68 E. Carrizosa, B. Martín-Barragán, F. Plastria, and D. Romero-Moralesand not nominal or ordinal. Moreover, no blanks are allowed, which exclu<strong>de</strong>s its direct usefor cases in which some measures are missing or simply do not apply. See e.g. Cristianini andShawe-Taylor [5], Freed and Glover [9], Gehrlein [10], Gochet et al. [12] and Mangasarian [20].A more flexible methodology, which just requires the knowledge of a metric (or, with moregenerality, a dissimilarity), is the Nearest Neighbor (NN) method [4, 6, 7, 14], which provi<strong>de</strong>s,as documented e.g. in [16], excellent results.In Nearest Neighbor methods, for each new entry i, the distances (or dissimilarities) d(i, j)to some individuals j in the database (called prototypes) are computed, and i is classified accordingto such set of distances. In particular, in the classical NN, [4], all individuals are prototypes,and i is classified as member of class c ∗ to which its closest prototype j ∗ (satisfyingd(i, j ∗ ) ≤ d(i, j) ∀j) belongs.A generalization of the NN is the k-NN, e.g. [7], which classifies each i in the class mostfrequently found in the set of k prototypes closest to i. In particular, the NN is the k-NN fork = 1.These classification rules, however, require distances to be calculated to all data in thedatabase for each new entry, involving high storage and time resources, making it impracticalto perform on-line queries.For these reasons, several variants have been proposed in the last three <strong>de</strong>ca<strong>de</strong>s, see e.g.[1,2,6,7,11,13,17,18] and the references therein. For instance, Hart [13] suggests the Con<strong>de</strong>nsedNearest Neighbor (CNN) rule, in which the full database J is replaced by a certain subset I,namely, a so-called minimal consistent subset: a subset of records such that, if the NN is usedwith I (instead of J) as set of prototypes, all points in J are classified in the correct classes.Since such minimal consistent subset can still be too large, several procedures have beensuggested to reduce its size. Although such procedures do not necessarily classify correctlyall the items in the database, (i.e., they are not consistent), they may have a similar or evenbetter behavior to predict class membership on future entries because they may reduce thepossible overfitting suffered by the CNN rule, see e.g. [3, 19].In this talk we propose a new mo<strong>de</strong>l, in which a set of prototypes I ⊂ J, of prespecifiedcardinality p is sought, minimizing an empirical misclassification cost. Hence, if p is takengreater or equal than the cardinality p ∗ of a minimal consistent subset, then all individualsin the training sample will be correctly classified, yielding an empirical misclassification costof zero. On the other hand, for p < p ∗ we allow some data in the training sample to beincorrectly classified with the hope of reducing the possible overfitting which the classifierbased on a minimal consistent subset might cause. Additionally, the effort nee<strong>de</strong>d to classifya new entry is directly proportional to p, which may therefore serve in practice to gui<strong>de</strong> thechoice of an upper bound on p.For simplicity we restrict ourselves to the classification rule based on the closest distance,and hence can be seen as a variant of the NN rule. However, the results <strong>de</strong>veloped here extenddirectly to the case in which the k closest distances, k ≥ 1, are consi<strong>de</strong>red in the classificationprocedure, leading to a variant of the k-NN method.The talk is structured as follows. First, the mathematical mo<strong>de</strong>l is introduced, showingthat it is N P-Hard. Two Integer Programming formulations are proposed and theoreticallycompared. Numerical results are given, showing that, when the optimization problems aresolved exactly (with a standard MIP solver) the classification rules for p < p ∗ behaves betterthan or equal to the CNN rule, but with enormous preprocessing times. For this reason, aheuristic procedure is also proposed, its quality and speed being also explored. It is shownthat the rules obtained with this heuristic procedure have similar behavior on testing samplesas the optimal ones.
