Introduction to Differential Evolution Rajib Kumar Bhattacharjya - - PowerPoint PPT Presentation

Introduction to Differential Evolution

Rajib Kumar Bhattacharjya

Department of Civil Engineering

Indian Institute of Technology Guwahtai

Differential Evolution

It is a stochastic, population-based optimization algorithm for solving nonlinear

• ptimization problem

Consider an optimization problem Minimize 𝑔 𝑌 Where 𝑌 = 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝐸 , 𝐸 is the number of variables The algorithm was introduced by Storn and Price in 1996

Evolutionary algorithms

Initialize population Mutation Recombination Selection Termination Criteria Next Generation Optimal Solution

Yes No This is a population based algorithm Consider a population size of 𝑂 The population matrix can be shown as 𝑦𝑜,𝑗

𝑕 = 𝑦𝑜,1 𝑕 , 𝑦𝑜,2 𝑕 , 𝑦𝑜,3 𝑕 , … , 𝑦𝑜,𝐸 𝑕

Where, 𝑕 is the Generation and 𝑜 = 1,2,3, … 𝑂

Initial population

Initial population is generated randomly between upper lower and upper bound 𝑦𝑜,𝑗 = 𝑦𝑜,𝑗

𝑀 + 𝑠𝑏𝑜𝑒

∗ 𝑦𝑜,𝑗

𝑉 − 𝑦𝑜,𝑗 𝑀

𝑗 = 1,2,3, … 𝐸 and 𝑜 = 1,2,3, … 𝑂 Where 𝑦𝑗

𝑀 is the lower bound of the variable 𝑦𝑗

Where 𝑦𝑗

𝑉 is the upper bound of the variable 𝑦𝑗

Mutation

From each parameter vector, select three other vectors 𝑦𝑠1𝑜

𝑕 , 𝑦𝑠2𝑜 𝑕

and 𝑦𝑠3𝑜

𝑕

randomly. Add the weighted difference of two of the vectors to the third 𝑤𝑜

𝑕+1 = 𝑦𝑠1𝑜 𝑕

+ 𝐺 𝑦𝑠2𝑜

𝑕

− 𝑦𝑠3𝑜

𝑕

𝑤𝑜

𝑕+1 is called donor vector

𝐺 is generally taken between 0 and 1

Recombination

A trial vector 𝑣𝑜,𝑗

𝑕+1 is developed from the target vector, 𝑦𝑜,𝑗 𝑕 , and the donor vector,

𝑤𝑜,𝑗

𝑕+1

𝑣𝑜,𝑗

𝑕+1 =

𝑤𝑜,𝑗

𝑕+1

𝑗𝑔 𝑠𝑏𝑜𝑒() ≤ 𝐷𝑞 𝑝𝑠 𝑗 = 𝐽𝑠𝑏𝑜𝑒 𝑦𝑜,𝑗

𝑕

𝑗𝑔 𝑠𝑏𝑜𝑒() > 𝐷𝑞 𝑏𝑜𝑒 𝑗 ≠ 𝐽𝑠𝑏𝑜𝑒 𝐽𝑠𝑏𝑜𝑒 is a integer random number between [1,D] 𝐷𝑞 is the recombination probability 𝑗 = 1,2,3, … 𝐸 and 𝑜 = 1,2,3, … 𝑂

Selection

The target vector 𝑦𝑜,𝑗

𝑕 is compared with the trial vector 𝑣𝑜,𝑗 𝑕+1 and the one with the

lowest function value is selected for the next generation 𝑦𝑜

𝑕+1 = 𝑣𝑜,𝑗 𝑕+1

𝑗𝑔 𝑔 𝑣𝑜

𝑕+1 < 𝑔 𝑦𝑜 𝑕

𝑦𝑜

𝑕

𝑃𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 𝑜 = 1,2,3, … 𝑂