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A novel modeling and smoothing technique in global optimization

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  • In this paper, we introduce a new methodology for modeling of the given data and finding the global optimum value of the model function. First, a new surface blending technique is offered by using Bezier curves and a smooth objective function is obtained with the help of this technique. Second, a new global optimization method followed by an adapted algorithm is presented to reach the global minimizer of the objective function. As an application of this new methodology, we consider energy conformation problem in Physical Chemistry as a very important real-world problem.

    Mathematics Subject Classification: Primary: 90C26; Secondary: 65D05, 65D17, 97M10.

    Citation:

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  • Figure 1.  The subregions of $\Omega = [0,360]\times[0,360]$

    Figure 2.  Constructed Bezier surfaces on the subregions $-438$ was taken as zero to remove the complexity

    Figure 3.  The graph of the function $\tilde{f}(x, y, \varepsilon, \delta)$ which is constructed by blending Bezier surfaces

    Table 1.  The list of test problems

    Problem No.Function NameDimension $n$RegionOptimum value
    1Two dimensional function $c=0.05$$2$$[-3, 3]^2$$0$
    2Two dimensional function $c=0.2$$2$$[-3, 3]^2$$0$
    3Two dimensional function $c=0.5$$2$$[-3, 3]^2$$0$
    43-hump function$2$$[-3, 3]^2$$0$
    56-hump function$2$$[-3, 3]^2$ $-1.0316$
    6Treccani function $2$$[-3, 3]^2$$0$
    7Goldstein-Price function $2$$[-3, 3]^2$$3.0000$
    8Shubert function$2$$[-10, 10]^2$$-186.73091$
    9Rastrigin function$2$$[-3, 3]^2$$-2.0000$
    10Branin function$2$$[-5, 10]\times[10],[15]$$0.3979$
    11(S5) Shekel function$4$$[0, 10]^4$$-10.1532$
    12(S7) Shekel function$4$$[0, 10]^4$$-10.4029$
    13(S10) Shekel function$4$$[0, 10]^4$$-10.5364$
    14, 15, 16, 17Sin-square I function $2, 3, 5, 7$$[-10, 10]^n$ $0$
    18, 19, 20, 21Sin-square I function$10, 20, 30, 50$$[-10, 10]^n$ $0$
     | Show Table
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    Table 2.  The numerical results of our method

    Problem No.niter-mf.eval-mf-meanf-bestSR
    1$2$$1.50004$$214$$5.9087e-15$$2.6630e-154$$8/10$
    2$2$$1.1250$$290.6250$$7.5789e-15$$3.4336e-16$$8/10$
    3$2$$1.7500$$414.2857$$4.0814e-15$$4.7243e-16$$8/10$
    4$2$$1.4000$$411$$4.8635e-15$$2.8802e-16$$10/10$
    5$2$$1.5000$$234$$-1.0316$$-1.0316$$10/10$
    6$2$$1.0000$$216.5000$$5.5963e-14$$1.6477e-15$$10/10$
    7$2$$1.2222$$487.8889$$3.0000$$3.0000$$9/10$
    8$2$$2.7000$$813.5000$$-186.7309$$-186.7309$$10/10$
    9$2$$3.4000$$501$$-2.0000$$-2.0000$$10/10$
    10$2$$1.0000$$222.3000$$0.3979$$0.3979$$10/10$
    11$4$$1.6667$$1001$$-10.1532$$-10.1532$$9/10$
    12$4$$1.7500$$1365.1000$$-10.4029$$-10.4029$$8/10$
    13$4$$1.2857$$1412$$-10.5321$$-10.5321$$7/10$
    14$2$$2.7500$$743.2500$$9.6751e-15$$9.4192e-15$$8/10$
    15$3$$1.9000$$3027$$1.3445e-14$$5.6998e-15$$10/10$
    16$5$$1.8000$$4999.3$$1.8351e-13$$3.7007e-15$$10/10$
    17$7$$1.7500$$8171$$1.7275e-14$$1.3790e-14$$8/10$
    18$10$$2.7778$$8895.4$$4.3639e-13$$3.0992e-14$$9/10$
    19$20$$2.7143$$18242$$2.2066e-12$$3.0016e-13$$7/10$
    20$30$$3.5000$$43232$$6.9372e-12$$1.7361e-12$$6/10$
    21$50$$2.5000$$83243$$7.0303e-12$$9.8531e-13$$6/10$
     | Show Table
    DownLoad: CSV

    Table 3.  The comparison of the results

    No n Our Method Ma et. al [16] El-Gindy et. al [5]
    iter-mf.eval-miter-mf.eval-miter-mf.eval-m
    1$2$$1.5$$214$$4$$5097$$2$$310$
    2$2$$1.13$$290.6$$3$$4012$$2$$778$
    3$2$$1.75$$414.3$$3$$2507$$3$$977$
    4$2$$1.4$$411$$3$$545$$2$$577$
    5$2$$1.5$$234$$3$$518$$2$$279$
    6$2$$1.2$$216.5$$1$$595$$2$$265$
    7$2$$2.7$$487.9$$3$$8140$$-$$ -$
    8$2$$3.4$$813.5$$3$$5280$$3$$635$
    9$2$$1$$501$$3$$337$$2$$315$
    10$2$$1$$222.3$$3$$1819$$-$$-$
    14$2$$2.75$$743.3$$3$$536$$3$$549$
    15$3$$1.9$$3027$$1$$6083$$2$$1283$
    16$5$$1.8$$4999.3$$1$$7839$$2$$5291$
    17$7$$1.75$$8171$$4$$10130$$2$$12793$
    18$10$$2.78$$8895.4$$2$$29463$$2$$33810$
    19$20$$2.71$$18242$$-$$-$$2$$96223$
    20$30$$3.5$$43232$$-$$-$$4$$376885$
    21$50$$2.5$$83243$$-$$-$$9$$>10^6$
     | Show Table
    DownLoad: CSV

    Table 4.  Numerical Results

    $k$$\alpha$$\beta$$x_0$$x_k^*$$f_k^*$
    1$0.5$$0.1$(160.0000,280.0000)$(190.2613,277.4205)$$-438.2412$
    2$0.5$$0.1$$(190.2613,277.4205)$$(329.0062,186.9678)$$-438.2625$
    3$0.5$$0.1$$(329.0062,186.9678)$$(181.6167,187.5836)$$-438.2678$
     | Show Table
    DownLoad: CSV
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