5.3.2. C 的QP建模和优化

在本节中，我们将使用 MindOpt C API，以按行输入的形式来建模以及求解 二次规划问题示例 中的问题。

首先，引入头文件：

27#include "Mindopt.h"

并创建优化模型：

67    CHECK_RESULT(MDOemptyenv(&env));
68    CHECK_RESULT(MDOstartenv(env));
69    CHECK_RESULT(MDOnewmodel(env, &model, MODEL_NAME, 0, NULL, NULL, NULL, NULL, NULL));

接下来，我们通过 MDOsetIntAttr() 将目标函数设置为 最小化，并调用 MDOaddvar() 来添加四个优化变量，定义其下界、上界、名称和类型，以及其在目标函数中线性项的系数（关于 MDOsetIntAttr() 和 MDOaddvar() 的详细使用方式，请参考 属性）：

74    /* Change to minimization problem. */
75    CHECK_RESULT(MDOsetintattr(model, MODEL_SENSE, MDO_MINIMIZE));
76
77    /* Add variables. */
78    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0,         10.0, MDO_CONTINUOUS, "x0"));
79    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x1"));
80    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x2"));
81    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x3"));

Note

矩阵的非零元随后将按 列 输入；因此，MDOaddvar() 中，与约束矩阵相关联的参数 size 、 indices 、 value 分别用 0 、 NULL 、 NULL 代替（换句话说，此时问题无约束）。

以下我们将开始添加线性约束中的的非零元及其上下界，我们使用以下四列数组来定义线性约束；其中， row1_idx 和 row2_idx 分别表示第一和第二个约束中非零元素的位置（索引），而 row1_val 和 row2_val 则是与之相对应的非零数值。

49    /* Linear part in the first constraint: 1 x0 + 1 x1 + 2 x2 + 3 x3 */
50    int    row1_nnz   = 4;
51    int    row1_idx[] = { 0,   1,   2,   3   };
52    double row1_val[] = { 1.0, 1.0, 2.0, 3.0 };
53    /* Linear part in the second constraint: 1 x0 - 1 x2 + 6 x3 */
54    int    row2_nnz   = 3;
55    int    row2_idx[] = { 0,    2,   3   };
56    double row2_val[] = { 1.0, -1.0, 6.0 };

我们调用 MDOaddconstr() 来输入约束：

83    /* Add constraints.
84     * Note that the nonzero elements are inputted in a row-wise order here.
85     */
86    CHECK_RESULT(MDOaddconstr(model, row1_nnz, row1_idx, row1_val, MDO_GREATER_EQUAL, 1.0, "c0"));
87    CHECK_RESULT(MDOaddconstr(model, row2_nnz, row2_idx, row2_val, MDO_EQUAL,         1.0, "c1"));

接下来我们将添加二次规划中的目标函数的二次项系数。我们使用以下三列数组来定义：其中 qo_col1 和 qo_col2 分别记录二次项中所有非零项的两个变量索引，而 qo_values 是与之相对应的非零系数值。

58    /* Quadratic part in objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
59    int    qo_nnz      = 5;
60    int    qo_col1[]   = { 0,   1,   2,   3,   0 };
61    int    qo_col2[]   = { 0,   1,   2,   3,   1 };
62    double qo_values[] = { 0.5, 0.5, 0.5, 0.5, 0.5 };

我们调用 MDOaddqpterms() 设置目标的二次项：

89    /* Add quadratic objective term. */
90    CHECK_RESULT(MDOaddqpterms(model, qo_nnz, qo_col1, qo_col2, qo_values));

问题输入完成后，再调用 MDOoptimize() 求解优化问题:

92    /*------------------------------------------------------------------*/
93    /* Step 3. Solve the problem and populate optimization result.                */
94    /*------------------------------------------------------------------*/
95    /* Solve the problem. */
96    CHECK_RESULT(MDOoptimize(model));

然后，我们可以通过获取属性值的方式来获取对应的优化目标值objective和变量的取值：

 98    CHECK_RESULT(MDOgetintattr(model, STATUS, &status));
 99    if (status == MDO_OPTIMAL) 
100    {
101        CHECK_RESULT(MDOgetdblattr(model, OBJ_VAL, &obj));
102        printf("The optimal objective value is: %f\n", obj);
103        for (int i = 0; i < 4; ++i) 
104        {
105            CHECK_RESULT(MDOgetdblattrelement(model, X, i, &x));
106            printf("x[%d] = %f\n", i, x);
107        }
108    } 
109    else 
110    {
111        printf("No feasible solution.\n");
112    }

最后，调用 MDOfreemodel() 和 MDOfreeenv() 来释放模型：

30#define RELEASE_MEMORY  \
31    MDOfreemodel(model);    \
32    MDOfreeenv(env);

