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12 Bernardetta Addis and Sven Leyfferx k ). The model in the subset of given radius ∆, B(x k , ∆), is solved by using a constrainedoptimization procedure. The use of a fixed value for ∆ is restrictive for the length of the stepswe are allowed to take. In addition, it is not clear a priori what value ∆ should take. Wedo know that a small radius results in small steps and, hence, in slow convergence, but largevalues of ∆ can result in poor agreement between the model and the function and, hence,in useless candidate points. Therefore, we propose to embed ALSO within a trust-regionframework that adjusts the radius ∆ automatically.2.2 Trust-Region FrameworkTo extend the trust-region framework to global optimization of L(x), we need to modify thesmooth local trust-region algorithm. Specifically,we have to account for the fact that L(x) isnonsmooth and that ˆL B g (x) does not agree with L(x) to first order. More important, we wishto avoid getting trapped in local minima. Clearly, it is not sufficient to simply reduce the trustregionradius if we reject the step. At each iteration we construct the model ˆL B g (x) around ourcurrent iterate x k . Specifically, we choose K samples (uniform) inside the currentB(x k , ∆ k )and perform a local minimization of f(x) from each sample. If we find a new best pointduring this phase, we simply move to the new point and construct a new model. Otherwise,we apply a local minimization to the model inside the trust region and obtainx + =argmin ˆL B k(x)g .x∈B(x k ,∆ k )To decide whether to accept a step, we compute the ratio of the actual to the predicted reduction,namely,ρ = L(x k) − L(x + )ˆL B kg (x k ) − ˆL B kg (x + ) ,noting that the predicted reduction ˆL B kg (x k ) − ˆL B kg (x + ) is always nonnegative. We accept thenew point x + if we observe sufficient decrease, that is ρ ≥ η 1 > 0. If the step is very successful,ρ ≥ η 2 > η 1 , and theis active, ‖x k − x + ‖ ≥ ∆, then we increase the trust region radius for thenext iteration. As long as ρ ≥ η 1 , we refer to the iteration as successful, otherwise (ρ < η 1 )the iteration is referred to as unsuccessful. Unsuccessfuliterations require special attention inthe global optimization setting. In smooth local optimization, reducing ∆ is guaranteed toimprove the agreement between the model and the objective function. The same is not true inthe global optimization context. Hence, we introduce a measure for the global quality of ourmodel ˆL B k(x)g ,max | {yq(ˆL B j : L(y j ) = L(y i )} |k(x) i∈Mg ) =, (7)Mwhere M is the set of collected samples, that is, the largest number of samples with the sameobjective value, divided by the total number of samples. Clearly, 0 ≤ q( ˆL B k(x)g ) ≤ 1, and avalue close to 1 means that a large number of samples have the same function value and stemfrom the same “flat region” of L(x). A smaller value of q( ˆL B k(x)g ) implies that the samplesrepresent the global nature of the function L(x) better.In our algorithm, we compute q( ˆL B k(x)g ) at every unsuccessful iteration. If it is larger than afixed value ¯q, we remove all but one sample from the largest set, increase theradius, and obtainnew uniform samples in B(x k , ∆ k+1 )\B(x k , ∆ k ). The motivation for this step is twofold: itimproves the global nature of the model ˆL B k(x)g , and it increases σ, thus smoothing the model.The increase of σ arises because we have adopted the following formula for calculating thesmoothing parameter, depending on theradius ∆ and the number of samples K:σ =∆ , (8)K1/n
A Trust-Region Algorithm for Global Optimization 13where n is the dimension of the problem.2.3 ConclusionsWe presented a two-phase procedure for the global optimization of funnel functions. Theapproach builds on ALSO [2] and combines sampling with local searches. ALSO constructsa local smooth model from the samples by applying Gaussian smoothing. We demonstratedhow to embed ALSO within a trust-region framework that adaptively updates the sampleradius.To extend the trust-region framework to global optimization, we introduced the conceptof global quality, which triggers a model improvement step. Global quality measures thelargest number of samples that have the same objective value and stem from the same basinof attraction. If global quality is large, then a model improvement step removes all but onesample from the largest set and generates a new set of uniform samples.We compared our algorithm to ALSO and variants of monotone basin hopping (MBH). Thenew algorithm is more robust than ALSO and MBH on a range of test problems, and faster interms of the number of local searches it requires per successful run.1log2 ( local searches / least local searcher )0.90.80.7% of problems0.60.50.40.30.2MBH0.1AMBHALSOTRF00 1 2 3 4 5 6 7 8solved within factor 2 x of best solverFigure 3.Performance profile of the average number of local searchesAcknowledgmentsThis work was supported by the Mathematical, Information, and Computational Sciences Divisionsubprogram of the Office of Advanced Scientific Computing Research, Office of Science,U.S. Department of Energy, under Contract W-31-109-ENG-38. The first author was also supportedby the Italian national research program “FIRB/RBNE01WBBBB Large Scale NonlinearOptimization”.
