多目标遗传算法 ------ NSGA-II （部分源码解析）目标函数 problemdef.c

/* Test problem definitions */

# include <stdio.h>

# include <stdlib.h>

# include <math.h>

# include "global.h"

# include "rand.h"

# define sch1

/* # define sch2 */

/* # define fon */

/* # define kur */

/* # define pol */

/* # define vnt */

/* # define zdt1 */

/* # define zdt2 */

/* # define zdt3 */

/* # define zdt4 */

/* # define zdt5 */

/* # define zdt6 */

/* # define bnh */

/* # define osy */

/* # define srn */

/* # define tnk */

/* # define ctp1 */

/* # define ctp2 */

/* # define ctp3 */

/* # define ctp4 */

/* # define ctp5 */

/* # define ctp6 */

/* # define ctp7 */

/* # define ctp8 */

/*  Test problem SCH1

    # of real variables = 1

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef sch1

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = pow(xreal[],2.0);

    obj[] = pow((xreal[]-2.0),2.0);

    return;

}

#endif

/*  Test problem SCH2

    # of real variables = 1

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef sch2

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    if (xreal[]<=1.0)

    {

        obj[] = -xreal[];

        obj[] = pow((xreal[]-5.0),2.0);

        return;

    }

    if (xreal[]<=3.0)

    {

        obj[] = xreal[]-2.0;

        obj[] = pow((xreal[]-5.0),2.0);

        return;

    }

    if (xreal[]<=4.0)

    {

        obj[] = 4.0-xreal[];

        obj[] = pow((xreal[]-5.0),2.0);

        return;

    }

    obj[] = xreal[]-4.0;

    obj[] = pow((xreal[]-5.0),2.0);

    return;

}

#endif

/*  Test problem FON

    # of real variables = n

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef fon

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double s1, s2;

    int i;

    s1 = s2 = 0.0;

    for (i=; i<nreal; i++)

    {

        s1 += pow((xreal[i]-(1.0/sqrt((double)nreal))),2.0);

        s2 += pow((xreal[i]+(1.0/sqrt((double)nreal))),2.0);

    }

    obj[] = 1.0 - exp(-s1);

    obj[] = 1.0 - exp(-s2);

    return;

}

#endif

/*  Test problem KUR

    # of real variables = 3

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef kur

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    int i;

    double res1, res2;

    res1 = -0.2*sqrt((xreal[]*xreal[]) + (xreal[]*xreal[]));

    res2 = -0.2*sqrt((xreal[]*xreal[]) + (xreal[]*xreal[]));

    obj[] = -10.0*( exp(res1) + exp(res2));

    obj[] = 0.0;

    for (i=; i<; i++)

    {

        obj[] += pow(fabs(xreal[i]),0.8) + 5.0*sin(pow(xreal[i],3.0));

    }

    return;

}

#endif

/*  Test problem POL

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef pol

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double a1, a2, b1, b2;

    a1 = 0.5*sin(1.0) - 2.0*cos(1.0) + sin(2.0) - 1.5*cos(2.0);

    a2 = 1.5*sin(1.0) - cos(1.0) + 2.0*sin(2.0) - 0.5*cos(2.0);

    b1 = 0.5*sin(xreal[]) - 2.0*cos(xreal[]) + sin(xreal[]) - 1.5*cos(xreal[]);

    b2 = 1.5*sin(xreal[]) - cos(xreal[]) + 2.0*sin(xreal[]) - 0.5*cos(xreal[]);

    obj[] = 1.0 + pow((a1-b1),2.0) + pow((a2-b2),2.0);

    obj[] = pow((xreal[]+3.0),2.0) + pow((xreal[]+1.0),2.0);

    return;

}

#endif

/*  Test problem VNT

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 3

    # of constraints = 0

    */

#ifdef vnt

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = 0.5*(xreal[]*xreal[] + xreal[]*xreal[]) + sin(xreal[]*xreal[] + xreal[]*xreal[]);

    obj[] = (pow((3.0*xreal[] - 2.0*xreal[] + 4.0),2.0))/8.0 + (pow((xreal[]-xreal[]+1.0),2.0))/27.0 + 15.0;

    obj[] = 1.0/(xreal[]*xreal[] + xreal[]*xreal[] + 1.0) - 1.1*exp(-(xreal[]*xreal[] + xreal[]*xreal[]));

    return;

}

#endif

/*  Test problem ZDT1

    # of real variables = 30

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt1

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double f1, f2, g, h;

    int i;

    f1 = xreal[];

    g = 0.0;

    for (i=; i<; i++)

    {

        g += xreal[i];

    }

    g = 9.0*g/29.0;

    g += 1.0;

    h = 1.0 - sqrt(f1/g);

