Test-Problem Generator for Unconstrained Global Optimization
by Chi-Kong Ng and Duan Li

Downloads:

Problem Characteristics:
Problem-IDs Dimension Difficulty Level
ngli001 - ngli010 2 0
ngli011 - ngli020 1
ngli021 - ngli030 2
ngli031 - ngli035 5 0
ngli036 - ngli040 1
ngli041 - ngli045 2
ngli046 - ngli050 3
ngli051 - ngli055 4
ngli056 - ngli060 5
ngli061 - ngli065 10 0
ngli066 - ngli070 1
ngli071 - ngli075 2
ngli076 - ngli080 3
ngli081 - ngli085 4
ngli086 - ngli090 5
ngli091 - ngli095 20 0
ngli096 - ngli100 1
ngli101 - ngli105 2
ngli106 - ngli110 3
ngli111 - ngli115 4
ngli116 - ngli120 5
ngli121 - ngli125 50 0
ngli126 - ngli130 1
ngli131 - ngli135 2
ngli136 - ngli140 3
ngli141 - ngli145 4
ngli146 - ngli150 5
ngli151 - ngli155 100 0
ngli156 - ngli160 1
ngli161 - ngli165 2
ngli166 - ngli170 3
ngli171 - ngli175 4
ngli176 - ngli180 5
ngli181 - ngli185 200 0
ngli186 - ngli190 1
ngli191 - ngli195 2
ngli196 - ngli200 3
ngli201 - ngli205 4
ngli206 - ngli210 5
ngli211 - ngli215 500 0
ngli216 - ngli220 1
ngli221 - ngli225 2
ngli226 - ngli230 3
ngli231 - ngli235 4
ngli236 - ngli240 5
ngli241 - ngli245 1000 0
ngli246 - ngli250 1
ngli251 - ngli255 2
ngli256 - ngli260 3
ngli261 - ngli265 4
ngli266 - ngli270 5
ngli271 - ngli275 2000 0
ngli276 - ngli280 1
ngli281 - ngli285 2
ngli286 - ngli290 3
ngli291 - ngli295 4
ngli296 - ngli300 5

Some Statistics:
Dimension Number of Global Minimizer Number of Non-global Minimizers Number of Saddle Points
2 1 3 4
5 1 31 210
10 1 1023 58024
20 1 2^20 O(3^20)
50 1 2^50 O(3^50)
100 1 2^100 O(3^100)
200 1 2^200 O(3^200)
500 1 2^500 O(3^500)
10001 2^1000 O(3^1000)
20001 2^2000 O(3^2000)

Stability Test Results:
Dimension Difficulty Level Problem-ID Norm of Gradient of the Objective Function at the Designate Global Minimizer Minimum Eigenvalue of the Hessian Matrix of the Objective Function at the Designate Global Minimizer
2 0 ngli001 1.179181e-010 5.945133e+004
ngli002 1.090190e-011 2.714310e+003
ngli003 4.706972e-013 3.946292e+002
ngli004 7.359673e-011 2.479789e+004
ngli005 8.030045e-011 9.297885e+003
ngli006 2.027888e-010 3.494996e+004
ngli007 8.288678e-012 6.528977e+000
ngli008 8.154974e-011 3.754143e+004
ngli009 7.102749e-010 3.664713e+003
ngli010 4.247892e-011 4.898305e+003
1 ngli011 9.458342e-012 2.609989e+003
ngli012 3.364232e-011 3.956561e+002
ngli013 3.903017e-011 2.908108e+002
ngli014 3.743822e-013 1.409491e+003
ngli015 1.643366e-010 1.532143e+004
ngli016 1.681788e-010 3.195770e+003
ngli017 3.047569e-011 1.090126e+003
ngli018 2.319667e-011 2.117231e+003
ngli019 4.084810e-010 1.508712e+004
ngli020 1.326449e-010 1.001710e+004
2 ngli021 4.345511e-010 4.417691e+003
ngli022 1.438950e-011 6.538590e+003
ngli023 5.131539e-012 4.502905e+003
ngli024 9.527419e-011 6.654707e+003
ngli025 1.243420e-010 3.051302e+004
ngli026 1.825684e-011 9.149805e+003
ngli027 1.940530e-011 1.165498e+002
ngli028 3.093428e-011 6.251220e+003
ngli029 5.520586e-011 8.649961e+003
ngli030 3.637498e-011 8.056468e+002
