5.6.5. Java 的MIQCP建模和优化¶
在本节中，我们将使用 MindOpt JAVA API，以按行输入的形式来建模以及求解 MIQCP题示例 中的问题。
首先，引入头文件：
25import com.alibaba.damo.mindopt.*;
并创建优化模型：
30 // Create model
31 MDOEnv env = new MDOEnv();
32 MDOModel model = new MDOModel(env);
33 model.set(MDO.StringAttr.ModelName, "MIQCP_01");
接下来，我们通过 MDOModel.set 将目标函数设置为 最小化，并调用 MDOModel.addVar 来添加四个优化变量，定义其下界、上界、名称和类型（有关 MDOModel.set 和 MDOModel.addVar 的详细使用方式，请参考 JAVA API）：
36 // Add variables.
37 MDOVar[] x = new MDOVar[4];
38 x[0] = model.addVar(0.0, 10.0, 0.0, 'I', "x0");
39 x[1] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x1");
40 x[2] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x2");
41 x[3] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x3");
接着我们开始设置二次目标函数。我们创建一个二次表达式 MDOQuadExpr , 再调用 MDOQuadExpr.addTerms 来设置目标函数线性部分。这里
obj_idx 表示线性部分的索引，obj_val 表示与 obj_idx 中的索引相对应的非零系数值。
44 /* Linear part in the objective: 1 x0 + 1 x1 + 1 x2 + 1 x3 */
45 int obj_nnz = 4;
46 MDOVar[] obj_idx = new MDOVar[] { x[0], x[1], x[2], x[3] };
47 double[] obj_val = new double[] { 1.0, 1.0, 1.0, 1.0 };
75 // Create a QuadExpr for quadratic objective
76 MDOQuadExpr obj = new MDOQuadExpr();
77 // Add objective linear term: 1 x0 + 1 x1 + 1 x2 + 1 x3
78 obj.addTerms(obj_val, obj_idx);
然后，调用 MDOQuadExpr.addTerms 来设置目标的二次项系数 \(Q\)。
其中，qo_values 表示要添加的二次项的系数，qo_col1 和 qo_col2 表示与qo_values相对应的二次项的第一个变量和第二个变量。
48 /* Quadratic part in the objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
49 int qo_nnz = 5;
50 MDOVar[] qo_col1 = new MDOVar[] { x[0], x[1], x[2], x[3], x[0] };
51 MDOVar[] qo_col2 = new MDOVar[] { x[0], x[1], x[2], x[3], x[1] };
52 double[] qo_values = new double[] { 0.5, 0.5, 0.5, 0.5, 0.5 };
79 // Add quadratic part in objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1]
80 obj.addTerms(qo_values, qo_col1, qo_col2);
最后，我们调用 MDOModel.setObjective 设定优化目标与方向。
82 model.setObjective(obj, MDO.MINIMIZE);
现在我们开始添加二次约束。构建二次约束中的二次表达式的方式与二次目标函数类似。
54 /* Linear part in the first constraint: 1 x0 + 1 x1 + 2 x2 + 3 x3 */
55 int c1_nnz = 4;
56 MDOVar[] c1_idx = new MDOVar[] { x[0], x[1], x[2], x[3] };
57 double[] c1_val = new double[] { 1.0, 1.0, 2.0, 3.0 };
58 /* Quadratic part in the first constraint: - 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
59 int qc1_nnz = 5;
60 MDOVar[] qc1_col1 = new MDOVar[] { x[0], x[1], x[2], x[3], x[0] };
61 MDOVar[] qc1_col2 = new MDOVar[] { x[0], x[1], x[2], x[3], x[1] };
62 double[] qc1_values = new double[] { -0.5, -0.5, -0.5, -0.5, -0.5 };
84 // Add 1st quadratic constraint.
85 MDOQuadExpr c1 = new MDOQuadExpr();
86 c1.addTerms(c1_val, c1_idx);
87 c1.addTerms(qc1_values, qc1_col1, qc1_col2);
64 /* Linear part in the second constraint: 1 x0 - 1 x2 + 6 x3 */
65 int c2_nnz = 3;
66 MDOVar[] c2_idx = new MDOVar[] { x[0], x[2], x[3] };
67 double[] c2_val = new double[] { 1.0, -1.0, 6.0 };
68 /* Quadratic part in the second constraint: 1/2 [x1^2] */
69 int qc2_nnz = 1;
70 MDOVar[] qc2_col1 = new MDOVar[] { x[1] };
71 MDOVar[] qc2_col2 = new MDOVar[] { x[1] };
72 double[] qc2_values = new double[] { 0.5 };
90 // Add 2nd quadratic constraint.
91 MDOQuadExpr c2 = new MDOQuadExpr();
92 c2.addTerms(c2_val, c2_idx);
93 c2.addTerms(qc2_values, qc2_col1, qc2_col2);
然后，我们调用 MDOModel.addQConstr 将二次约束添加至模型中。
88 model.addQConstr(c1, MDO.GREATER_EQUAL, 1.0, "c0");
94 model.addQConstr(c2, MDO.LESS_EQUAL, 1.0, "c1");
问题输入完成后，调用 MDOModel.optimize 求解优化问题：
96 // Solve the problem and populate optimization result.
97 model.optimize();
示例 MdoMIQCPEx1.java 提供了完整源代码：
1/**
2 * Description
3 * -----------
4 *
5 * Mixed Integer Quadratically constrained quadratic optimization (row-wise input).