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem ZDT2

    # of real variables = 30

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt2

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double f1, f2, g, h;

    int i;

    f1 = xreal[];

    g = 0.0;

    for (i=; i<; i++)

    {

        g += xreal[i];

    }

    g = 9.0*g/29.0;

    g += 1.0;

    h = 1.0 - pow((f1/g),2.0);

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem ZDT3

    # of real variables = 30

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt3

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double f1, f2, g, h;

    int i;

    f1 = xreal[];

    g = 0.0;

    for (i=; i<; i++)

    {

        g += xreal[i];

    }

    g = 9.0*g/29.0;

    g += 1.0;

    h = 1.0 - sqrt(f1/g) - (f1/g)*sin(10.0*PI*f1);

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem ZDT4

    # of real variables = 10

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt4

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double f1, f2, g, h;

    int i;

    f1 = xreal[];

    g = 0.0;

    for (i=; i<; i++)

    {

        g += xreal[i]*xreal[i] - 10.0*cos(4.0*PI*xreal[i]);

    }

    g += 91.0;

    h = 1.0 - sqrt(f1/g);

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem ZDT5

    # of real variables = 0

    # of bin variables = 11

    # of bits for binvar1 = 30

    # of bits for binvar2-11 = 5

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt5

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    int i, j;

    int u[];

    int v[];

    double f1, f2, g, h;

    for (i=; i<; i++)

    {

        u[i] = ;

    }

    for (j=; j<; j++)

    {

        if (gene[][j] == )

        {

            u[]++;

        }

    }

    for (i=; i<; i++)

    {

        for (j=; j<; j++)

        {

            if (gene[i][j] == )

            {

                u[i]++;

            }

        }

    }

    f1 = 1.0 + u[];

    for (i=; i<; i++)

    {

        if (u[i] < )

        {

            v[i] =  + u[i];

        }

        else

        {

            v[i] = ;

        }

    }

    g = ;

    for (i=; i<; i++)

    {

        g += v[i];

    }

    h = 1.0/f1;

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem ZDT6

    # of real variables = 10

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 0

    */

#ifdef zdt6

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double f1, f2, g, h;

    int i;

    f1 = 1.0 - (exp(-4.0*xreal[]))*pow((sin(4.0*PI*xreal[])),6.0);

    g = 0.0;

    for (i=; i<; i++)

    {

        g += xreal[i];

    }

    g = g/9.0;

    g = pow(g,0.25);

    g = 1.0 + 9.0*g;

    h = 1.0 - pow((f1/g),2.0);

    f2 = g*h;

    obj[] = f1;

    obj[] = f2;

    return;

}

#endif

/*  Test problem BNH

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 2

    */

#ifdef bnh

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = 4.0*(xreal[]*xreal[] + xreal[]*xreal[]);

    obj[] = pow((xreal[]-5.0),2.0) + pow((xreal[]-5.0),2.0);

    constr[] = 1.0 - (pow((xreal[]-5.0),2.0) + xreal[]*xreal[])/25.0;

    constr[] = (pow((xreal[]-8.0),2.0) + pow((xreal[]+3.0),2.0))/7.7 - 1.0;

    return;

}

#endif

/*  Test problem OSY

    # of real variables = 6

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 6

    */

#ifdef osy

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = -(25.0*pow((xreal[]-2.0),2.0) + pow((xreal[]-2.0),2.0) + pow((xreal[]-1.0),2.0) + pow((xreal[]-4.0),2.0) + pow((xreal[]-1.0),2.0));

    obj[] = xreal[]*xreal[] +  xreal[]*xreal[] + xreal[]*xreal[] + xreal[]*xreal[] + xreal[]*xreal[] + xreal[]*xreal[];

    constr[] = (xreal[]+xreal[])/2.0 - 1.0;

    constr[] = 1.0 - (xreal[]+xreal[])/6.0;

    constr[] = 1.0 - xreal[]/2.0 + xreal[]/2.0;

    constr[] = 1.0 - xreal[]/2.0 + 3.0*xreal[]/2.0;

    constr[] = 1.0 - (pow((xreal[]-3.0),2.0))/4.0 - xreal[]/4.0;

    constr[] = (pow((xreal[]-3.0),2.0))/4.0 + xreal[]/4.0 - 1.0;

    return;

}

#endif

/*  Test problem SRN

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 2

    */

#ifdef srn

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = 2.0 + pow((xreal[]-2.0),2.0) + pow((xreal[]-1.0),2.0);

    obj[] = 9.0*xreal[] - pow((xreal[]-1.0),2.0);

    constr[] = 1.0 - (pow(xreal[],2.0) + pow(xreal[],2.0))/225.0;

    constr[] = 3.0*xreal[]/10.0 - xreal[]/10.0 - 1.0;

    return;