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 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 32 and 33: 24 Charles Audet, Pierre Hansen, an
Page 38 and 39: 30 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 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
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64 Emilio Carrizosa, José Gordillo
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Proceedings of GO 2005, pp. 67 - 69
Page 77:
Globally optimal prototypes 69Refer
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72 Leocadio G. Casado, Eligius M.T.
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874 Leocadio G. Casado, Eligius M.T
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76 Leocadio G. Casado, Eligius M.T.
Page 86 and 87:
78 András Erik Csallner, Tibor Cse
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80 András Erik Csallner, Tibor Cse
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82 Tibor Csendes, Balázs Bánhelyi
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84 Tibor Csendes, Balázs Bánhelyi
Page 94 and 95:
86 Bernd DachwaldFor spacecraft wit
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88 Bernd Dachwaldcurrent target sta
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90 Bernd Dachwaldreference launch d
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92 Mirjam Dür and Chris TofallisMo
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94 Mirjam Dür and Chris Tofallis2.
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96 Mirjam Dür and Chris Tofallis[3
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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
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Page 116 and 117:
Page 118 and 119:
110 Juergen Garloff and Andrew P. S
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112 Juergen Garloff and Andrew P. S
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Global multiobjective optimization
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Global multiobjective optimization
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Page 131 and 132:
Conditions for ε-Pareto Solutions
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Conditions for ε-Pareto Solutions
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128 Eligius M.T. Hendrix1.1 Effecti
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130 Eligius M.T. Hendrix4h(x)3.532.
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132 Eligius M.T. Hendrixneighbourho
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134 Kenneth Holmströmcomputed by R
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136 Kenneth Holmströmα(x) =∑i=1
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138 Kenneth HolmströmGL-step Phase
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140 Kenneth Holmströmsurrogate mod
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142 Dario Izzo and Mihály Csaba Ma
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Page 154 and 155:
Page 156 and 157:
148 Leo Liberti and Milan DražićV
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150 Leo Liberti and Milan Dražićs
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Set-covering based p-center problem
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Set-covering based p-center problem
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Page 169 and 170:
On the Solution of Interplanetary T
Page 171 and 172:
On the Solution of Interplanetary T
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Page 175 and 176:
Parametrical approach for studying
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Parametrical approach for studying
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Page 181 and 182:
An Interval Branch-and-Bound Algori
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An Interval Branch-and-Bound Algori
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Page 187 and 188:
A New approach to the Studyof the S
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A New approach to the Studyof the S
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184 Katharine M. Mullen, Mikas Veng
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Page 196 and 197:
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190 Niels J. Olieman and Eligius M.
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Page 204 and 205:
196 Andrey V. Orlovwhere A is (m 1
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198 Andrey V. OrlovStep 4. Beginnin
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200 Blas Pelegrín, Pascual Fernán
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Page 212 and 213:
Page 214 and 215:
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208 Deolinda M. L. D. Rasteiro and
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214 José-Oscar H. Sendín, Antonio
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220 Ya. D. Sergeyev and D. E. Kvaso
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226 J. Cole Smith, Fransisca Sudarg
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Page 238 and 239:
Page 240 and 241:
232 Fazil O. Sonmezcost of a config
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234 Fazil O. Sonmezhere f h is the
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236 Fazil O. SonmezThe optimal shap
Page 246 and 247:
238 Alexander S. StrekalovskyDevelo
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Page 251 and 252:
PSfrag replacementsPruning a box fr
Page 253 and 254:
Pruning a box from Baumann point in
Page 255 and 256:
Page 257 and 258:
A Hybrid Multi-Agent Collaborative
Page 259 and 260:
A Hybrid Multi-Agent Collaborative
Page 261 and 262:
Page 263:
Improved lower bounds for optimizat
Page 266 and 267:
258 Graham R. Wood, Duangdaw Sirisa
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Page 270 and 271:
Page 272 and 273:
264 Yinfeng Xu and Wenqiang Daiopti
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266 Yinfeng Xu and Wenqiang DaiThe
Page 277 and 278:
Author IndexAddis, BernardettaDipar
Page 279:
Author Index 271Nasuto, S.J.Departm
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