5 0 ngli031 3.948051e-010 2.713497e+004
ngli032 1.420059e-010 1.580679e+004
ngli033 4.702555e-010 5.914334e+002
ngli034 8.935383e-011 8.808330e+002
ngli035 7.083527e-010 3.381759e+003
1 ngli036 1.180954e-010 1.585926e+003
ngli037 2.182373e-010 6.590464e+002
ngli038 8.586162e-011 7.960701e+003
ngli039 3.455259e-011 7.026489e+002
ngli040 5.487109e-011 8.864607e+001
2 ngli041 4.300989e-010 4.705353e+003
ngli042 6.827111e-010 7.564721e+002
ngli043 3.265941e-011 2.351228e+003
ngli044 4.102548e-010 2.192293e+003
ngli045 8.792301e-011 5.203118e+001
3 ngli046 3.836892e-010 6.859967e+003
ngli047 4.044399e-009 8.368493e+003
ngli048 2.220664e-010 1.055613e+004
ngli049 2.989856e-011 3.213966e+003
ngli050 2.663283e-010 1.813269e+002
4 ngli051 2.162463e-010 1.148641e+004
ngli052 1.709274e-010 1.257821e+003
ngli053 1.805894e-010 1.550652e+004
ngli054 1.811464e-010 7.141301e+003
ngli055 1.579626e-010 2.141827e+004
5 ngli056 1.909877e-009 3.543918e+003
ngli057 4.461647e-011 2.213941e+002
ngli058 1.620698e-010 1.821974e+003
ngli059 8.479608e-010 1.137057e+004
ngli060 2.474976e-010 3.418149e+002
10 0 ngli061 1.178132e-009 3.259954e+003
ngli062 4.777439e-009 1.455257e+003
ngli063 3.279634e-010 9.257143e+002
ngli064 1.628992e-009 6.508877e+003
ngli065 3.147563e-009 3.302927e+002
1 ngli066 5.945607e-009 7.668123e+003
ngli067 6.733594e-010 3.211749e+002
ngli068 2.617045e-010 3.522417e+003
ngli069 1.814814e-009 4.623952e+003
ngli070 7.969575e-010 9.425546e+002
2 ngli071 3.891332e-010 4.118896e+003
ngli072 9.042843e-010 1.977443e+003
ngli073 7.875400e-010 1.673720e+002
ngli074 8.513152e-010 3.664025e+003
ngli075 9.593347e-010 2.746697e+000
3 ngli076 7.480492e-011 6.492159e+002
ngli077 1.025276e-009 9.588299e+003
ngli078 6.818021e-010 2.226905e+003
ngli079 5.378529e-010 4.364216e+001
ngli080 2.504314e-009 4.295435e+002
4 ngli081 1.203763e-010 9.040600e-001
ngli082 1.483334e-009 1.793215e+002
ngli083 1.205116e-009 4.329398e+003
ngli084 9.206337e-010 1.648235e+002
ngli085 7.384248e-010 5.254863e+003
5 ngli086 6.937813e-010 8.568127e+003
ngli087 1.082938e-009 4.198202e+002
ngli088 4.797339e-010 1.368677e+002
ngli089 2.537978e-010 3.585406e+002
ngli090 3.531723e-010 1.059982e+001
20 0 ngli091 8.337039e-010 9.404658e+002
ngli092 2.813727e-009 1.564874e+003
ngli093 3.472956e-009 3.473023e+002
ngli094 1.306572e-009 2.029080e+003
ngli095 1.652328e-009 2.134082e+002
1 ngli096 2.230213e-009 5.863154e+000
ngli097 1.369158e-009 8.037758e+002
ngli098 1.628208e-009 5.398060e+003
ngli099 1.594054e-009 1.343440e+002
ngli100 1.021970e-009 5.087592e+001
2 ngli101 2.177293e-009 2.424622e+003
ngli102 7.793951e-010 1.887221e+002
ngli103 2.315593e-009 3.176301e+003
ngli104 2.910832e-009 2.413381e+003
ngli105 6.302261e-010 3.757387e+003
3 ngli106 1.444324e-009 1.758578e+003
ngli107 9.128211e-010 4.514536e+002
ngli108 1.365457e-009 9.246073e+002
ngli109 1.326278e-009 1.412045e+002
ngli110 8.199722e-010 7.925892e+001
4 ngli111 9.482289e-010 4.738288e+001
ngli112 2.603755e-009 3.079295e+002
