差分進化算法 DE-Differential Evolution

時間 2019-11-06

標籤差分進化算法 differential evolution 简体版

原文原文鏈接

差分進化算法 (Differential Evolution)

Differential Evolution（DE）是由Storn等人於1995年提出的，和其它演化算法同樣，DE是一種模擬生物進化的隨機模型，經過反覆迭代，使得那些適應環境的個體被保存了下來。但相比於進化算法，DE保留了基於種羣的全局搜索策略，採用實數編碼、基於差分的簡單變異操做和一對一的競爭生存策略，下降了遺傳操做的複雜性。同時，DE特有的記憶能力使其能夠動態跟蹤當前的搜索狀況，以調整其搜索策略，具備較強的全局收斂能力和魯棒性，且不須要藉助問題的特徵信息，適於求解一些利用常規的數學規劃方法所沒法求解的複雜環境中的優化問題。目前，DE已經在許多領域獲得了應用，譬如人工神經元網絡、化工、電力、機械設計、機器人、信號處理、生物信息、經濟學、現代農業、食品安全、環境保護和運籌學等。

DE算法-做者網站: http://www1.icsi.berkeley.edu/~storn/code.html
維基百科資料庫 : https://en.wikipedia.org/wiki/Differential_evolution

DE 算法主要用於求解連續變量的全局優化問題，其主要工做步驟與其餘進化算法基本一致，主要包括變異(Mutation)、交叉(Crossover)、選擇(Selection)三種操做。算法的基本思想是從某一隨機產生的初始羣體開始，利用從種羣中隨機選取的兩個個體的差向量做爲第三個個體的隨機變化源，將差向量加權後按照必定的規則與第三個個體求和而產生變異個體，該操做稱爲變異。而後，變異個體與某個預先決定的目標個體進行參數混合，生成試驗個體，這一過程稱之爲交叉。若是試驗個體的適應度值優於目標個體的適應度值，則在下一代中試驗個體取代目標個體，不然目標個體仍保存下來，該操做稱爲選擇。在每一代的進化過程當中，每個體矢量做爲目標個體一次，算法經過不斷地迭代計算，保留優良個體，淘汰劣質個體，引導搜索過程向全局最優解逼近。

算法圖解:

算法僞代碼:

算法C代碼:

  1 //********************************************************/
  2 //      DE/rand/1/bin --差分進化算法-(基本類型)       
  3 //********************************************************/
  4 
  5 
  6 #include <stdlib.h>
  7 #include <stdio.h>
  8 #include <time.h>
  9 #include <float.h>
 10 
 11 /* Function definitions        */
 12 
 13 double func(double *);
 14 int usage(char *);
 15 
 16 /* Random number generator defined by URAND should return
 17 double-precision floating-point values uniformly distributed
 18 over the interval [0.0, 1.0)                    */
 19 
 20 #define URAND    ((double)rand()/((double)RAND_MAX + 1.0))
 21 
 22 /* Definition for random number generator initialization    */
 23 
 24 #define INITRAND srand(time(0))
 25 
 26 /* Usage for the program    */
 27 
 28 int usage(char *str)
 29 {
 30     fprintf(stderr, "Usage: %s [-h] [-u] [-s] [-N NP (20*D)] ", str);
 31     fprintf(stderr, "[-G Gmax (1000)]\n");
 32     fprintf(stderr, "\t[-C crossover constant, CR (0.9)]\n");
 33     fprintf(stderr, "\t[-F mutation scaling factor, F (0.9)]\n");
 34     fprintf(stderr, "\t[-o <outputfile>]\n\n");
 35     fprintf(stderr, "\t-s does not initialize random number generator\n");
 36     exit(-1);
 37 }
 38 
 39 
 40 int main(int argc, char **argv)
 41 {
 42     register int i, j, k, r1, r2, r3, jrand, numofFE = 0;
 43     extern int D;
 44     extern double Xl[], Xu[];
 45 
 46     int NP = 20 * D, Gmax = 1000, c, index = -1, s = 1;
 47 
 48     double **popul, **next, **ptr, *iptr, *U, CR = 0.9, F = 0.9,
 49 
 50         min_value = DBL_MAX, totaltime = 0.0;
 51 
 52     char *ofile = NULL;
 53 
 54     FILE *fid;
 55     clock_t starttime, endtime;
 56 
 57 
 58     /* Parse command line arguments given by user    */
 59 
 60     for (i = 1; i < argc; i++)
 61     {
 62         if (argv[i][0] != '-')
 63             usage(argv[0]);
 64 
 65         c = argv[i][1];
 66 
 67         switch (c)
 68         {
 69         case 'N':
 70             if (++i >= argc)
 71                 usage(argv[0]);
 72 
 73             NP = atoi(argv[i]);
 74             break;
 75         case 'G':
 76             if (++i >= argc)
 77                 usage(argv[0]);
 78 
 79             Gmax = atoi(argv[i]);
 80             break;
 81         case 'C':
 82             if (++i >= argc)
 83                 usage(argv[0]);
 84 
 85             CR = atof(argv[i]);
 86             break;
 87         case 'F':
 88             if (++i >= argc)
 89                 usage(argv[0]);
 90 
 91             F = atof(argv[i]);
 92             break;
 93         case 'o':
 94             if (++i >= argc)
 95                 usage(argv[0]);
 96 
 97             ofile = argv[i];
 98             break;
 99         case 's':    /* Flag for using same seeds for        */
100             s = 0;    /* different runs                */
101             break;
102         case 'h':
103         case 'u':
104         default:
105             usage(argv[0]);
106         }
107     }
108 
109     if (s) INITRAND;
110 
111     /* Printing out information about optimization process for the user    */
112 
113     printf("Program parameters: ");
114     printf("NP = %d, Gmax = %d, CR = %.2f, F = %.2f\n",
115         NP, Gmax, CR, F);
116 
117     printf("Dimension of the problem: %d\n", D);
118 
119 
120     /* Starting timer    */
121 
122     starttime = clock();
123 
124 
125     /* Allocating memory for current and next populations, intializing
126     current population with uniformly distributed random values and
127     calculating value for the objective function    */
128 
129 
130     // NP:種羣大小, Gmax:迭代次數, CR:交叉機率, F:擾動向量的縮放因子
131 
132     //當前種羣
133     popul = (double **)malloc(NP*sizeof(double *));
134     if (popul == NULL) perror("malloc");
135 
136     //下代種羣
137     next = (double **)malloc(NP*sizeof(double *));
138     if (next == NULL) perror("malloc");
139 
140     //當前種羣popul[NP][D+1] 
141     for (i = 0; i < NP; i++)
142     {
143         //個體維度空間分配
144         popul[i] = (double *)malloc((D + 1)*sizeof(double));
145         if (popul[i] == NULL) perror("malloc");
146 
147         //初始化維度值
148         for (j = 0; j < D; j++)
149             popul[i][j] = Xl[j] + (Xu[j] - Xl[j])*URAND;
150 
151         //最後的元素內存放該個體的適應度值
152         popul[i][D] = func(popul[i]);
153 
154         numofFE++;//統計評估次數
155 
156         //下一代個體空間分配
157         next[i] = (double *)malloc((D + 1)*sizeof(double));
158         if (next[i] == NULL) perror("malloc");
159     }
160 
161     /* 爲實驗向量分配空間--Allocating memory for a trial vector U    */
162 
163     U = (double *)malloc((D + 1)*sizeof(double));
164     if (U == NULL) perror("malloc");
165 
166 
