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：
91 CHECK_RESULT(MDOemptyenv(&env));
92 CHECK_RESULT(MDOstartenv(env));
93 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（关于 MDOsetIntAttr() 和 MDOaddvar() 的详细使用方式，请参考 属性）：
99 /* Change to minimization problem. */
100 CHECK_RESULT(MDOsetintattr(model, MODEL_SENSE, MDO_MINIMIZE));
101
102 /* Add variables. */
103 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, 10.0, MDO_CONTINUOUS, "x0"));
104 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 2.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x1"));
105 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x2"));
106 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。
48 /* Model data. */
49 int row1_idx[] = { 0, 1, 2, 3 };
50 double row1_val[] = { 1.0, 1.0, 2.0, 3.0 };
51 int row2_idx[] = { 0, 2, 3 };
52 double row2_val[] = { 1.0, -1.0, 6.0 };
我们调用 MDOaddconstr() 来输入约束：
108 /* Add constraints.
109 * Note that the nonzero elements are inputted in a row-wise order here.
110 */
111 CHECK_RESULT(MDOaddconstr(model, 4, row1_idx, row1_val, MDO_GREATER_EQUAL, 1.0, "c0"));
112 CHECK_RESULT(MDOaddconstr(model, 3, 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 \(Q\)。我们使用以下3个参数来定义：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。
Note
为了确保 \(Q\) 的对称性，用户只需要输入其下三角形部分， MindOpt 在求解器内部会乘以 1/2.
54 /* Quadratic objective matrix Q.
55 *
56 * Note.
57 * 1. The objective function is defined as c^Tx + 1/2 x^TQx, where Q is stored with coordinate format.
58 * 2. Q will be scaled by 1/2 internally.
59 * 3. To ensure the symmetricity of Q, user needs to input only the lower triangular part.
60 *
61 * Q = [ 1.0 0.5 0 0 ]
62 * [ 0.5 1.0 0 0 ]
63 * [ 0.0 0.0 1.0 0 ]
64 * [ 0 0 0 1.0 ]
65 */
66 int qo_col1[] =
67 {
68 0,
69 1, 1,
70 2,
71 3
72 };
73 int qo_col2[] =
74 {
75 0,
76 0, 1,
77 2,
78 3
79 };
80 double qo_values[] =
81 {
82 1.0,
83 0.5, 1.0,
84 1.0,
85 1.0
86 };
我们调用 MDOaddqpterms() 设置目标的二次项：
114 /* Add quadratic objective term. */
115 CHECK_RESULT(MDOaddqpterms(model, 5, qo_col1, qo_col2, qo_values));
问题输入完成后，再调用 MDOoptimize() 求解优化问题:
117 /*------------------------------------------------------------------*/
118 /* Step 3. Solve the problem and populate optimization result. */
119 /*------------------------------------------------------------------*/
120 /* Solve the problem. */
121 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：
124 CHECK_RESULT(MDOgetintattr(model, STATUS, &status));
125 if (status == MDO_OPTIMAL)
126 {
127 CHECK_RESULT(MDOgetdblattr(model, OBJ_VAL, &obj));
128 printf("The optimal objective value is: %f\n", obj);
129 for (int i = 0; i < 4; ++i)
130 {
131 CHECK_RESULT(MDOgetdblattrelement(model, X, i, &x));
132 printf("x[%d] = %f\n", i, x);
133 }
134 }
135 else
136 {
137 printf("No feasible solution.\n");
138 }
最后，调用 MDOfreemodel() 和 MDOfreeenv() 来释放模型：
30#define RELEASE_MEMORY \
31 MDOfreemodel(model); \
32 MDOfreeenv(env);
143 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 int row1_idx[] = { 0, 1, 2, 3 };
50 double row1_val[] = { 1.0, 1.0, 2.0, 3.0 };
51 int row2_idx[] = { 0, 2, 3 };
52 double row2_val[] = { 1.0, -1.0, 6.0 };
53
54 /* Quadratic objective matrix Q.
55 *
56 * Note.
57 * 1. The objective function is defined as c^Tx + 1/2 x^TQx, where Q is stored with coordinate format.
58 * 2. Q will be scaled by 1/2 internally.
59 * 3. To ensure the symmetricity of Q, user needs to input only the lower triangular part.
60 *
61 * Q = [ 1.0 0.5 0 0 ]
62 * [ 0.5 1.0 0 0 ]
63 * [ 0.0 0.0 1.0 0 ]
64 * [ 0 0 0 1.0 ]
65 */
66 int qo_col1[] =
67 {
68 0,
69 1, 1,
70 2,
71 3
72 };
73 int qo_col2[] =
74 {
75 0,
76 0, 1,
77 2,
78 3
79 };
80 double qo_values[] =
81 {
82 1.0,
83 0.5, 1.0,
84 1.0,
85 1.0
86 };
87
88 /*------------------------------------------------------------------*/
89 /* Step 1. Create environment and model. */
90 /*------------------------------------------------------------------*/
91 CHECK_RESULT(MDOemptyenv(&env));
92 CHECK_RESULT(MDOstartenv(env));
93 CHECK_RESULT(MDOnewmodel(env, &model, MODEL_NAME, 0, NULL, NULL, NULL, NULL, NULL));
94
95
96 /*------------------------------------------------------------------*/
97 /* Step 2. Input model. */
98 /*------------------------------------------------------------------*/
99 /* Change to minimization problem. */
100 CHECK_RESULT(MDOsetintattr(model, MODEL_SENSE, MDO_MINIMIZE));
101
102 /* Add variables. */
103 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, 10.0, MDO_CONTINUOUS, "x0"));
104 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 2.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x1"));
105 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x2"));
106 CHECK_RESULT(MDOaddvar(model, 0, NULL, NULL, 1.0, 0, MDO_INFINITY, MDO_CONTINUOUS, "x3"));
107
108 /* Add constraints.
109 * Note that the nonzero elements are inputted in a row-wise order here.
110 */
111 CHECK_RESULT(MDOaddconstr(model, 4, row1_idx, row1_val, MDO_GREATER_EQUAL, 1.0, "c0"));
112 CHECK_RESULT(MDOaddconstr(model, 3, row2_idx, row2_val, MDO_EQUAL, 1.0, "c1"));
113
114 /* Add quadratic objective term. */
115 CHECK_RESULT(MDOaddqpterms(model, 5, qo_col1, qo_col2, qo_values));
116
117 /*------------------------------------------------------------------*/
118 /* Step 3. Solve the problem and populate optimization result. */
119 /*------------------------------------------------------------------*/
120 /* Solve the problem. */
121 CHECK_RESULT(MDOoptimize(model));
122
123
124 CHECK_RESULT(MDOgetintattr(model, STATUS, &status));
125 if (status == MDO_OPTIMAL)
126 {
127 CHECK_RESULT(MDOgetdblattr(model, OBJ_VAL, &obj));
128 printf("The optimal objective value is: %f\n", obj);
129 for (int i = 0; i < 4; ++i)
130 {
131 CHECK_RESULT(MDOgetdblattrelement(model, X, i, &x));
132 printf("x[%d] = %f\n", i, x);
133 }
134 }
135 else
136 {
137 printf("No feasible solution.\n");
138 }
139
140 /*------------------------------------------------------------------*/
141 /* Step 4. Free the model. */
142 /*------------------------------------------------------------------*/
143 RELEASE_MEMORY;
144
145 return 0;
146}