差分進化算法(DE)求函數最小值

差分進化算法求函數 Z = 3 * cos(X .* Y) + X + Y , -4 <= X <= 4, -4 <= Y <= 4。

計算目標函數值

計算目標函數值的函數：

function z = calobj (pop)% 計算目標函數值% pop       input  種羣% z         output 目標函數值z = 3 * cos(pop(:,1) .* pop(:,2)) + pop(:,1) + pop(:,2);end

MATLAB

初始化種羣

目標函數有兩個參數，生成每個個體有兩個基因的種羣：

function pop = initpop(popsize, chromlength, xl, xu)% 生成初始種羣% popsize           input  種羣規模% chromlengt        input  染色體長度% xl                input  x下限% xu                input  x上限% pop               output 種羣pop = rand(popsize, chromlength) * (xu - xl) + xl;end

MATLAB

變異

變異函數如下：

function mutationpop = mutation (pop, F)% 變異操作% pop           input  種羣% F             input  縮放因子% mutationpop   output 變異後種羣[popsize, chromlength] = size(pop);mutationpop = zeros(popsize, chromlength);for i = 1:popsize    % 取3個互異的索引 r0 r1 r2
    r = randperm(popsize);
    index = find (r ~= i);
    rn = r(index(1:3));
    r0 = rn(1); r1 = rn(2); r2 = rn(3);%     fprintf('i = %d, r0 = %d, r1 = %d, r2 = %d\n', i, r0, r1, r2);

    mutationpop(i,:) = pop(r0,:) + F .* (pop(r1,:) - pop(r2,:));endend

MATLAB

交叉

交叉函數如下：

function crossoverpop = crossover(pop, mpop, cr)% 交叉% pop           input  種羣% mpop          input  變異後的種羣% cr            input  交叉概率% crossoverpop  output 交叉後的種羣[popsize, chromlength] = size(pop);crossoverpop = mpop;r = rand(popsize, chromlength);index = find (r > cr);crossoverpop(index) = pop(index);jrand = randi(chromlength, 1, popsize);crossoverpop(sub2ind(size(crossoverpop), [1:popsize], jrand)) ...
    = mpop(sub2ind(size(mpop), [1:popsize], jrand));end

MATLAB

在交叉操作之後，應約束邊界：

function newpop = constrictboundary(pop, xl, xu)% 約束邊界（邊界吸收）% pop       input  種羣% xl        input  自變量最小值（包含）% xu        input  自變量最大值（包含）% newpop    output 約束邊界後的種羣newpop = pop;newpop(newpop < xl) = xl;newpop(newpop > xu) = xu;end

MATLAB

選擇

function newpop = selection(pop, npop)% 選擇（小值優化）% pop           input  種羣1（原始種羣）% pop           input  種羣2（變異-交叉種羣）% newpop        output 選擇後的種羣newpop = pop;index = find(calobj(npop) <= calobj(pop));newpop(index, :) = npop(index, :);end

MATLAB

主程序

主程序如下：

clc;clear;NP  = 20;       % 種羣規模D   = 2;        % 參數個數G   = 30;       % 最大進化代數F   = 0.5;      % 縮放因子Cr  = 0.8;      % 交叉因子xl  = -4;       % x下限（也是y下限）xu  = 4;        % x上限（也是y上限）bestvalue = zeros(3, G);% 優化gen = 0;pop = initpop(NP, D, xl, xu);objvalue = calobj(pop);while gen < G
    mpop = mutation(pop, F);                    % 變異
    cpop = crossover(pop, mpop, Cr);            % 交叉
    cpop = constrictboundary(cpop, xl, xu);     % 約束邊界
    pop = selection(pop, cpop);                 % 選擇
    objvalue = calobj(pop);
    gen = gen + 1;

    % 記錄最優
    [~, index] = min(objvalue);
    bestvalue(1:2, gen) = pop(index,:)';
    bestvalue(3,gen) = objvalue(index);endfprintf('bestX = %f, bestY = %f, bestZ = %f\n', ...
    bestvalue(1,end), bestvalue(2,end), bestvalue(3,end));% 繪圖figure(1);x = [-4:0.1:4]; y = [-4:0.1:4];[X, Y] = meshgrid(x, y);Z = 3 * cos(X .* Y) + X + Y;surf(X, Y, Z);hold on;scatter3(bestvalue(1,:), bestvalue(2,:), bestvalue(3,:), ...
    'MarkerEdgeColor','k', 'MarkerFaceColor',[0 .75 .75]);xlabel('x'); ylabel('y'); zlabel('z'); title('函數圖');hold off;figure(2);plot(bestvalue(3,:));xlabel('進化代數'); ylabel('最優目標函數值'); title('目標函數值變化圖');

MATLAB

執行結果

bestX = -3.947841, bestY = -4.000000, bestZ = -10.937414