差分进化算法 DE-Differential Evolution

差分进化算法 (Differential Evolution)

 

Differential Evolution（DE）是由Storn等人于1995年提出的，和其它演化算法一样，DE是一种模拟生物进化的随机模型，通过反复迭代，使得那些适应环境的个体被保存了下来。但相比于进化算法，DE保留了基于种群的全局搜索策略，采用实数编码、基于差分的简单变异操作和一对一的竞争生存策略，降低了遗传操作的复杂性。同时，DE特有的记忆能力使其可以动态跟踪当前的搜索情况，以调整其搜索策略，具有较强的全局收敛能力和鲁棒性，且不需要借助问题的特征信息，适于求解一些利用常规的数学规划方法所无法求解的复杂环境中的优化问题。目前，DE已经在许多领域得到了应用，譬如人工神经元网络、化工、电力、机械设计、机器人、信号处理、生物信息、经济学、现代农业、食品安全、环境保护和运筹学等。

 

• DE算法-作者网站: http://www1.icsi.berkeley.edu/~storn/code.html
• 维基百科资料库  : https://en.wikipedia.org/wiki/Differential_evolution

 

DE 算法主要用于求解连续变量的全局优化问题，其主要工作步骤与其他进化算法基本一致，主要包括变异(Mutation)、交叉(Crossover)、选择(Selection)三种操作。算法的基本思想是从某一随机产生的初始群体开始，利用从种群中随机选取的两个个体的差向量作为第三个个体的随机变化源，将差向量加权后按照一定的规则与第三个个体求和而产生变异个体，该操作称为变异。然后，变异个体与某个预先决定的目标个体进行参数混合，生成试验个体，这一过程称之为交叉。如果试验个体的适应度值优于目标个体的适应度值，则在下一代中试验个体取代目标个体，否则目标个体仍保存下来，该操作称为选择。在每一代的进化过程中，每一个体矢量作为目标个体一次，算法通过不断地迭代计算，保留优良个体，淘汰劣质个体，引导搜索过程向全局最优解逼近。

 

算法图解:

 算法伪代码: 

 

 

 

 

算法C代码:

 

//********************************************************/
//      DE/rand/1/bin --差分进化算法-(基本类型)       
//********************************************************/


#include <stdlib.h>
#include <stdio.h>
#include <time.h>
#include <float.h>

/* Function definitions        */

double func(double *);
int usage(char *);

/* Random number generator defined by URAND should return
double-precision floating-point values uniformly distributed
over the interval [0.0, 1.0)                    */

#define URAND    ((double)rand()/((double)RAND_MAX + 1.0))

/* Definition for random number generator initialization    */

#define INITRAND srand(time(0))

/* Usage for the program    */

int usage(char *str)
{
    fprintf(stderr, "Usage: %s [-h] [-u] [-s] [-N NP (20*D)] ", str);
    fprintf(stderr, "[-G Gmax (1000)]
");
    fprintf(stderr, "	[-C crossover constant, CR (0.9)]
");
    fprintf(stderr, "	[-F mutation scaling factor, F (0.9)]
");
    fprintf(stderr, "	[-o <outputfile>]

");
    fprintf(stderr, "	-s does not initialize random number generator
");
    exit(-1);
}


int main(int argc, char **argv)
{
    register int i, j, k, r1, r2, r3, jrand, numofFE = 0;
    extern int D;
    extern double Xl[], Xu[];

    int NP = 20 * D, Gmax = 1000, c, index = -1, s = 1;

    double **popul, **next, **ptr, *iptr, *U, CR = 0.9, F = 0.9,

        min_value = DBL_MAX, totaltime = 0.0;

    char *ofile = NULL;

    FILE *fid;
    clock_t starttime, endtime;


    /* Parse command line arguments given by user    */

    for (i = 1; i < argc; i++)
    {
        if (argv[i][0] != '-')
            usage(argv[0]);

        c = argv[i][1];

        switch (c)
        {
        case 'N':
            if (++i >= argc)
                usage(argv[0]);