Globally optimal prototypes 69References[1] M.V. Bennett and T.R. Willemain. The Filtered Nearest Neighbor Method for Generating Low-DiscrepancySequences. INFORMS Journal on Computing 16:68–72, 2004.[2] J.C. Bez<strong>de</strong>k and L.I. Kuncheva. Nearest Prototype Classifier Designs: an Experimental Study. InternationalJournal of Intelligent Systems, 16:1445–1473, 2001.[3] H. Brighton and C. Mellish. Advances in Instance Selection for Instance-Based Learning algorithms. DataMining and Knowledge Discovery, 6:153–172, 2002.[4] T.M. Cover and P.E. Hart. Nearest Neighbor Pattern Classification. IEEE Transactions on Information Theory,13:21–27, 1967.[5] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-based LearningMethods. Cambridge University Press, 2000.[6] B.V. Dasarathy. Nearest Neighbor(NN) Norms: NN Pattern Classification Techniques. IEEE Computer SocietyPress, 1991.[7] L. Devroye, L. Györfi and G. Lugosi. A probabilistic Theory of Pattern Recognition. Springer, 1996.[8] R.A. Fisher. The Use of Multiple Measurements in Taxonomy Problems. Annals of Eugenics, 7:179–188, 1936.[9] N. Freed and F. Glover. Simple but Powerful Goal Programming Mo<strong>de</strong>ls for Discriminant Problems. EuropeanJournal of Operational Research, 7:44–60, 1981.[10] W.V. Gehrlein. General Mathematical Programming Formulations for the Statistical Classification Problem.Operations Research Letters, 5(6):299–304, 1986.[11] S. Geva and J. Sitte. Adaptive Nearest Neighbor Pattern Classifier. IEEE Trans Neural Networks, 2(2):318–322,1991.[12] W. Gochet, A. Stam, V. Srinivasan and S.X. Chen. Multigroup Discriminant Analysis Using Linear Programming.Operations Research, 45:213–225, 1997.[13] P.E. Hart. The Con<strong>de</strong>nsed Nearest Neighbor Rule. IEEE Transactions on Information Theory, 14:515–516, 1968.[14] T. Hastie, R. Tibshirani and J. Friedman. The Elements of Statistical Learning. Springer, 2001.[15] L. Kaufman and P.J. Rousseeuw. Finding Groups in Data. An Introduction to Cluster Analysis. Wiley, 1990.[16] R.D. King, C. Feng and A. Sutherland. Statlog: Comparision of Classification Algorithm in Large Real-wordProblems. Applied Artificial Intelligence, 9(3):289–333, 1995.[17] L.I. Kuncheva. Fitness Function in Editing k-NN Reference Set by Genetic Algorithms. Pattern Recognition,30(6):1041–1049, 1997.[18] L.I. Kuncheva and J.C. Bez<strong>de</strong>k. Nearest Prototype Classification: Clustering, Genetic Algorithm or RandomSearch? IEEE Transactions on Systems, Man, and Cybernetics – Part C, 28(1):160–164, 1998.[19] U. Lipowezky. Selection of the Optimal Prototype Subset for 1-NN Classification. Pattern Recognition Letters,19:907–918, 1998.[20] O.L. Mangasarian. Missclasification Minimization. Journal of Global Optimization, 5:309–323, 1994.[21] G.J. McLachlan. Discriminant Analysis and Statistical Pattern Recognition. Wiley, 1992.
- Page 1:
PROCEEDINGS OF THEINTERNATIONAL WOR
- Page 5 and 6:
ContentsPrefaceiiiPlenary TalksYaro
- Page 7 and 8:
ContentsviiFuh-Hwa Franklin Liu, Ch
- Page 9:
PLENARY TALKS
- Page 12 and 13:
4 Yaroslav D. Sergeyevto work with
- Page 15:
EXTENDED ABSTRACTS
- Page 18 and 19:
10 Bernardetta Addis and Sven Leyff
- Page 20 and 21:
12 Bernardetta Addis and Sven Leyff
- Page 22 and 23:
14 Bernardetta Addis and Sven Leyff
- Page 24 and 25:
16 Bernardetta Addis, Marco Locatel
- Page 26 and 27: 18 April K. Andreas and J. Cole Smi
- Page 28 and 29: 20 April K. Andreas and J. Cole Smi
- Page 30 and 31: 22 April K. Andreas and J. Cole Smi
- Page 32 and 33: 24 Charles Audet, Pierre Hansen, an
- Page 34 and 35: 26 Charles Audet, Pierre Hansen, an
- Page 36 and 37: 28 Charles Audet, Pierre Hansen, an
- Page 38 and 39: 30 János Balogh, József Békési,
- Page 40 and 41: 32 János Balogh, József Békési,
- Page 42 and 43: 34 János Balogh, József Békési,
- Page 44 and 45: 36 Balázs Bánhelyi, Tibor Csendes
- Page 47 and 48: Proceedings of GO 2005, pp. 39 - 45
- Page 49 and 50: MGA Pruning Technique 41Figure 1. A
- Page 51 and 52: MGA Pruning Technique 43O(n) = k 2
- Page 53: MGA Pruning Technique 45one), while
- Page 56 and 57: 48 Edson Tadeu Bez, Mirian Buss Gon
- Page 58 and 59: 50 Edson Tadeu Bez, Mirian Buss Gon
- Page 60 and 61: 52 Edson Tadeu Bez, Mirian Buss Gon
- Page 62 and 63: 54 R. Blanquero, E. Carrizosa, E. C
- Page 64 and 65: 56 R. Blanquero, E. Carrizosa, E. C
- Page 66 and 67: 58 Sándor Bozókiwhere for any i,
- Page 68 and 69: 60 Sándor Bozóki[6] Budescu, D.V.