ngli113 1.164682e-009 8.160758e+002
ngli114 4.296650e-009 1.534882e+002
ngli115 1.876080e-009 1.436195e+003
5 ngli116 1.031516e-009 5.140927e+002
ngli117 1.117914e-009 1.061572e+003
ngli118 3.637452e-009 1.252094e+002
ngli119 2.545798e-009 4.423332e+003
ngli120 1.471980e-009 6.051778e+003
50 0 ngli121 2.128212e-009 1.608410e+002
ngli122 2.657850e-009 1.940221e+002
ngli123 1.437502e-009 2.284634e+001
ngli124 3.628747e-009 2.118696e+002
ngli125 2.297668e-009 9.194798e+001
1 ngli126 2.778588e-009 4.588976e+002
ngli127 2.076832e-009 5.162916e+001
ngli128 2.715797e-009 5.438020e+001
ngli129 2.487963e-009 4.275446e+001
ngli130 3.260167e-009 5.421542e+002
2 ngli131 2.426925e-009 3.395040e+002
ngli132 2.625414e-009 3.567990e+001
ngli133 1.146181e-009 5.137272e+000
ngli134 2.000520e-009 8.434331e-002
ngli135 5.377794e-009 1.871104e+002
3 ngli136 2.961034e-009 4.605267e+002
ngli137 8.140191e-009 6.033994e+002
ngli138 2.669844e-009 3.711341e+002
ngli139 1.220789e-009 3.357719e+001
ngli140 2.680165e-009 7.723051e+002
4 ngli141 1.994687e-009 1.443084e+002
ngli142 2.478099e-009 1.866807e+002
ngli143 1.463760e-009 1.379816e+002
ngli144 2.185743e-009 3.270905e+002
ngli145 2.811522e-009 9.093475e+001
5 ngli146 3.009154e-009 2.057779e+001
ngli147 1.947023e-009 6.771334e+001
ngli148 2.229407e-009 9.398992e+002
ngli149 3.072486e-009 1.802879e+003
ngli150 1.937940e-009 1.821875e+002
100 0 ngli151 6.038127e-009 1.529896e+002
ngli152 5.507216e-009 1.045062e+002
ngli153 4.223201e-009 1.267372e+002
ngli154 4.790942e-009 2.722792e+002
ngli155 4.386126e-009 3.837847e+000
1 ngli156 5.536385e-009 6.982340e+002
ngli157 4.173700e-009 1.327999e+001
ngli158 8.037845e-009 3.525305e+002
ngli159 4.711028e-009 1.336314e+002
ngli160 5.752860e-009 1.117724e+002
2 ngli161 4.247809e-009 3.002967e+002
ngli162 7.137064e-009 4.482124e+001
ngli163 4.643278e-009 2.562333e+001
ngli164 5.427028e-009 1.453580e+002
ngli165 3.841592e-009 8.933531e+000
3 ngli166 3.970440e-009 2.945316e+001
ngli167 5.091798e-009 2.843132e+002
ngli168 4.409243e-009 8.627649e+001
ngli169 2.539551e-009 8.869207e+002
ngli170 5.607381e-009 1.364618e+001
4 ngli171 3.278504e-009 1.152808e+002
ngli172 3.612723e-009 6.256252e+002
ngli173 3.352111e-009 1.820589e+002
ngli174 3.412711e-009 2.592114e+002
ngli175 8.321621e-009 4.425397e+002
5 ngli176 5.221485e-009 7.525716e+001
ngli177 4.112531e-009 1.087092e+001
ngli178 6.005534e-009 2.558574e+002
ngli179 5.884179e-009 7.690487e+002
ngli180 5.287459e-009 5.726660e+002
200 0 ngli181 9.832861e-009 5.661342e+000
ngli182 7.790494e-009 3.563625e+001
ngli183 7.110034e-009 8.445753e+002
ngli184 7.677759e-009 4.405225e+001
ngli185 7.596129e-009 1.950304e+001
1 ngli186 9.066188e-009 2.086952e+001
ngli187 7.311931e-009 3.599021e+000
ngli188 8.760098e-009 2.609130e+001
ngli189 1.026604e-008 9.640326e+001
ngli190 6.425679e-009 5.133279e+001
2 ngli191 7.544400e-009 7.121822e+001
ngli192 6.139495e-009 6.981576e+001
ngli193 8.191638e-009 3.847271e+001