            NP = atoi(argv[i]);
            break;
        case 'G':
            if (++i >= argc)
                usage(argv[0]);

            Gmax = atoi(argv[i]);
            break;
        case 'C':
            if (++i >= argc)
                usage(argv[0]);

            CR = atof(argv[i]);
            break;
        case 'F':
            if (++i >= argc)
                usage(argv[0]);

            F = atof(argv[i]);
            break;
        case 'o':
            if (++i >= argc)
                usage(argv[0]);

            ofile = argv[i];
            break;
        case 's':    /* Flag for using same seeds for        */
            s = 0;    /* different runs                */
            break;
        case 'h':
        case 'u':
        default:
            usage(argv[0]);
        }
    }

    if (s) INITRAND;

    /* Printing out information about optimization process for the user    */

    printf("Program parameters: ");
    printf("NP = %d, Gmax = %d, CR = %.2f, F = %.2f
",
        NP, Gmax, CR, F);

    printf("Dimension of the problem: %d
", D);


    /* Starting timer    */

    starttime = clock();


    /* Allocating memory for current and next populations, intializing
    current population with uniformly distributed random values and
    calculating value for the objective function    */


    // NP:种群大小, Gmax:迭代次数, CR:交叉概率, F:扰动向量的缩放因子

    //当前种群
    popul = (double **)malloc(NP*sizeof(double *));
    if (popul == NULL) perror("malloc");

    //下代种群
    next = (double **)malloc(NP*sizeof(double *));
    if (next == NULL) perror("malloc");

    //当前种群popul[NP][D+1] 
    for (i = 0; i < NP; i++)
    {
        //个体维度空间分配
        popul[i] = (double *)malloc((D + 1)*sizeof(double));
        if (popul[i] == NULL) perror("malloc");

        //初始化维度值
        for (j = 0; j < D; j++)
            popul[i][j] = Xl[j] + (Xu[j] - Xl[j])*URAND;

        //最后的元素内存放该个体的适应度值
        popul[i][D] = func(popul[i]);

        numofFE++;//统计评估次数

        //下一代个体空间分配
        next[i] = (double *)malloc((D + 1)*sizeof(double));
        if (next[i] == NULL) perror("malloc");
    }

    /* 为实验向量分配空间--Allocating memory for a trial vector U    */

    U = (double *)malloc((D + 1)*sizeof(double));
    if (U == NULL) perror("malloc");


    /* The main loop of the algorithm    */

    for (k = 0; k < Gmax; k++)
    {

        for (i = 0; i < NP; i++)    /* Going through whole population    */
        {

            /* Selecting random indeces r1, r2, and r3 to individuls of
            the population such that i != r1 != r2 != r3    */

            //1.选择三个互不相同的随机个体r1,r2,r3
            do
            {
                r1 = (int)(NP*URAND);
            } while (r1 == i);

            do
            {
                r2 = (int)(NP*URAND);
            } while (r2 == i || r2 == r1);
            do
            {
                r3 = (int)(NP*URAND);
            } while (r3 == i || r3 == r1 || r3 == r2);

            jrand = (int)(D*URAND);

            /* Mutation and crossover    */
            //2. 执行变异和交叉操作
            for (j = 0; j < D; j++)
            {
                //执行二项式交叉
                if (URAND < CR || j == jrand)
                {
                    //试验向量部分来自变异后的向量
                    U[j] = popul[r3][j] + F*(popul[r1][j] - popul[r2][j]);
                }
                else
                    //试验向量部分来自个体i
                    U[j] = popul[i][j];
            }
            //3. 计算新生成向量的适应度值
            U[D] = func(U);

            numofFE++;

            /* Comparing the trial vector 'U' and the old individual
            'next[i]' and selecting better one to continue in the
            next population.注意:空间的交替变换和使用    */

            //贪婪策略从试验向量U和当前个体i中选择一个好的放入到下一代个体中
            if (U[D] <= popul[i][D])//新向量好
            {

