19 11 V o l. 19 N o. 11 Con trol nd D ecision 2004 11 N ov. 2004 : 100120920 (2004) 1121213205, (, 100084) :,,,,, ; ;. : ; ; ; : O 232: A Fuzzy subgrd ien t lgor ith for solv ing Lgrng in relx tion du l proble ZH OU W ei, J IN Y i2hu i (D eprt ent of A uto tion, T singhu U niversity, Beijing 100084, Ch in. Co rrespondent: ZHOU W ei, E2 il: zhouw 99@ ils. tsinghu. edu. cn) Abstrct: To the p roble of zigzging hppened in so lving the undifferentil L grngin dul p roble s by subgrdi2 ent lgo rith, subgrdient lgo rith bsed on fuzzy theo ry is p resented. In th is ethod, the resulting subgrdient direction is ttined by co bining ll h isto ry subgrdient directions, w h ich re ch ieved in the itertion p rocess, fo l2 low ing si p le e bersh ip function. T he resulting subgrdient direction uses the h isto ry info r tion suitbly, thereby significntly reduces the so lution zigzgging difficulty w ithout uch dditionl co puttionl require ents. T he convergence of the lgo rith is p roved. T h is ethod is then pp lied in the trveling sles n p roble, nd the results show tht th is ethod leds to significnt i p rove ent over the trditionl subgrdient lgo rith. Key words: L grngin relxtion; subgrdient lgo rith ; fuzzy theo ry; dul 1 (L R ),,.,,., [1, 2 ],., : [ 2 ] ; [ 3 ], ; [ 4 ],,.,., : 2003212216; : 2004203211. : (60174046). : (1977),,,, ; (1936),,,,.
1214 19., T SP.,. 2 (P) : in f (x), s. t. g (x) 0, x D. : D, f g D, g (x) r 1., : L (x, u) = f (x) + g T (x) u, q (u) = inf L (x, u), x D (D ) : x q (u), s. t. u 0. : L (x, u) ; q (u) f (x), (D ) (P) ; u 1 r,,, L (x, u) x. L R (D ) (P). q (u), (D ), k : u k = u k- 1 + tkg k. (1) : g k = g (x k) k, tk k.,,,., [15 ]., [ 2 ], u k = u k- 1 + tkd k (2) (1),. d k = k w k j g j, (3) g j j, w k j k j. [1, 5 ],. [ 6 ] Bundle,.,,., ( j = 1,, k ) : j L (u j, x j) = in x D L (u j, x) = g T j u j + cj. (4) : g j = g (x j), cj = f (x j). (D ) L 3 = L (u 3, x 3 ) = x inl (u, x) = g 3 T u 3 + c 3. (5) u0 x D : g 3 = g (x 3 ), c 3 = f (x 3 ). w k j = w { k j k w { k j. (6) w { k j : w { k j = A (x j) = (L (u k, x k) + Ε- L (u k, x j) ) gε, L (u k, x j) < L (u k, x k) + Ε L (u k, x j) L 3 ; 0,. (7) 0 Ε(L 3 - L (u k, x k) ) g. (8) tk 0 tk 2 ( - 1) (L 3 - L (u k, x k) ) d k 2, > 1. (9), (7),. : Step 1: : k = 0, u 0 0; Step 2: u k (D )., Step 5;, k = k + 1; Step 3: (6) (8), (9) ; Step 4: (2) u k, Step 2; Step 5: x k u k ;., O (X ), k, O (X + k). X,, k,, O (X )., k 200,
11 : 1215. 3 1 Π j = 1, 2,, k, L (u k, x j) L 3, : 0 L 3 - L (u k, x j) g T j (u 3 - u k). (10), L 3 - L (u k, x j) 0,. (4) : L (u k, x k) = in L (u k, x), x D L (u k, x k) L (u k, x j). (11) L 3 = L (u 3, x 3 ) L (u 3, x j), (12) (4) (12) L 3 - L (u k, x j) L (u 3, x j) - L (u k, x j) = g T j u 3 + cj - (g T j u k + cj) = g T j (u 3 - u k). 1,,. 2 > 1, : 0 L 3 - L (u k - x k) < d T j (u 3 - u k). (13) (13), : L (u k, x j) < L (u k, x k) + = 1, 2,, k (8), L (u k, x j) < L (u k, x k) + Ε L (u k, x k) + (L 3 - L (u k, x k) ) g, L 3 - L (u k, x j) > L 3 - [L (u k, x k) + (L 3 - L (u k, x k) ) g ] = ( - 1) (L 3 - L (u k, x k) ) g, 1, ( - 1) (L 3 - L (u k, x k) ) g < L 3 - L (u k, x j). ( - 1) (L 3 - L (u k, x k) ) L 3 - L (u k, x j) g T j (u 3 - u k). (3) (6), ( - 1) (L 3 - L (u k, x k) ) < < Ε, Π j w k j g T j (u 3 - u k) = d T k (u 3 - u k). 