ngli194 9.852317e-009 1.268104e+002
ngli195 1.048281e-008 8.661538e+001
3 ngli196 7.969558e-009 1.647485e+002
ngli197 6.544431e-009 4.007696e+000
ngli198 6.306665e-009 2.569059e+002
ngli199 7.754453e-009 8.579522e+000
ngli200 5.879602e-009 2.323061e+001
4 ngli201 8.467372e-009 1.294914e+002
ngli202 6.398541e-009 3.733837e+001
ngli203 6.632303e-009 3.556399e+001
ngli204 8.366781e-009 2.780851e+002
ngli205 5.755296e-009 7.404366e+001
5 ngli206 7.871840e-009 8.540007e-001
ngli207 7.747364e-009 7.801089e+000
ngli208 1.033596e-008 5.846626e+001
ngli209 5.009844e-009 8.023555e+001
ngli210 6.432893e-009 5.783022e+001
500 0 ngli211 1.511844e-008 3.160845e+001
ngli212 1.713306e-008 1.164261e+002
ngli213 1.885018e-008 1.111674e+001
ngli214 1.686437e-008 3.215160e+001
ngli215 1.532109e-008 2.070678e+001
1 ngli216 1.163831e-008 1.645804e+001
ngli217 1.357721e-008 1.474801e+001
ngli218 1.323698e-008 1.354500e+001
ngli219 1.235690e-008 1.517557e+000
ngli220 1.198265e-008 5.004262e+001
2 ngli221 1.485850e-008 2.661878e+001
ngli222 1.402858e-008 1.924329e+001
ngli223 1.252846e-008 1.634234e+001
ngli224 1.453601e-008 2.484561e+001
ngli225 1.079571e-008 1.999721e+001
3 ngli226 1.193650e-008 6.160755e+001
ngli227 1.136955e-008 4.740936e+000
ngli228 1.316219e-008 2.455887e+001
ngli229 1.169212e-008 8.868721e+001
ngli230 1.781543e-008 3.765523e+001
4 ngli231 9.168718e-009 9.729751e+000
ngli232 1.266432e-008 1.337760e+001
ngli233 1.453429e-008 9.835623e+000
ngli234 1.416150e-008 9.807350e+001
ngli235 1.137799e-008 2.326575e+001
5 ngli236 1.145506e-008 1.976468e+001
ngli237 1.204937e-008 5.992291e+000
ngli238 1.363104e-008 2.933369e+001
ngli239 1.416445e-008 1.190197e+001
ngli240 1.281439e-008 8.830246e-001
10000 ngli241 2.411679e-008 6.696600e+000
ngli242 2.474625e-008 1.981390e+001
ngli243 1.999490e-008 9.406030e+000
ngli244 2.382628e-008 4.798141e+000
ngli245 2.559739e-008 9.052023e+000
1 ngli246 2.490689e-008 2.329624e+001
ngli247 2.440709e-008 1.761387e+001
ngli248 2.428036e-008 6.317560e+000
ngli249 2.146559e-008 1.079325e+000
ngli250 2.220662e-008 3.059731e+001
2 ngli251 2.137817e-008 5.936885e+001
ngli252 1.970386e-008 4.013476e+000
ngli253 1.950734e-008 1.557495e+001
ngli254 1.958103e-008 5.828476e+001
ngli255 1.735882e-008 5.619161e+000
3 ngli256 2.097303e-008 7.434352e-001
ngli257 2.102524e-008 2.425078e+001
ngli258 1.925380e-008 1.379674e+000
ngli259 2.633102e-008 3.222796e+000
ngli260 2.228967e-008 2.647733e+001
4 ngli261 1.872353e-008 3.598863e+001
ngli262 1.652692e-008 3.010941e+001
ngli263 2.069270e-008 5.486643e+000
ngli264 1.648263e-008 1.969517e+001
ngli265 1.877279e-008 1.086158e+001
5 ngli266 2.307572e-008 4.413625e-001
ngli267 2.041608e-008 1.143337e+001
ngli268 1.842302e-008 1.627757e+001
ngli269 1.939068e-008 9.454285e+000
ngli270 1.893303e-008 4.659243e+000
2000ngli271 3.844368e-008 3.590733e+000
ngli272 4.322934e-008 1.036114e+001
ngli273 4.222056e-008 4.983569e+000
ngli274 4.102671e-008 1.191457e+001
