微分进化(Difference Evolution，DE)算法是一种优化算法，据称其比GA(遗传算法)等更为优秀。

借鉴网上实现的DE算法，用Matlab实现了对若干函数优化问题的解法，代码如下：

function [] = de_test

clear all;

close all;

clc;

%解：X = [0, 0, ...]

%nVar = 30

%dims: [-30, 30]

function fitness = sphere(vals)

prod = vals .* vals;

fitness = sum(prod, 2);

end

%f(x) = 1/4000 * sum^n_1(x_i)^2 - prod^n_1 * cos(x_i/sqrt(i)) + 1

%f* = 0, x* = [0, 0, ...];

%nVar = 30

%dims: [-600, 600]

function fitness = griewank(vals)

[h w] = size(vals);

p1 = vals.^2;

p1 = 1/4000 .* sum(p1, 2);

t = sqrt([1:w]);

p2 = vals ./ repmat(t, h, 1);

p2 = cos(p2);

p2 = prod(p2, 2);

fitness = p1 - p2 + 1;

end

%解: X = [1, 1, 1, ...]

function fitness = RosenBroek(vals)

[k n] = size(vals);

vals1 = vals(:, 1:n-1);

vals2 = vals(:, 2:n);

tmp = vals2 - vals1 .* vals1;

tmp = 100 * tmp .* tmp;

tmp1 = (vals1 - 1).^2;

fitness = sum(tmp, 2) + sum(tmp1, 2);

end

%许多局部最小值，最优解：X = [0, 0], 固定2个变量

function fitness = Rastrigin (X)

v = X.^2 - 10 .* cos(2 * pi .* X) + 10;

fitness = sum(v, 2);

%fitness = 20 + X(:, 1).^2 + X(:, 2).^2 - 10 .* (cos(2 * pi * X(:, 1)) + cos(2 * pi * X(:, 2)));

end

%参数

popSize = 100; % 群规模

F = 0.5;                    % 突变因子

C = 0.9;                    % 交叉率

iterTimes = 300; % 迭代次数

Run = 4;        % 测试轮次

id = [1:popSize];

%结果

bestFit = 1e5;  %设取最小值

bestGene = [];

for r = 1:Run  %轮次

Func = r;     % 测试适应度函数

switch Func

case 1

Func = @sphere;

Xmin = -100;

Xmax = 100;

nVar = 30;

case 2

Func = @griewank;

Xmin = -600;

Xmax = 600;

nVar = 2;

case 3

nVar = 2;

Func = @RosenBroek;

Xmin = -100;

Xmax = 100;

case 4

nVar = 2;

Func = @Rastrigin

Xmin = -5.12;

Xmax = 5.12;

end;

Func

tic;

%1.初始化种群

X = Xmin + rand(popSize, nVar) * Xmax;

%2.每轮迭代

for i = 1:iterTimes

X0 = X;

%             F = 2 * (1 - (i-1)/iterTimes);

%个体突变得到V

for j = 1:popSize

ids = id;

ids(j) = [];

rids = randperm(popSize - 1, 3);

rids = ids(rids);

V = X(rids(1), :) + F * (X(rids(2), :) - X(rids(3), :));

%对V(j, :)值域检查

ids = find(V < Xmin);

if length(ids) > 0

V(ids) = Xmin;

end;

ids = find(V > Xmax);

if length(ids) > 0

V(ids) = Xmax;

end;

%对每个X和V的配对，进行交叉操作，结果存在U中

jrand = floor(rand() * nVar + 1);  %必交叉项，保证至少一个交叉

for n = 1:nVar

R1 = rand();

if (R1 < C || n == jrand)

U = V;    %保留子代基因

else

U = X(j, :);    %保留父代基因

end

end

%在子代和父代间做选择运算

if Func(U) < Func(X(j, :))

Tr = U;

else

Tr=X(j, :);

end

% 更新种群基因

X(j,:) = Tr;

%计算新的适应度

fitness = Func(X(j,:));

%记录全局最优解

if fitness < bestFit

bestFit = fitness;

bestGene = X(j, :);

end;

end  %结束个体更新

end

bestGene

bestFit

toc

end

end运行结果如下：

Func =

@de_test/sphere

bestGene =

Columns 1 through 15

-7.5570  -11.9099    9.9957  -37.8403  -17.9445   -7.1438  -21.4304  -33.6260  -22.1812  -66.1438    0.9014  -39.6724   -0.3175  -56.4815   26.5422

Columns 16 through 30

6.5446  -31.9653   -9.3640  -37.1629  -23.7325    2.7271   -6.3413  -21.3204  -13.1450   28.7402  -28.8170  -22.6226   10.9031  -16.6128  -14.7637

bestFit =

2.0583e+04  (从此结果看，算法效果不佳)

时间已过 2.285354 秒。

Func =

@de_test/griewank

bestGene =

1.0e-08 *

0.1702    0.1424

bestFit =

0

时间已过 3.337347 秒。

Func =

@de_test/RosenBroek

bestGene =

1.0e-08 *

0.1702    0.1424

bestFit =

0

时间已过 1.747765 秒。

Func =

@de_test/Rastrigin

Func =

@de_test/Rastrigin

bestGene =

1.0e-08 *

0.1702    0.1424

bestFit =

0

时间已过 1.403871 秒。