}

#endif

/*  Test problem TNK

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 2

    */

#ifdef tnk

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    obj[] = xreal[];

    obj[] = xreal[];

    if (xreal[] == 0.0)

    {

        constr[] = -1.0;

    }

    else

    {

        constr[] = xreal[]*xreal[] + xreal[]*xreal[] - 0.1*cos(16.0*atan(xreal[]/xreal[])) - 1.0;

    }

    constr[] = 1.0 - 2.0*pow((xreal[]-0.5),2.0) + 2.0*pow((xreal[]-0.5),2.0);

    return;

}

#endif

/*  Test problem CTP1

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 2

    */

#ifdef ctp1

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*exp(-obj[]/g);

    constr[] = obj[]/(0.858*exp(-0.541*obj[]))-1.0;

    constr[] = obj[]/(0.728*exp(-0.295*obj[]))-1.0;

    return;

}

#endif

/*  Test problem CTP2

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp2

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = -0.2*PI;

    a = 0.2;

    b = 10.0;

    c = 1.0;

    d = 6.0;

    e = 1.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP3

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp3

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = -0.2*PI;

    a = 0.1;

    b = 10.0;

    c = 1.0;

    d = 0.5;

    e = 1.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP4

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp4

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = -0.2*PI;

    a = 0.75;

    b = 10.0;

    c = 1.0;

    d = 0.5;

    e = 1.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP5

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp5

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = -0.2*PI;

    a = 0.1;

    b = 10.0;

    c = 2.0;

    d = 0.5;

    e = 1.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP6

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp6

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = 0.1*PI;

    a = 40.0;

    b = 0.5;

    c = 1.0;

    d = 2.0;

    e = -2.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP7

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 1

    */

#ifdef ctp7

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    theta = -0.05*PI;

    a = 40.0;

    b = 5.0;

    c = 1.0;

    d = 6.0;

    e = 0.0;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

/*  Test problem CTP8

    # of real variables = 2

    # of bin variables = 0

    # of objectives = 2

    # of constraints = 2

    */

#ifdef ctp8

void test_problem (double *xreal, double *xbin, int **gene, double *obj, double *constr)

{

    double g;

    double theta, a, b, c, d, e;

    double exp1, exp2;

    g = 1.0 + xreal[];

    obj[] = xreal[];

    obj[] = g*(1.0  - sqrt(obj[]/g));

    theta = 0.1*PI;

    a = 40.0;

    b = 0.5;

    c = 1.0;

    d = 2.0;

    e = -2.0;

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    theta = -0.05*PI;

    a = 40.0;

    b = 2.0;

    c = 1.0;

    d = 6.0;

    e = 0.0;

    exp1 = (obj[]-e)*cos(theta) - obj[]*sin(theta);

    exp2 = (obj[]-e)*sin(theta) + obj[]*cos(theta);

    exp2 = b*PI*pow(exp2,c);

    exp2 = fabs(sin(exp2));

    exp2 = a*pow(exp2,d);

    constr[] = exp1/exp2 - 1.0;

    return;

}

#endif

以上，为NSGA-II 源码中给出的几个测试函数，其中无限制条件的测试函数不需解释，对有限制条件的做一定说明。

根据比对，可以直到算法中，将所有的约束条件都转换为 >=0 。

所以，违反限制条件的约束函数值均小于 0 。

TNK 测试函数有一定特殊性，因为其中存在无意义的个体，即 X2 为0 ，这里对该情况下的 约束函数1 直接赋值为 -1 。

代码，如下：

    if (xreal[] == 0.0)

    {

        constr[] = -1.0;

    }

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/* Crowding distance computation routines */ # include <stdio.h> # include <stdlib.h> # ...
多目标遗传算法 ------ NSGA-II （部分源码解析）README 算法的部分英文解释
This is the Readme file for NSGA-II code. About the Algorithm--------------------------------------- ...
多目标遗传算法 ------ NSGA-II （部分源码解析）实数、二进制编码的变异操作 mutation.c
遗传算法的变异操作 /* Mutation routines */ # include <stdio.h> # include <stdlib.h> # include < ...
多目标遗传算法 ------ NSGA-II （部分源码解析）两个个体支配判断 dominance.c
/* Domination checking routines */ # include <stdio.h> # include <stdlib.h> # include &l ...
多目标遗传算法 ------ NSGA-II （部分源码解析）二元锦标赛选择 tourselect.c
tourselect.c 文件中共有两个函数: selection (population *old_pop, population *new_pop) individual* tournament ...
多目标遗传算法 ------ NSGA-II （部分源码解析）临时种群生成新父代种群 fillnds.c
/* Nond-domination based selection routines */ # include <stdio.h> # include <stdlib.h> ...

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多目标遗传算法 ------ NSGA-II （部分源码解析）目标函数 problemdef.c

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