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An Adaptive Radial Basis Algorithm (ARBF) for Mixed-Integer Expensive Constrained Global Optimization139function evaluations. For almost all the failures a local minimum was found. For the Shekel5 problem, the number of G-step Phase 1 and GL-step Phase 1 that produced interior pointsare shown (row OK in the table). Also shown is the iteration from where these steps never gotan interior point (row Iter in the table). Only very few iterations at the start actually producedinterior points, and the convergence is entirely dependent on if these steps produced goodenough points to find the steep global minima. The Results show that ARBFMIP is abouttwice as fast rbfSolve and solves slightly more of the problems. ARBFMIP is also much better ingetting local convergence and achieving higher accuracy in the global minimum, as expected.5. ConclusionsThe global search with target values on the RBF surface often does not produce interior points.The cause must be further investigated. If it is not possible to add, delete or change the RBFinterpolation or problem formulation in a way that produce interior points, then some other0−2−4Function Value−6−8Shekel5Shekel7Shekel10−10−120 2 4 6 8 10xFigure 1. Plot of the Shekel 5, Shekel 7 and Shekel 10, functions along the line segment from (0, 0, 0, 0) to(10, 10, 10, 10).Table 2.Test of Shekel problems for rbfSolve and ARBFMIPrand(’state’) ɛ 0 1 2 3 4 5 6 7 8 9 Mean FailShekel 5 with 10 −3 74 62 66 - 77 - - 87 54 - 70 4ARBFMIP 10 −5 84 72 90 - 89 - - 105 68 - 85 4G,GL Phase 1 OK 7 7 6 4 5 6 5 6 5 5 5.6G,GL bad from Iter 20 18 11 11 8 60 16 12 23 8 18.7rbfSolve 10 −3 153 153 - - 171 - - 141 129 - 149 5rbfSolve 10 −5 - 195 - - - - - 165 171 - 177 7Shekel 7 with 10 −3 - 66 69 78 78 - 66 63 65 - 69 3ARBFMIP 10 −5 - 87 96 89 87 - 72 84 74 - 84 3rbfSolve 10 −3 135 123 123 - 123 129 135 141 - 123 129 2rbfSolve 10 −5 - - - - - 171 171 151 - 164 7Shekel 10 with 10 −3 66 - 66 66 72 90 72 75 - 72 72 2ARBFMIP 10 −5 75 - 72 78 90 96 80 89 - 87 87 2rbfSolve 10 −3 123 123 177 117 117 147 111 159 - 123 133 1rbfSolve 10 −5 159 147 - 171 - - 153 195 - 141 161 4