- Page 70 and 71: 62 Emilio Carrizosa, José Gordillo
- Page 72 and 73: 64 Emilio Carrizosa, José Gordillo
- Page 75: Proceedings of GO 2005, pp. 67 - 69
- Page 80 and 81: 72 Leocadio G. Casado, Eligius M.T.
- Page 82 and 83: 874 Leocadio G. Casado, Eligius M.T
- Page 84 and 85: 76 Leocadio G. Casado, Eligius M.T.
- Page 86 and 87: 78 András Erik Csallner, Tibor Cse
- Page 88 and 89: 80 András Erik Csallner, Tibor Cse
- Page 90 and 91: 82 Tibor Csendes, Balázs Bánhelyi
- Page 92 and 93: 84 Tibor Csendes, Balázs Bánhelyi
- Page 94 and 95: 86 Bernd DachwaldFor spacecraft wit
- Page 96 and 97: 88 Bernd Dachwaldcurrent target sta
- Page 98 and 99: 90 Bernd Dachwaldreference launch d
- Page 100 and 101: 92 Mirjam Dür and Chris TofallisMo
- Page 102 and 103: 94 Mirjam Dür and Chris Tofallis2.
- Page 104 and 105: 96 Mirjam Dür and Chris Tofallis[3
- Page 106 and 107: 98 José Fernández and Boglárka T
- Page 108 and 109: 100 José Fernández and Boglárka
- Page 110 and 111: 102 José Fernández and Boglárka
- Page 112 and 113: 104 Erika R. Frits, Ali Baharev, Zo
- Page 114 and 115: 106 Erika R. Frits, Ali Baharev, Zo
- Page 116 and 117: 108 Erika R. Frits, Ali Baharev, Zo
- Page 118 and 119: 110 Juergen Garloff and Andrew P. S
- Page 120 and 121: 112 Juergen Garloff and Andrew P. S
- Page 123 and 124: Proceedings of GO 2005, pp. 115 - 1
- Page 125 and 126: Global multiobjective optimization
- Page 127 and 128:
Global multiobjective optimization
- Page 129 and 130:
Proceedings of GO 2005, pp. 121 - 1
- Page 131 and 132:
Conditions for ε-Pareto Solutions
- Page 133:
Conditions for ε-Pareto Solutions
- Page 136 and 137:
128 Eligius M.T. Hendrix1.1 Effecti
- Page 138 and 139:
