DIFFERENTIAL EVOLUTION

By Fakhroddin Noorbehbahani

EA course, Dr. Mirzaee

December, 20101

AGENDA

Preface

Basic Differential Evolution

Difference Vectors

Mutation

Crossover

Selection

General Differential Evolution Algorithm

Control Parameters

Geometrical Illustration

DE/x/y/z

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AGENDA

Variations to Basic Differential Evolution

Hybrid Differential Evolution Strategies

Population-Based Differential Evolution

Self-Adaptive Differential Evolution

Differential Evolution for Discrete-Valued Problems

Constraint Handling Approaches

Comparison with other algorithms

Applications3

PREFACE

Price and Storn in 1995 Chebychev Polynomial fitting

Problem

3rd place at the First International Contest on

evolutionary Computation (1stICEO) 1996, the best

genetic type of algorithm for solving the real-valued test

function suite.

stochastic, population-based search strategy

Main characteristics

Guide search with distance and direction information from the

current population

original DE strategies for continuous-valued landscapes

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BASIC DIFFERENTIAL EVOLUTION

mutation is applied first to generate a trial vector, which is then used within the crossover operator to produce one offspring,

mutation step sizes are not sampled from a prior known probability distribution function.

mutation step sizes are influenced by differences between individuals of the current population

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DIFFERENCE VECTORS

Position of individuals and fitness

Over time, as the search progresses, the distances between

individuals become smaller

The magnitude of the initial distances between individuals is

influenced by the size of the population

Distances between individuals are a very good indication of the

diversity of the current population

Use difference vector to determine the step size

total number of differential perturbations

nv is the number of differentials used

ns is the population size

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MUTATION

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CROSSOVER

Binomial crossover Exponential crossover

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SELECTION

Random Selection To select the individuals from which difference vectors

are calculated. The target vector is either randomly selected or the

best individual is selected

Deterministic Selection To construct the population for the next generation,

the offspring replaces the parent if the fitness of the offspring is better than its parent; otherwise the parent survives to the next generation.

This ensures that the average fitness of the population does not deteriorate. 9

GENERAL DIFFERENTIAL EVOLUTION ALGORITHM

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CONTROL PARAMETERS

population size,ns

scale factor, β

probability of recombination,Pr

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POPULATION SIZE

The size of the population has a direct influence on the exploration ability of DE algorithms.

The more individuals there are in the population, the more differential vectors are available, and the more directions can be explored

The computational complexity per generation increases with the size of the population.

Empirical studies provide the guideline that ns ≈ 10nx 13

SCALE FACTOR

The scaling factor, β ∈ (0,∞), controls the amplification of the differential variations, (xi2−xi3 ).

The smaller the value of β, the smaller the mutation step sizes Smaller step sizes can be used to explore local

areas. slower convergence Larger values for β facilitate exploration, but may cause

the algorithm to overshoot optima

As the population size increases, the scaling factor should decrease. 14

RECOMBINATION PROBABILITY

This parameter controls the number of elements of the parent, xi(t), that will change.

The higher the probability of recombination, the more variation is introduced in the new population, thereby increasing diversity and increasing exploration.

Increasing pr often results in faster convergence, while decreasing pr increases search robustness

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GEOMETRICAL ILLUSTRATION

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GEOMETRICAL ILLUSTRATION

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GEOMETRICAL ILLUSTRATION

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EXAMPLE:PEAK FUNCTION

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GEOMETRICAL ILLUSTRATION

Generation 1: DE’s population and difference vector distributions

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GEOMETRICAL ILLUSTRATION

Generation 6: The population coalesces around the two main minima

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GEOMETRICAL ILLUSTRATION

Generation 12: The difference vector distribution contains three main clouds – one for local searches and two for moving between the two main minima.

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GEOMETRICAL ILLUSTRATION

Generation 16: The population is concentrated on the main minimum

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GEOMETRICAL ILLUSTRATION

Generation 20: Convergence is imminent. The difference vectors automatically shorten for a fine-grained, local search.

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GEOMETRICAL ILLUSTRATION

Generation 26: The population has almost converged.

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GEOMETRICAL ILLUSTRATION

Generation 34: DE finds the global minimum.