2,,. 1,. u 3 - u k+ 1< u 3 - u k, Π k. (14) u 3 - u k+ 1 2 = u 3 - u k - tkd k 2 u 3 - u k 2-2tkd T k (u 3 - u k) + t 2 kd k 2. 2 u 3 - u k+ 1 2 < u 3 - u k 2 - tk t 2 kd k 2 = u 2 - u k 2 - tk 2 ( - 1) (L 3 - L (u k, x k) ) (9), 2 ( - 1) (L 3 - L (u k, x k) ) - tkd k 2. u 3 - u k+ 1 2 < u 3 - u k 2. 4 (TSP). : cijx ij, i= 1, ji in s. t. x ij + x j i = 2, i = 1,,., ji, ji : i j ; ; X ; cij i j ; i j x ij = 1,, x ij = 0., q (u) = - 2 u i + in{ i= 1 u i + u j) x ij: x X }. (cij + i= 1, ji XPR ESS, 33, 42 53 3, 110. : gp 0. 1, gp = (L 3 - L ) gl 3 100. (S) (FS), tk = Χk (L 3 - L (u k, x k) ) gg k 2 (15). : Χk = 1, = 2., L 3 +,, [1 ], [ 7 ]..,, XPR ESS, L 3. PC, CPU 850
1216 19 M H z, 128 M. 1 3.,.,,. 133,. 20,. 10,, 1 2 3 4 5 150. 61 128. 382 159. 867 142. 017 149. 012 86 52 312 115 272 87 71 55 182 88 10 146. 483 146. 709 124. 39 123. 368 3 155. 972 156. 146 138. 468 138. 574 142. 831 143. 524 20 148. 153 148. 78 125. 918 126. 277 157. 758 158. 371 140. 269 140. 895 145. 170 145. 978 30 148. 877 149. 168 125. 858 126. 768 158. 038 158. 754 140. 828 141. 182 146. 741 147. 469 40 148. 927 149. 673 126. 133 127. 169 158. 254 158. 846 141. 169 141. 455 147. 207 147. 856 50 149. 797 150. 110 127. 098 127. 386 158. 442 159. 122 141. 503 141. 884 147. 348 148. 024 75 150. 375 126. 971 127. 953 158. 601 159. 599 147. 957 148. 620 100 127. 428 128. 155 159. 059 148. 372 125 127. 662 159. 201 148. 652 150 127. 791 159. 434 148. 683 175 127. 751 159. 397 148. 817 242 1 2 3 4 5 175. 387 177. 43 183. 063 174. 908 176. 34 129 52 126 66 94 52 60 41 72 48 10 170. 431 170. 995 171. 928 172. 981 176. 442 177. 452 169. 300 168. 910 3 171. 432 171. 843 20 173. 405 173. 861 174. 513 174. 770 181. 51 181. 666 172. 918 172. 975 173. 416 174. 351 30 173. 745 174. 563 175. 953 176. 073 182. 208 182. 445 173. 914 174. 169 174. 548 175. 219 40 174. 151 174. 946 176. 385 176. 632 182. 449 182. 657 174. 441 174. 635 175. 5 175. 924 50 174. 611 175. 127 176. 571 176. 92 182. 531 182. 826 174. 614 175. 855 75 174. 898 176. 945 182. 717 100 174. 954 177. 113 125 175. 211 177. 231 150 175 353 1 2 3 4 5 219. 314 222. 459 207. 226 233. 896 216. 189 53 43 164 82 94 64 190 94 48 48 10 212. 592 213. 531 213. 587 212. 300 3 200. 211 201. 547 225. 981 227. 169 213. 535 213. 619 20 217. 534 217. 554 217. 868 218. 665 204. 226 204. 689 230. 738 230. 975 215. 484 215. 551 30 218. 204 218. 679 219. 767 220. 575 205. 394 206. 030 232. 12 232. 411 215. 829 215. 924 40 218. 718 219. 009 220. 679 221. 48 206. 284 206. 662 232. 793 233. 222 215. 896 50 219. 021 221. 18 221. 853 206. 847 206. 917 233. 174 233. 275 75 221. 777 222. 192 206. 992 233. 231 233. 526 100 221. 865 233. 292 125 222. 012 233. 403 150 222. 121 233. 258 175 233. 598
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