167     /* The main loop of the algorithm    */
168 
169     for (k = 0; k < Gmax; k++)
170     {
171 
172         for (i = 0; i < NP; i++)    /* Going through whole population    */
173         {
174 
175             /* Selecting random indeces r1, r2, and r3 to individuls of
176             the population such that i != r1 != r2 != r3    */
177 
178             //1.選擇三個互不相同的隨機個體r1,r2,r3
179             do
180             {
181                 r1 = (int)(NP*URAND);
182             } while (r1 == i);
183 
184             do
185             {
186                 r2 = (int)(NP*URAND);
187             } while (r2 == i || r2 == r1);
188             do
189             {
190                 r3 = (int)(NP*URAND);
191             } while (r3 == i || r3 == r1 || r3 == r2);
192 
193             jrand = (int)(D*URAND);
194 
195             /* Mutation and crossover    */
196             //2. 執行變異和交叉操做
197             for (j = 0; j < D; j++)
198             {
199                 //執行二項式交叉
200                 if (URAND < CR || j == jrand)
201                 {
202                     //試驗向量部分來自變異後的向量
203                     U[j] = popul[r3][j] + F*(popul[r1][j] - popul[r2][j]);
204                 }
205                 else
206                     //試驗向量部分來自個體i
207                     U[j] = popul[i][j];
208             }
209             //3. 計算新生成向量的適應度值
210             U[D] = func(U);
211 
212             numofFE++;
213 
214             /* Comparing the trial vector 'U' and the old individual
215             'next[i]' and selecting better one to continue in the
216             next population.注意:空間的交替變換和使用    */
217 
218             //貪婪策略從試驗向量U和當前個體i中選擇一個好的放入到下一代個體中
219             if (U[D] <= popul[i][D])//新向量好
220             {
221 
222                 //試驗向量U牛逼, next指向當前的試驗向量U,u指向next, 方法:指針交換
223                 iptr = U;
224                 U = next[i];
225                 next[i] = iptr;
226             }
227             else//原始向量牛逼, next指向個體i, 方法: 直接拷貝
228             {
229                 for (j = 0; j <= D; j++)
230                     next[i][j] = popul[i][j];
231             }
232 
233         }    /* End of the going through whole population    */
234 
235 
236         /* Pointers of old and new populations are swapped    */
237         //指針交換,各指針指向的空間發生變化
238         ptr = popul;
239         popul = next;
240         next = ptr;
241 
242     }    /* End of the main loop        */
243 
244 
245     /* Stopping timer    */
246 
247     endtime = clock();
248     totaltime = (double)(endtime - starttime);
249 
250 
251     /* If user has defined output file, the whole final population is
252     saved to the file                        */
253 
254     if (ofile != NULL)
255     {
256         if ((fid = (FILE *)fopen(ofile, "a")) == NULL)
257         {
258             fprintf(stderr, "Error in opening file %s\n\n", ofile);
259             usage(argv[0]);
260         }
261 
262         for (i = 0; i < NP; i++)
263         {
264             for (j = 0; j <= D; j++)
265                 fprintf(fid, "%.15e ", popul[i][j]);
266             fprintf(fid, "\n");
267         }
268         fclose(fid);
269     }
270 
271     /* Finding best individual    */
272 
273     for (i = 0; i < NP; i++)
274     {
275         if (popul[i][D] < min_value)
276         {
277             min_value = popul[i][D];
278             index = i;
279         }
280     }
281 
282     /* Printing out information about optimization process for the user    */
283 
284     printf("Execution time: %.3f s\n", totaltime / (double)CLOCKS_PER_SEC);
285     printf("Number of objective function evaluations: %d\n", numofFE);
286 
287     printf("Solution:\nValues of variables: ");
288     for (i = 0; i < D; i++)
289         printf("%.15f ", popul[index][i]);
290 
291     printf("\nObjective function value: ");
292     printf("%.15f\n", popul[index][D]);
293 
294 
295     /* Freeing dynamically allocated memory    */
296 
297     for (i = 0; i < NP; i++)
298     {
299         free(popul[i]);
300         free(next[i]);
301     }
302     free(popul);
303     free(next);
304     free(U);
305 
306     return(0);
307 }

經典文獻: html

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

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

[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.安全

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

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

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

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

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

[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.優化

[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").