ngli275 3.734690e-008 1.091047e+000
1 ngli276 3.993637e-008 3.173355e+001
ngli277 3.680115e-008 2.391573e+001
ngli278 4.482871e-008 1.252477e+000
ngli279 3.426137e-008 1.751484e+000
ngli280 3.989597e-008 1.842013e+000
2 ngli281 3.427234e-008 1.103038e+001
ngli282 3.479838e-008 3.676681e+001
ngli283 3.678980e-008 6.614935e+000
ngli284 3.333589e-008 2.777476e+000
ngli285 3.432731e-008 3.126471e+000
3 ngli286 3.370308e-008 2.838101e+001
ngli287 3.676761e-008 7.808213e+000
ngli288 3.974301e-008 2.565523e+000
ngli289 3.153263e-008 3.632465e+000
ngli290 3.369057e-008 4.904377e+000
4 ngli291 3.313337e-008 7.402129e+000
ngli292 4.125296e-008 2.910998e+000
ngli293 3.600839e-008 4.165238e+000
ngli294 3.396852e-008 2.478023e+000
ngli295 3.003292e-008 6.824446e+000
5 ngli296 3.310540e-008 1.215420e+001
ngli297 3.606808e-008 2.814987e+001
ngli298 2.856022e-008 2.075598e+000
ngli299 3.010818e-008 1.294041e+001
ngli300 2.915999e-008 2.062085e+000

Some Computational Results (BARON version 9.0.2 under GAMS Distribution 23.3):
Dimension Difficulty Level Problem-ID Number of Iterations CPU Time in Seconds (Intel Core2Duo E6700)
2 0 ngli001 11 0.05
ngli002 11 0.06
ngli003 9 0.05
ngli004 15 0.06
ngli005 19 0.05
ngli006 23 0.08
ngli007 1 0.03
ngli008 19 0.06
ngli009 9 0.06
ngli010 9 0.08
1 ngli011 21 0.06
ngli012 7 0.05
ngli013 9 0.05
ngli014 57 0.11
ngli015 19 0.06
ngli016 7 0.08
ngli017 29 0.09
ngli018 35 0.11
ngli019 11 0.06
ngli020 29 0.08
2 ngli021 29 0.08
ngli022 31 0.06
ngli023 25 0.06
ngli024 9 0.08
ngli025 42 0.08
ngli026 23 0.09
ngli027 35 0.08
ngli028 29 0.09
ngli029 23 0.09
ngli030 * 22 * 0.06
5 0 ngli031 359 1.02
ngli032 297 1.20
ngli033 179 0.77
ngli034 111 0.48
ngli035 219 0.84
1 ngli036 * 361 * 1.20
ngli037 525 1.81
ngli038 421 1.38
ngli039 143 0.48
ngli040 43 0.19
2 ngli041 497 1.66
ngli042 731 2.19
ngli043 185 0.81
ngli044 113 0.39
ngli045 389 1.42
3 ngli046 373 1.09
ngli047 759 2.25
ngli048 333 1.33
ngli049 861 2.91
ngli050 183 0.62
4 ngli051 525 1.75
ngli052 1093 3.36
ngli053 841 3.48
ngli054 979 2.69
ngli055 949 2.98
5 ngli056 725 2.53
ngli057 579 1.89
ngli058 1423 4.16
ngli059 1395 4.67
ngli060 * 1205 * 3.94
10 0 ngli061 49449 314.73
ngli062 12373 61.62
ngli063 62279 316.09
ngli064 58487 316.47
ngli065 42093 248.27
1 ngli066 172619 974.09
ngli067 16751 84.44
ngli068 * 31416 * 205.47
ngli069 85143 398.61
ngli070 13161 79.38
2 ngli071 145942 754.14
ngli072 35077 184.62
ngli073 25143 150.94
ngli074 61233 341.44
ngli075 4898 36.48
3 ngli076 83354 408.56
ngli077 213761 1196.19
ngli078 64162 379.55
ngli079 7519 46.62
ngli080 172431 912.58
4 ngli081 * 6393 * 38.89
ngli082 71796 375.98
ngli083 123958 699.36
ngli084 51873 287.38
ngli085 110193 568.27
5 ngli086 524352 2517.80
ngli087 59084 324.19
ngli088 107397 571.02
ngli089 278871 1458.22
ngli090 70097 370.50
20 0 ngli095 1280823 18214.59
1 ngli100 ** 1377500 ** 19837.00
* with "EpsR = 0.01"
** failed (run out of memory)
More Information:
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