130 Eligius M.T. Hendrix4h(x)3.532.
- Page 140 and 141:
132 Eligius M.T. Hendrixneighbourho
- Page 142 and 143:
134 Kenneth Holmströmcomputed by R
- Page 144 and 145:
136 Kenneth Holmströmα(x) =∑i=1
- Page 146 and 147:
138 Kenneth HolmströmGL-step Phase
- Page 148 and 149:
140 Kenneth Holmströmsurrogate mod
- Page 150 and 151:
142 Dario Izzo and Mihály Csaba Ma
- Page 152 and 153:
144 Dario Izzo and Mihály Csaba Ma
- Page 154 and 155:
146 Dario Izzo and Mihály Csaba Ma
- Page 156 and 157:
148 Leo Liberti and Milan DražićV
- Page 158 and 159:
150 Leo Liberti and Milan Dražićs
- Page 161 and 162:
Proceedings of GO 2005, pp. 153 - 1
- Page 163 and 164:
Set-covering based p-center problem
- Page 165 and 166:
Set-covering based p-center problem
- Page 167 and 168:
Proceedings of GO 2005, pp. 159 - 1
- Page 169 and 170:
On the Solution of Interplanetary T
- Page 171 and 172:
On the Solution of Interplanetary T
- Page 173 and 174:
Proceedings of GO 2005, pp. 165 - 1
- Page 175 and 176:
Parametrical approach for studying
- Page 177 and 178:
Parametrical approach for studying
- Page 179 and 180:
Proceedings of GO 2005, pp. 171 - 1
- Page 181 and 182:
An Interval Branch-and-Bound Algori
- Page 183 and 184:
An Interval Branch-and-Bound Algori
- Page 185 and 186:
Proceedings of GO 2005, pp. 177 - 1
- Page 187 and 188:
A New approach to the Studyof the S
- Page 189:
A New approach to the Studyof the S
- Page 192 and 193:
184 Katharine M. Mullen, Mikas Veng
- Page 194 and 195:
186 Katharine M. Mullen, Mikas Veng
- Page 196 and 197:
188 Katharine M. Mullen, Mikas Veng
- Page 198 and 199:
190 Niels J. Olieman and Eligius M.
- Page 200 and 201:
192 Niels J. Olieman and Eligius M.
- Page 202 and 203:
194 Niels J. Olieman and Eligius M.
- Page 204 and 205:
196 Andrey V. Orlovwhere A is (m 1
- Page 206 and 207:
198 Andrey V. OrlovStep 4. Beginnin
- Page 208 and 209:
200 Blas Pelegrín, Pascual Fernán
- Page 210 and 211:
202 Blas Pelegrín, Pascual Fernán
- Page 212 and 213:
204 Blas Pelegrín, Pascual Fernán
- Page 214 and 215:
206 Blas Pelegrín, Pascual Fernán
- Page 216 and 217:
208 Deolinda M. L. D. Rasteiro and
- Page 218 and 219:
210 Deolinda M. L. D. Rasteiro and
- Page 220 and 221:
212 Deolinda M. L. D. Rasteiro and
- Page 222 and 223:
214 José-Oscar H. Sendín, Antonio
- Page 224 and 225:
216 José-Oscar H. Sendín, Antonio
- Page 226 and 227:
218 José-Oscar H. Sendín, Antonio
- Page 228 and 229:
220 Ya. D. Sergeyev and D. E. Kvaso
- Page 230 and 231:
222 Ya. D. Sergeyev and D. E. Kvaso
- Page 232 and 233:
224 Ya. D. Sergeyev and D. E. Kvaso
- Page 234 and 235:
226 J. Cole Smith, Fransisca Sudarg
- Page 236 and 237:
228 J. Cole Smith, Fransisca Sudarg
- Page 238 and 239:
230 J. Cole Smith, Fransisca Sudarg
- Page 240 and 241:
232 Fazil O. Sonmezcost of a config
- Page 242 and 243:
234 Fazil O. Sonmezhere f h is the
- Page 244 and 245:
236 Fazil O. SonmezThe optimal shap
- Page 246 and 247:
238 Alexander S. StrekalovskyDevelo
- Page 249 and 250:
Proceedings of GO 2005, pp. 241 - 2
- Page 251 and 252:
PSfrag replacementsPruning a box fr
- Page 253 and 254:
Pruning a box from Baumann point in
- Page 255 and 256:
Proceedings of GO 2005, pp. 247 - 2
- Page 257 and 258:
A Hybrid Multi-Agent Collaborative
- Page 259 and 260:
A Hybrid Multi-Agent Collaborative
- Page 261 and 262:
Proceedings of GO 2005, pp. 253 - 2
- Page 263:
Improved lower bounds for optimizat
- Page 266 and 267:
258 Graham R. Wood, Duangdaw Sirisa
- Page 268 and 269:
260 Graham R. Wood, Duangdaw Sirisa
- Page 270 and 271:
262 Graham R. Wood, Duangdaw Sirisa
- Page 272 and 273:
264 Yinfeng Xu and Wenqiang Daiopti
- Page 274 and 275:
266 Yinfeng Xu and Wenqiang DaiThe
- Page 277 and 278:
Author IndexAddis, BernardettaDipar
- Page 279:
Author Index 271Nasuto, S.J.Departm