 (bit wise mutation)

Evolutionary Algorithms

• Tournament selection

Evolutionary Algorithms

23 30

24 11

37 9

24 24

11 9

9 11 23

37 24

30

11

9

Tournament 1 Tournament 2

37 30 Deleted from population

• Roulette wheel selection (proportional selection)

– Weaker solutions can survive.

Evolutionary Algorithms

• Concept of exploration vs exploitation.

• Exploration – Search for promising solutions • Crossover and mutation operators

• Exploitation – preferring the good solutions • Selection operator

• Excessive exploration – Random search.

• Excessive exploitation – Premature convergence.

Evolutionary Algorithms

Good evolutionary algorithm

Exploitation Exploration

Evolutionary Algorithms

Classical gradient based algorithms

• Convergence to an optimal solution usually depends on the starting solution.

• Most algorithms tend to get stuck to a locally optimal solution.

• An algorithm efficient in solving one class of optimization problem may not be efficient in solving others.

• Algorithms cannot be easily parallelized.

Evolutionary algorithms

• Convergence to an optimal solution is designed to be independent of initial population.

• A search based algorithm. Population helps not to get stuck to locally optimal solution.

• Can be applied to wide class of problems without major change in the algorithm.

• Can be easily parallelized.

Evolutionary Algorithms

Fitness Landscapes

x

f(x) Ideal and best case

f(x)

x

Multimodal

Nightmare

f(x)

x

f(x)

x

Teaser

Using traditional gradient based methods

Fitness Landscapes

x

f(x) Ideal and best case

f(x)

x

Multimodal

Nightmare

f(x)

x

f(x)

x

Teaser

Using population based algorithms

• GA: John Holland in 1962 (UMich) • Evolutionary Strategy: Rechenberg and Schwefel in 1965

(Berlin) • Evolutionary Programming: Larry Fogel in 1965 (California) • First ICGA: 1985 in Carnegie Mellon University • First GA book: Goldberg (1989) • First FOGA workshop: 1990 in Indiana (Theory) • First Fusion: 1990s (Evolutionary Algorithms) • Journals: ECJ (MIT Press), IEEE TEC, Natural Computation

(Elsevier) • GECCO and CEC since 1999, PPSN since 1990 • About 20 major conferences each year

History of Evolutionary Algorithms

• Proposed by R. Storn and K. Price (1997) – Storn, R., Price, K. (1997). "Differential evolution - a simple and efficient heuristic for global optimization over continuous

spaces“, Journal of Global Optimization 11: 341–359.

• A population based approach for minimization of functions – A maximization function is converted to a

minimization function (max f(x) = -min(f(x))

• Parameters to be set – NP, Population size (5 – 10) x number of variables – F, Scaling factor [0,2] – CR, Crossover ratio [0,1] – NGEN, Maximum number of generations

Differential Evolution

Differential Evolution

x4-x5

v1 = x3 + F(x4-x5)

X1 x2 x3 x4 x5

f1 f2 f3 f4 f5

P1 P2 P3 P4 P5

X

Y

Z

fI fII fI < fII

X1

fI Y

X1

fII

N

X1 x2 x3 x4 x5

f1 f2 f3 f4 f5

C5 C1 C2 C3 C4

Target vector

Trial vector

Y

Z

Rand < CR

Mutation

Crossover

• DE Scheme

– DE/x/y/z

• x: specifies the vector to be mutated which currently is “rand”.

• y: number of difference vectors used.

• z: denotes the crossover scheme. The current variant is “bin”. Also ‘exp’ is available.

• DE/rand/1/bin

Differential Evolution

Differential Evolution

x4 x5

Minimum

v1 = x3 + F(x4-x5)

x3 v1

x1

x2

Mutation

t1

v1

c2

c1

x1

x2

Crossover

Constraint Handling

• Penalty parameter-less approach

– A feasible solution is preferred to infeasible solution

– When both solutions feasible, choose the solution with better function value

– When both solutions are infeasible, choose the solution with lower constraint violation

• Box constraints

– If variable is lower/higher than lower/upper bound,

• set to lower/upper bound

• A random value inside the bounds

Constraint Handling

• No guarantee of finding an optimal solution in finite time – Asymptotic convergence

• Containing a number of parameters – Sometimes the result is highly dependent on the

parameters set

– Self-adaptive parameters are commonly used

• Computationally very expensive – Metamodels of functions are commonly used

Limitations of Evolutionary Algorithms

• Application 1

– Tracking suspect

• Caldwell and Johnston, 1991

• Objective function: fitness rating on a nine point scale

Applications

• Optimization (Min/Max) of functions

• Airfoil optimization

• Evolving optimal structure

• Games

Applications