                //试验向量U牛逼, next指向当前的试验向量U,u指向next, 方法:指针交换
                iptr = U;
                U = next[i];
                next[i] = iptr;
            }
            else//原始向量牛逼, next指向个体i, 方法: 直接拷贝
            {
                for (j = 0; j <= D; j++)
                    next[i][j] = popul[i][j];
            }

        }    /* End of the going through whole population    */


        /* Pointers of old and new populations are swapped    */
        //指针交换,各指针指向的空间发生变化
        ptr = popul;
        popul = next;
        next = ptr;

    }    /* End of the main loop        */


    /* Stopping timer    */

    endtime = clock();
    totaltime = (double)(endtime - starttime);


    /* If user has defined output file, the whole final population is
    saved to the file                        */

    if (ofile != NULL)
    {
        if ((fid = (FILE *)fopen(ofile, "a")) == NULL)
        {
            fprintf(stderr, "Error in opening file %s

", ofile);
            usage(argv[0]);
        }

        for (i = 0; i < NP; i++)
        {
            for (j = 0; j <= D; j++)
                fprintf(fid, "%.15e ", popul[i][j]);
            fprintf(fid, "
");
        }
        fclose(fid);
    }

    /* Finding best individual    */

    for (i = 0; i < NP; i++)
    {
        if (popul[i][D] < min_value)
        {
            min_value = popul[i][D];
            index = i;
        }
    }

    /* Printing out information about optimization process for the user    */

    printf("Execution time: %.3f s
", totaltime / (double)CLOCKS_PER_SEC);
    printf("Number of objective function evaluations: %d
", numofFE);

    printf("Solution:
Values of variables: ");
    for (i = 0; i < D; i++)
        printf("%.15f ", popul[index][i]);

    printf("
Objective function value: ");
    printf("%.15f
", popul[index][D]);


    /* Freeing dynamically allocated memory    */

    for (i = 0; i < NP; i++)
    {
        free(popul[i]);
        free(next[i]);
    }
    free(popul);
    free(next);
    free(U);

    return(0);
}

 经典文献: 

[1] Storn, R., "Designing Nonstandard Filters with Differential Evolution, IEEE Signal Processing Magazine, january 2005, pp. 103 - 106.

[2] Storn, R., "Sytem Design by Constraint Adaptation and Differential Evolution", IEEE Trans. on Evolutionary Computation, 1999, Vol. 3, No. 1, pp. 22 - 34.

[3] Storn, R. and Price, K., "Differential Evolution - a Simple and Efficient Heuristic for Global Optimization over Continuous Spaces", Journal of Global Optimization, Kluwer Academic Publishers, 1997, Vol. 11, pp. 341 - 359.

[4] Gitsels, M. and Storn, R., Internet-Videotelephonie nach dem H.323-Standard, ITG-Fachbericht 144, 7. Dortmunder Fernsehseminar, pp. 87 - 92.

[5] Storn, R., "Echo Cancellation Techniques for Multimedia Applications - A Survey" , Technical Report TR-96-046, ICSI, November 1996, ftp.icsi.berkeley.edu.

[6] Storn, R., "System Design by Constraint Adaptation and Differential Evolution" , Technical Report TR-96-039, ICSI, November 1996, ftp.icsi.berkeley.edu.

[7] Price, K. and Storn, R., "Differential Evolution: Numerical Optimization Made Easy", Dr. Dobb's Journal, April 97, pp. 18 - 24.

[8] Storn, R., "On the Usage of Differential Evolution for Function Optimization" NAFIPS 1996, Berkeley, pp. 519 - 523.

[9] Storn, R. and Price, K., "Minimizing the real functions of the ICEC'96 contest by Differential Evolution" IEEE Conference on Evolutionary Computation, Nagoya, 1996, pp. 842 - 844.

[10] Storn, R., "Efficient Input Reordering for the DCT Based on a Real-Valued Decimation in Time FFT" (IEEE Signal Processing Letters, Vol. 3, No. 8, August 1996, pp. 242 - 244), Technical Report TR-95-061, ICSI, September 1995, ftp.icsi.berkeley.edu.

