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An Interval Branch-and-Bound Algorithm based on the General Quadratic Form 175Functions Initial Domain of Research Global minimumf 1(x) = x 3 1x 2 + x 2 2x 3x 2 4 − 2x 2 5x 1 + 3x 2x 2 4x 5 X = [−10, 10] 5 f1 ∗ = −1042000f 2(x) = [1 + (x 1 + x 2 + 1) 2 X = [−2, 2] 2 f2 ∗ = 3(19 − 14x 1 + 3x 2 1 − 14x 2 + 6x 1x 2 + 3x 2 2)]×[30 + (2x 1 − 3x 2) 2(18 − 32x 1 + 12x 2 1 + 48x 2 − 36x 1x 2 + 27x 2 2)]f 3(x) = 4x 2 1 − 2x 1x 2 + 4x 2 2 − 2x 2x 3 + 4x 2 3 − 2x 3x 4 X = [−1, 3] × [−10, 10]× f3 ∗ = 5.77+4x 2 4 + 2x 1 − x 2 + 3x 3 + 5x 4 [1, 4] × [−1, 5]f 4(x) = x 3 1x 2 + x 2 2x 3x 2 4 − 2x 2 5x 1 + 3x 2x 2 4x 5 X = [−10, 10] 5 f4 ∗ = −1667708716.3372− 1 6 x5 5x 3 4x 2 3f 5(x) = 4x 2 1 − 2.1x 4 1 + 1 3 x6 1 + x 1x 2 − 4x 2 2 + 4x 4 2 X = [−1000, 1000] 2 f8 ∗ = −1.03162845348366Table 1.Test Functions.Pbs NE T 1 AF GQFCPU ɛ p #its CPU ɛ p #its CPU ɛ p #its CPU ɛ p #itsf 1 0.61 10 −8 248 146.1 10 −8 9148 1.01 10 −8 314 1.97 10 −8 269f 2 612.0 1 117552 9.88 10 −12 6536 6.12 10 −12 1849 3.23 10 −12 1071f 3 307.6 10 −1 47738 135.8 10 −12 10980 13.41 10 −13 5495 15.76 10 −13 4121f 4 0.60 10 −5 257 — — — 17.1 10 −4 3278 7.55 10 −5 626f 5 3.93 10 −2 13125 2.61 10 −14 4431 8.52 10 −14 3009 4.24 10 −14 1553avg 184.8s 35784 73.59s 7774 9.2s 2789 6.5s 1528Table 2.Comparative Tests between Different Reliable Inclusion Functionsfirst order T 1 , AF and GQF. All these inclusion functions use the double precision floatingpoint representation.The first thing that we must compare is the accuracy (ɛ p ) obtained by each method. Therefore,considering equal accuracy, the CPU-times (CPU) and the number of iterations (# its)can then be compared.Table 2 shows that, the obtained accuracy (ɛ p ) are quiet similar for all the considered inclusionfunctions (GQF is always the most efficient) except for the natural extension NE whichis clearly less efficient (for f 2 , f 3 and f 5 ); this remark is not true when the functions f 1 andf 4 are considered, because those functions are particular examples constructed to show thatsometimes the inclusion function T 1 can be inefficient (here the global optimum is on a sideof the initial domain). — in Table 2 means that the inclusion function T 1 cannot produce aresult with an accuracy 1 in 100000 iterations; this case is not taken into account for the averageof T 1 . Furthermore, the inclusion function T 1 is less efficient when the problems havea relatively high number of variables (4 or 5) because the computations of enclosures of thegradient become very expensive, considering the CPU-time, by using an interval automaticdifferentiation code, [7].Considering the average in Table 2, the interval Branch-and-Bound algorithm associatedwith GQF proves is own efficiency on these 5 examples providing the best accuracy for allcases, the best CPU-time average and the best average for the number of iterations. A gainaround 30% is denoted comparing both CPU-time and number of iterations between GQFand AF. The gains induced by the GQF method are important even if the polynomial functionsare quiet simple in the sense that the so-generated quadratic forms are not dense (a lotof elements of the matrix are equal to 0).