117    RELEASE_MEMORY;

示例 MdoQoEx1.c 提供了完整源代码：

  1/**
  2 *  Description
  3 *  -----------
  4 *
  5 *  Quadratic optimization (row-wise input).
  6 *
  7 *  Formulation
  8
  9 *  -----------
 10 *
 11 *  Minimize
 12 *    obj: 1 x0 + 1 x1 + 1 x2 + 1 x3 
 13 *         + 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1]
 14 *  Subject To
 15 *   c0 : 1 x0 + 1 x1 + 2 x2 + 3 x3 >= 1
 16 *   c1 : 1 x0 - 1 x2 + 6 x3 = 1
 17 *  Bounds
 18 *    0 <= x0 <= 10
 19 *    0 <= x1
 20 *    0 <= x2
 21 *    0 <= x3
 22 *  End
 23 */
 24
 25#include <stdio.h>
 26#include <stdlib.h>
 27#include "Mindopt.h"
 28
 29/* Macro to check the return code */
 30#define RELEASE_MEMORY  \
 31    MDOfreemodel(model);    \
 32    MDOfreeenv(env);
 33#define CHECK_RESULT(code) { int res = code; if (res != 0) { fprintf(stderr, "Bad code: %d\n", res);  RELEASE_MEMORY; return (res); } }
 34#define MODEL_NAME  "QP_01"
 35#define MODEL_SENSE "ModelSense"
 36#define STATUS      "Status"
 37#define OBJ_VAL     "ObjVal"
 38#define X           "X"
 39
 40int main(void)
 41{
 42    /* Variables. */
 43    MDOenv *env;
 44    MDOmodel *model;
 45    double obj, x;
 46    int status, i;
 47
 48    /* Model data. */
 49    /* Linear part in the first constraint: 1 x0 + 1 x1 + 2 x2 + 3 x3 */
 50    int    row1_nnz   = 4;
 51    int    row1_idx[] = { 0,   1,   2,   3   };
 52    double row1_val[] = { 1.0, 1.0, 2.0, 3.0 };
 53    /* Linear part in the second constraint: 1 x0 - 1 x2 + 6 x3 */
 54    int    row2_nnz   = 3;
 55    int    row2_idx[] = { 0,    2,   3   };
 56    double row2_val[] = { 1.0, -1.0, 6.0 };
 57
 58    /* Quadratic part in objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
 59    int    qo_nnz      = 5;
 60    int    qo_col1[]   = { 0,   1,   2,   3,   0 };
 61    int    qo_col2[]   = { 0,   1,   2,   3,   1 };
 62    double qo_values[] = { 0.5, 0.5, 0.5, 0.5, 0.5 };
 63
 64    /*------------------------------------------------------------------*/
 65    /* Step 1. Create environment and model.                            */
 66    /*------------------------------------------------------------------*/
 67    CHECK_RESULT(MDOemptyenv(&env));
 68    CHECK_RESULT(MDOstartenv(env));
 69    CHECK_RESULT(MDOnewmodel(env, &model, MODEL_NAME, 0, NULL, NULL, NULL, NULL, NULL));
 70
 71    /*------------------------------------------------------------------*/
 72    /* Step 2. Input model.                                             */
 73    /*------------------------------------------------------------------*/
 74    /* Change to minimization problem. */
 75    CHECK_RESULT(MDOsetintattr(model, MODEL_SENSE, MDO_MINIMIZE));
 76
 77    /* Add variables. */
 78    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0,         10.0, MDO_CONTINUOUS, "x0"));
 79    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x1"));
 80    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x2"));
 81    CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x3"));
 82
 83    /* Add constraints.
 84     * Note that the nonzero elements are inputted in a row-wise order here.
 85     */
 86    CHECK_RESULT(MDOaddconstr(model, row1_nnz, row1_idx, row1_val, MDO_GREATER_EQUAL, 1.0, "c0"));
 87    CHECK_RESULT(MDOaddconstr(model, row2_nnz, row2_idx, row2_val, MDO_EQUAL,         1.0, "c1"));
 88
 89    /* Add quadratic objective term. */
 90    CHECK_RESULT(MDOaddqpterms(model, qo_nnz, qo_col1, qo_col2, qo_values));
 91    
 92    /*------------------------------------------------------------------*/
 93    /* Step 3. Solve the problem and populate optimization result.                */
 94    /*------------------------------------------------------------------*/
 95    /* Solve the problem. */
 96    CHECK_RESULT(MDOoptimize(model));
 97        
 98    CHECK_RESULT(MDOgetintattr(model, STATUS, &status));
 99    if (status == MDO_OPTIMAL) 
100    {
101        CHECK_RESULT(MDOgetdblattr(model, OBJ_VAL, &obj));
102        printf("The optimal objective value is: %f\n", obj);
103        for (int i = 0; i < 4; ++i) 
104        {
105            CHECK_RESULT(MDOgetdblattrelement(model, X, i, &x));
106            printf("x[%d] = %f\n", i, x);
107        }
108    } 
109    else 
110    {
111        printf("No feasible solution.\n");
112    }
113 
114    /*------------------------------------------------------------------*/
115    /* Step 4. Free the model.                                          */
116    /*------------------------------------------------------------------*/
117    RELEASE_MEMORY;
118       
119    return 0;
120}