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DE/X/Y/Z

DE/best/1/z

DE/x/nv/z

DE/rand-to-best/nv/z

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DE/X/Y/Z

DE/current-to-best/1+nv/z

DE/rand/1/bin vs. DE/current-to-best/2/bin DE/rand/1/bin maintains good diversity DE/current-to-best/2/bin shows good convergence

characteristics Dynamically switch between these two

strategies

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VARIATIONS TO BASIC DIFFERENTIAL EVOLUTION

Hybrid Differential Evolution Strategies Gradient-Based Hybrid Differential Evolution

Acceleration operator : to improve convergence speed Migration operator : to improve ability for escaping

local optima Acceleration operator

uses gradient descent to adjust the best individual toward obtaining a better position if the mutation and crossover operators failed to improve

x(t), replaces the worst individual in the new population, C(t+1).

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MIGRATION OPERATOR Gradient decent speed up but local minima Migration operator

increase population diversity Generate new individual from best individuals

Applied when diversity is too small i.e.:

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HYBRID DIFFERENTIAL EVOLUTION WITH ACCELERATION AND MIGRATION

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EVOLUTIONARY ALGORITHM-BASED HYBRIDS DE reproduction process as a crossover

operator in a simple GA

Rank-Based Crossover Operator for Differential Evolution To select individuals to calculate difference vectors xi1 (t) precedes xi2 (t) if f(xi1(t)) > f(xi2(t)).

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RANK-BASED MUTATION OPERATOR FOR DIFFERENTIAL EVOLUTION

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OTHER VARIATIONS TO BASIC DE

Population-Based Differential Evolution Improve exploration by using 2 population set Initialize with ns pairs Rejected individual by selection put in auxiliary pop

Self-Adaptive Differential Evolution Dynamic Parameters

Self-Adaptive Parameters

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DIFFERENTIAL EVOLUTION FOR DISCRETE-VALUED PROBLEMS

Angle Modulated DE

where x is a single element from a set of evenly separated intervals determined by the required number of bits that need to be generated

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ANGLE MODULATED DIFFERENTIAL EVOLUTION

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BINARY DIFFERENTIAL EVOLUTION

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CONSTRAINT HANDLING APPROACHES

Penalty methods Converting the constrained problem to an

unconstrained problem By changing the selection operator of DE,

infeasible solutions can be rejected

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COMPARISON WITH GA AND PSO

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COMPARISON

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APPLICATIONS 1) General Optimization Framework "Mystic" by Mike McKerns,

Caltech. 2) Multiprocessor synthesis. 3) Neural network learning. 4) Chrystallographic characterization. 5) Synthesis of modulators. 6) Heat transfer parameter estimation in a trickle bed reactor. 7) Scenario-Integrated Optimization of Dynamic Systems. 8) Optimal Design of Shell-and-Tube Heat Exchangers. 9) Optimization of an Alkylation Reaction. 10) Optimization of Thermal Cracker Operation. 11) Optimization of Non-Linear Chemical Processes. 12) Optimum planning of cropping patterns. 13) Optimization of Water Pumping System . 14) Optimal Design of Gas Transmission Network . 15) Differential Evolution for Multi-Objective Optimization 16) Physiochemistry of Carbon Materials. 17) Radio Network Design. 18) Reflectivity Curve Simulation.

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COMMERCIAL SOFT

1) Built in optimizer in MATHEMATICA's function Nminimize (since version 4.2).

2) MATLAB's GA toolbox contains a variant of DE.

3) Digital Filter Design. 4) Diffraction grating design. 5) Electricity market simulation. 6) Auto2Fit. 7) LMS Virtual Lab Optimization. 8) Optimization of optical systems. 9) Finite Element Design. 46

APPLICATION : BUMP PROBLEM

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APPLICATION : BUMP PROBLEM

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REFERENCES [1] http://www.icsi.berkeley.edu/~storn/code.html [2] Andries P. Engelbrecht ,(2007),Computational Intelligence:

An Introduction, 2nd Edition., ISBN: 978-0-470-03561-0. [3] Price, K.; Storn, R.M.; Lampinen, J.A. (2005).

Differential Evolution: A Practical Approach to Global Optimization. Springer. ISBN 978-3-540-20950-8. http://www.springer.com/computer/theoretical+computer+science/foundations+of+computations/book/978-3-540-20950-8.

[4] Feoktistov, V. (2006). Differential Evolution: In Search of Solutions. Springer. ISBN 978-0-387-36895-5. http://www.springer.com/mathematics/book/978-0-387-36895-5.

[5] J. Vesterstrom and R. Thomson, A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems, Proc. of IEEE Congress on Evolutionary Computation, 2004,pp. 1980–1987.

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در حالي که تو سرگرم برنامه ريزي هاي ديگري

هستي

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THANKS FOR YOUR ATTENTION

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