[11] Storn, R., "Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay" IEEE International Conference on Evolutionary Computation ICEC 96, pp. 268 - 273, Technical Report TR-95-026, ICSI, May 1995, ftp.icsi.berkeley.edu.

[12] Storn, R., "Modeling and Optimization of PET-Redundancy Assignment for MPEG Sequences" , Technical Report TR-95-018, ICSI, May 1995, ftp.icsi.berkeley.edu.

[13] Storn, R. and Price, K., "Differential Evolution - a Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces" , Technical Report TR-95-012, ICSI, March 1995, ftp.icsi.berkeley.edu. Anyone who is interested in trying Differential Evolution (DE) might access the source code.

[14] Storn, R., "A Debug/Trace Tool for C SW Projects", Dr. Dobb's Journal, February 1997, pp. 22 - 26.

[15] Storn, R., "Constrained Optimization", Dr. Dobb's Journal, May 1995, pp. 119 - 123.

[16] Christ, J., Storn, R. and Lueder, E., " New Shortlength DFTs for the Prime Factor Implementation on DSP Architectures", Frequenz, 1995, Band 49, Issue 1-2, pp. 8 - 10.

[17] Ballay, H. and Storn, R., "A Tool for Checking C Coding Conventions", C User's Journal, july 94, pp. 41 - 50..

[18] Storn, R., "A Hashing Function Based on Algebraic Coding", submitted for publication in the I.E.E. Proceedings~E, Computers and Digital Techniques.

[19] Storn, R., "A Radix-2 FFT-Pipeline Architecture With Reduced Noise to Signal Ratio", I.E.E. Proceedings~F, Radar and Signal Processing, 1994.

[20] Storn, R. , "Datensicherung mit Prüfsummen", ST-Computer, 1994.

[21] Storn, R., "Some Results in Fixed Point Error Analysis of the Bruun-FFT Algorithm, IEEE Trans. on Signal Processing, Vol. 41, No. 7, July 93, pp. 2371 - 2375.

[22] Storn, R. , "Statistische Optimierung", ST-Computer, Issues 12/1992 and 1/1993.

[23] Storn, R. , "On the Bruun Algorithm and its Inverse", Frequenz, Vol. 3-4, 1992, pp. 110 -116.

[24] Storn, R. , "Logische Schaltungen und deren Vereinfachung nach Quine-McCluskey", ST-Computer, Issues 3, 4 and 5, 1990.

[25] Storn, R. , "A novel Radix-2 Pipeline Architecture for the Computation of the DFT", IEEE Proc. of the ISCAS 1988, pp. 1899 -1902.

[26] Storn, R. , "On the Reduction of Arithmetic Complexity in the Chirp-Transform", Proc. ECCTD, 1987, pp. 239 -244.

[27] Storn, R. , "Ein Primfaktor-Algorithmus für die diskrete Hartley-Transformation", 9. DFG-Kolloquium über digitale Signalverarbeitung, 1986, pp. 79 -82.

[28] Storn, R. , "Fast Algorithms for the Discrete Hartley Transform", AEÜ, Band 40, Heft 4, 1986, pp. 233 -240.

[29] Storn, R. , "Dreieck-Quadratur-Oszillator. Nur ein zeitbestimmendes Glied erforderlich", Elektronik, Issue 5, 1982, p. 74.

[30] Storn, R. , "Constant Current Adapter", Elektor, Issue 7/8, 1981.

[31] Storn, R. , "De Luxe Transistor Tester", Elektor, Issue 7/8, 1979. (The corresponding circuit was among the winners of the european circuit design contest "EUROTRONIK").

BOOKS

[1] Price K., Storn R., Lampinen J., Differential Evolution - A Practical Approach to Global Optimization, Springer, Berlin, 2005.

[2] Contributor for Babu, B.V., Onwubolu, G. (Editors), New Optimization Techniques in Engineering, Springer, Berlin, 2004.

[3] Contributor for Corne, D., Dorigo., M, and Glover., F. (Editors), New Ideas in Optimization, McGraw-Hill, 1999.