6
7 * Formulation
8 * -----------
9 * Minimize
10 * obj: 1 x0 + 1 x1 + 1 x2 + 1 x3
11 * + 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1]
12 *
13 * Subject To
14 * c0 : 1 x0 + 1 x1 + 2 x2 + 3 x3 - 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] >= 1
15 * c1 : 1 x0 - 1 x2 + 6 x3 + 1/2 [x1^2] <= 1
16 * Bounds
17 * 0 <= x0 <= 10
18 * 0 <= x1
19 * 0 <= x2
20 * 0 <= x3
21 * x0 integer
22 * End
23 */
24
25import com.alibaba.damo.mindopt.*;
26import java.util.*;
27
28public class MdoQcoEx1 {
29 public static void main(String[] args) throws MDOException {
30 // Create model
31 MDOEnv env = new MDOEnv();
32 MDOModel model = new MDOModel(env);
33 model.set(MDO.StringAttr.ModelName, "MIQCP_01");
34
35 try {
36 // Add variables.
37 MDOVar[] x = new MDOVar[4];
38 x[0] = model.addVar(0.0, 10.0, 0.0, 'I', "x0");
39 x[1] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x1");
40 x[2] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x2");
41 x[3] = model.addVar(0.0, MDO.INFINITY, 0.0, 'C', "x3");
42
43 /* Prepare model data. */
44 /* Linear part in the objective: 1 x0 + 1 x1 + 1 x2 + 1 x3 */
45 int obj_nnz = 4;
46 MDOVar[] obj_idx = new MDOVar[] { x[0], x[1], x[2], x[3] };
47 double[] obj_val = new double[] { 1.0, 1.0, 1.0, 1.0 };
48 /* Quadratic part in the objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
49 int qo_nnz = 5;
50 MDOVar[] qo_col1 = new MDOVar[] { x[0], x[1], x[2], x[3], x[0] };
51 MDOVar[] qo_col2 = new MDOVar[] { x[0], x[1], x[2], x[3], x[1] };
52 double[] qo_values = new double[] { 0.5, 0.5, 0.5, 0.5, 0.5 };
53
54 /* Linear part in the first constraint: 1 x0 + 1 x1 + 2 x2 + 3 x3 */
55 int c1_nnz = 4;
56 MDOVar[] c1_idx = new MDOVar[] { x[0], x[1], x[2], x[3] };
57 double[] c1_val = new double[] { 1.0, 1.0, 2.0, 3.0 };
58 /* Quadratic part in the first constraint: - 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1] */
59 int qc1_nnz = 5;
60 MDOVar[] qc1_col1 = new MDOVar[] { x[0], x[1], x[2], x[3], x[0] };
61 MDOVar[] qc1_col2 = new MDOVar[] { x[0], x[1], x[2], x[3], x[1] };
62 double[] qc1_values = new double[] { -0.5, -0.5, -0.5, -0.5, -0.5 };
63
64 /* Linear part in the second constraint: 1 x0 - 1 x2 + 6 x3 */
65 int c2_nnz = 3;
66 MDOVar[] c2_idx = new MDOVar[] { x[0], x[2], x[3] };
67 double[] c2_val = new double[] { 1.0, -1.0, 6.0 };
68 /* Quadratic part in the second constraint: 1/2 [x1^2] */
69 int qc2_nnz = 1;
70 MDOVar[] qc2_col1 = new MDOVar[] { x[1] };
71 MDOVar[] qc2_col2 = new MDOVar[] { x[1] };
72 double[] qc2_values = new double[] { 0.5 };
73
74 /* Construct model. */
75 // Create a QuadExpr for quadratic objective
76 MDOQuadExpr obj = new MDOQuadExpr();
77 // Add objective linear term: 1 x0 + 1 x1 + 1 x2 + 1 x3
78 obj.addTerms(obj_val, obj_idx);
79 // Add quadratic part in objective: 1/2 [ x0^2 + x1^2 + x2^2 + x3^2 + x0 x1]
80 obj.addTerms(qo_values, qo_col1, qo_col2);
81
82 model.setObjective(obj, MDO.MINIMIZE);
83
84 // Add 1st quadratic constraint.
85 MDOQuadExpr c1 = new MDOQuadExpr();
86 c1.addTerms(c1_val, c1_idx);
87 c1.addTerms(qc1_values, qc1_col1, qc1_col2);
88 model.addQConstr(c1, MDO.GREATER_EQUAL, 1.0, "c0");
89
90 // Add 2nd quadratic constraint.
91 MDOQuadExpr c2 = new MDOQuadExpr();
92 c2.addTerms(c2_val, c2_idx);
93 c2.addTerms(qc2_values, qc2_col1, qc2_col2);
94 model.addQConstr(c2, MDO.LESS_EQUAL, 1.0, "c1");
95
96 // Solve the problem and populate optimization result.
97 model.optimize();
98
99 if (model.get(MDO.IntAttr.Status) == MDO.OPTIMAL) {
100 System.out.println("Optimal objective value is: " + model.get(MDO.DoubleAttr.ObjVal));
101 System.out.println("Decision variables: ");
102 for (int i = 0; i < 4; i++) {
103 System.out.println( "x[" + i + "] = " + x[i].get(MDO.DoubleAttr.X));
104 }
105 }
106 else {
107 System.out.println("No feasible solution.");
108 }
109 } catch (Exception e) {
110 System.out.println("Exception during optimization");
111 e.printStackTrace();
112 } finally {
113 model.dispose();
114 env.dispose();
115 }
116 }
117}