Differential Evolution

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Differential EvolutionHossein TalebiHassan nikoo

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Variations to Basic Differential Evolution Hybrid Differential Evolution Strategies

Gradient-Based Hybrid Differential Evolution

Evolutionary Algorithm-Based Hybrids DE Reproduction Process used as Cross over

Operation of Simple GA Ranked-Based Cross Over Operator for DE

Particle Swarm Optimization Hybrids Population-Based Differential Evolution Self-Adaptive Differential Evolution

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Variations to Basic Differential Evolution Differential Evolution for Discrete-Valued

Problems Angle Modulated Differential Evolution Binary Differential Evolution

Constraint Handling Approaches Multi-Objective Optimization Dynamic Environments Applications

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Gradient-Based Hybrid Differential Evolution acceleration operator to improve

convergence speed – without decreasing diversity

migration operator The Acceleration Operator uses gradient

descent to adjust the best individual toward obtaining a better position

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Gradient-Based Hybrid Differential Evolution(Cont.) Using Gradient Descent may results in

getting stuck in a local optima or premature convergence

We can increase Population diversity by Migration Operator

This operator spawns new individuals from the best individual and replaces the current population with these new individuals

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Gradient-Based Hybrid Differential Evolution(Cont.) The Migration Operator is applied only

when diversity of population becomes too small

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Gradient-Based Hybrid Differential Evolution(Cont.)

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Using Stochastic Gradient Descent and DE for Neural Networks Training

Stochastic Gradient Descent

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Evolutionary Algorithm-Based Hybrids Hrstka and Kucerov´a used the DE

reproduction process as a crossover operator in a simple GA

Chang and Chang used standard mutation operators to increase DE population diversity by adding noise to the created trial vectors.

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Evolutionary Algorithm-Based Hybrids(Cont.) Sarimveis and Nikolakopoulos [758] use

rank-based selection to decide which individuals will take part to calculate difference vectors

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Ranked-base Mutation

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Particle Swarm Optimization Hybrids Hendtlass proposed that the DE

reproduction process be applied to the particles in a PSO swarm at specified intervals.

Kannan et al. apply DE to each particle for a number of iterations and replaces the best with particle

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Particle Swarm Optimization Hybrids(Cont.) Another approach is to change only

change best particle using

Where sigma is general difference vector

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Population-Based Differential Evolution Ali and T¨orn proposed to use an

auxiliary population For each offspring created, if the fitness

of the offspring is not better than the parent, instead of discarding the offspring, it is considered for inclusion in the auxiliary Population

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DETVSF (DE with Time Varying Scale Factor) During the later stages it is important to

adjust the movements of trial solutions finely so that they can explore the interior of a relatively small space in which the suspected global optimum lies

We can reduce the scale factor linearly with time from a (predetermined) maximum to a (predetermined) minimum value

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DETVSF (DE with Time Varying Scale Factor)(Cont.)

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Parameter Control in DE Dynamic Parameters

Self-Adaptive

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Self-Adaptive Parameters probability of recombination be self –

adapted

Mu is the average of successful probablities

Abbass Proposed to use this formula :

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Self-Adaptive Parameters(Cont.) Omran et al. propose a self-adaptive DE

strategy that makes use of this formula for scale factor

For mutation operator

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Angle Modulated Differential Evolution Pampar´a et al. proposed a DE

algorithm to evolve solutions to binary-valued optimization problems, without having to change the operation of the original DE

They use a mapping between binary-valued and continuous-valued space to solve the problem in binary space

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Angle Modulated Differential Evolution(Cont.) The objective is to evolve, in the

abstracted continues space, a bitstring generating function will be used in the original space to produce bit-vector solutions

‘a’, ’b’, ‘c’ and ‘d’ are continues space problem parameter

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Angle Modulated Differential Evolution(Cont.) ‘a=0’ ‘b=1’ ‘c=1’ ‘d=0’

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Binary Differential Evolution binDE borrows concepts from the binary

particle swarm optimizer binPSO binDE uses the floating-point DE

individuals to determine a probability for each component

the corresponding bitstring solution will be calculated as follow :

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Binary Differential Evolution

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Constraint Handling Approaches

Penalty methods adding a function to penalize solutions

that violate constraints Using F(x, t) = f(x, t) + λp(x, t) where λ is

the penalty coefficient and p is time dependent penalty function

Converting the constrained problem to an unconstrained problem

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Constraint Handling Approaches(Cont.) We can convert constrained problem to

an unconstrained problem by defining the Lagrangian for the constrained problem

If primal problem is convex then defining dual problem and solving minmax problem

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Constraint Handling Approaches(Cont.) By changing selection operator ,

infeasible solutions can be rejected and we can use a method for repairing of the infeasible solution

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Constraint Handling Approaches(Cont.) Boundary constraints are easily

enforced by clamping offspring to remain within the given boundaries

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Multi-Objective Optimization

Converting the problem into the Weighted Aggregation Methods

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Multi-Objective Optimization(Cont.) This method intends to define an

aggregate objective function as a weighted sum of the objectives

Usually assumed that

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Multi-Objective Optimization(Cont.) There is no guarantee that different

solutions will be found A niching strategy can be used to find

multiple solutions It is difficult to get the best weight

values, ωk, since these are problem-dependent

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Multi-Objective Optimization(Cont.) Vector evaluated DE is a population based

method for MOO If K objectives have to be optimized, K

sub-populations are used, where each subpopulation optimizes one of the objectives.

Sub-populations are organized in a ring topology

The best individual of sub-population Ck migrates to population Ck+1 to produce the trial vectors for that population

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Dynamic Environments Assumptions

the number of peaks, nX , to be found are know and these peaks are evenly distributed through the search space

Changes are small and gradual DynDE uses multiple populations, with

each population maintaining one of the peaks

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Dynamic Environments(Cont.) At each iteration, the best individuals of

each pair of sub-populations are compared if these global best positions are too close to one another, the sub-population with the worst global best solution is re-initialized

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Dynamic Environments(Cont.) The following diversity increasing

strategies Re-initialize the sub-populations Use quantum individuals :Some of the

individuals are re-initialized to random points inside a ball centered at the global best individual

Use Brownian individuals: Some positions are initialized to random positions around global best individual

Some individuals are simply added noise

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Dynamic Environments(Cont.)

Initialization of Quantum Individuals

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Applications Mostly applied to optimize functions

defined over continuous-valued landscapes

Clustering Controllers Filter design Image analysis Integer-Programming Model selection NN training

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References1. Computational Intelligence, an introduction,2nd

edition, Andries Engelbercht, Wiley2. Differential Evolution - A simple and efficient

adaptive scheme for global optimization over continuous spaces, Rainer Storn,Kenneth Price,1995

3. Particle Swarm Optimization and Differential Evolution Algorithms: Technical Analysis, Applications and Hybridization Perspectives, Swagatam Das1, Ajith Abraham2, and Amit Konar1,Springer 2008.

4. Differential Evolution, homepage http://www.icsi.berkeley.edu/~storn/code.html

Differential Evolution

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Differential Evolution