非线性约束转线性约束后CPLEX代码报错，求解决方案

问题描述 #

将非线性表达式转换为线性约束11、12和13后，运行CPLEX代码时出现以下错误：

• constraint labeling not supported for dimensions with variable size, use named constraints instead
• CPLEX cannot extract expression
• Element "cons12" not defined
• Invalid initialization expression for element "cons12"

原代码 #

using CPLEX;

//Total nodes number. 
range Nodes = 1..9;
//{int} Nodes = {1,2,3,4,5,6,7,8,9};
//................................................................................

//Total links number

//two_directed
tuple edge{
  int node_out;
  int node_in;
          };
 {edge} L with node_out, node_in in Nodes = {<1,3>, <3,1>, <2,3>, <3,2>, <3,4>, <4,3>, <3,5>, 
                                             <5,3>, <3,6>, <6,3>, <4,5>, <5,4>, <4,6>, <6,4>, 
                                             <4,8>, <8,4>, <5,6>, <6,5>, <6,7>, <7,6>, <6,9>, 
                                             <9,6>};
  {edge} Lout[Nodes] = [{<1,3>},//node1
                 {<2,3>},//node2
                 {<3,1>, <3,2>, <3,4>, <3,5>, <3,6>},//node3
                 {<4,3>, <4,5>, <4,6>, <4,8>},//node4
                 {<5,3>, <5,4>, <5,6>},//node5
                 {<6,3>, <6,4>, <6,5>, <6,7>, <6,9>},//node6
                 {<7,6>},//node7
                 {<8,4>},//node8
                 {<9,6>}];//node9 
 //Flows
 tuple cflow{
   int origin;
   int destination;
           }
 {cflow} F with origin,destination in Nodes = {<1,2>, <1,3>, <1,4>, <1,5>, <1,6>, <1,7>, 
                      <1,8>, <1,9>, <2,1>, <2,3>, <2,4>, <2,5>, <2,6>, <2,7>, <2,8>, <2,9>, 
                      <3,1>, <3,2>, <3,4>, <3,5>, <3,6>, <3,7>, <3,8>, <3,9>, 
                      <4,1>, <4,2>, <4,3>, <4,5>, <4,6>, <4,7>, <4,8>, <4,9>, 
                      <5,1>, <5,2>, <5,3>, <5,4>, <5,6>, <5,7>, <5,8>, <5,9>,
                      <6,1>, <6,2>, <6,3>, <6,4>, <6,5>, <6,7>, <6,8>, <6,9>, <7,1>, <7,2>};
 
float landa_f[f in F]=[0.86, 0.3, 0.75, 0.23, 0.32, 0.4, 0.5, 0.6, 0.22, 0.14, 
                       0.23, 0.42, 0.33, 0.5, 0.62, 0.36, 0.42, 0.35, 0.2, 0.16,
                       0.33, 0.9, 0.41, 0.51, 0.61, 0.33, 0.42, 0.51, 0.87, 0.96,
                       0.31, 0.55, 0.91, 0.36, 0.32, 0.72, 0.76, 0.32, 0.45, 0.64,
                       0.38, 0.71, 0.43, 0.55, 0.53, 0.9, 0.58, 0.97, 0.5, 0.33 ];
 
   {string} V = {"IDS", "DPI", "NAT", "Proxy", "Firewall"};
 
  //MAIN DECISION VARIABLES
  dvar int I[v in V][n in Nodes][f in F][j in 1..2] in 0..1; //denotes that an NF instance v 
  hosted at node n is used by the j-th service on the service chain of flow f.

  dvar int IL[l in L][f in F][j in 1..2][n in Nodes] in 0..1;//denotes that link l is used by 
  flow f to route from the j-th to (j + 1)-th NF service, hosted at node nj and nj+1.

  dvar int Y[v in V][n in Nodes];

  //Decision variables related  with non linear equations
  dvar int z[l in L][f in F][j in 1..2][n in Nodes][v in V] in 0..1;

subject to{
     //convert non_linear_equations to new linear constraints
     forall (f in F, j in 1..2, v in V) 
     cons11: sum( l in Lout[item(Routes[f],j-1)] ) z[l][f][j][item(Routes[f],j-1)][v] == 1;
                        
   forall (f in F, j in 1..2, l in Lout[item(Routes[f],j-1)], v in V) {
    
    cons12: 3 * z[l][f][j][item(Routes[f],j-1)][v] <= ( IL[l][f][j][item(Routes[f],j-1)] + 
                 I[v][item(Routes[f],j-1)][f][j] );     
    cons13:   z[l][f][j][item(Routes[f],j-1)][v] >= ( IL[l][f][j][item(Routes[f],j-1)] + 
                 I[v][item(Routes[f],j-1)][f][j] ) - 2;  }
       }

错误原因分析 #

1. 约束标签语法错误：在包含动态大小集合（如Lout[item(Routes[f],j-1)]）的forall循环中，直接使用cons11:这类标签的方式不被CPLEX支持，因为动态集合的维度无法提前确定，导致约束标签无法正确绑定。
2. 未定义Routes对象：代码中多次引用item(Routes[f],j-1)，但未定义Routes集合或数组，CPLEX无法解析该表达式，导致提取失败。
3. 嵌套循环中的约束标签重复定义：在嵌套forall的代码块内定义cons12和cons13标签，OPL不允许在循环块内重复声明同名约束，引发未定义和初始化错误。

解决方法 #

1. 调整约束标签定义方式 #

要么直接移除约束标签（OPL会自动生成约束），要么使用约束数组的方式命名约束，确保维度匹配：

subject to {
  // 提前定义约束数组，覆盖所有循环变量维度
  con cons11[F, 1..2, V];
  con cons12[F, 1..2, L, V];
  con cons13[F, 1..2, L, V];

  forall (f in F, j in 1..2, v in V) 
    cons11[f,j,v]: sum( l in Lout[item(Routes[f],j-1)] ) z[l][f][j][item(Routes[f],j-1)][v] == 1;
                        
  forall (f in F, j in 1..2, l in Lout[item(Routes[f],j-1)], v in V) {
    cons12[f,j,l,v]: 3 * z[l][f][j][item(Routes[f],j-1)][v] <= ( IL[l][f][j][item(Routes[f],j-1)] + I[v][item(Routes[f],j-1)][f][j] );     
    cons13[f,j,l,v]: z[l][f][j][item(Routes[f],j-1)][v] >= ( IL[l][f][j][item(Routes[f],j-1)] + I[v][item(Routes[f],j-1)][f][j] ) - 2;  
  }
}

2. 补充Routes定义 #

必须添加Routes的定义，示例如下（根据实际业务逻辑调整）：

// 示例：定义每个流对应的路径节点数组，j的范围需匹配代码中的1..2
int Routes[F][1..2] = [
  // 为每个流f填充对应的路径节点，比如<1,2>流的路径节点为[1,3]
  [1,3], [1,3], ... // 按F集合的顺序依次填充
];

3. 简化约束定义（无标签方式） #

如果不需要命名约束，直接移除所有约束标签，OPL会自动处理：

subject to{
  //convert non_linear_equations to new linear constraints
  forall (f in F, j in 1..2, v in V) 
    sum( l in Lout[item(Routes[f],j-1)] ) z[l][f][j][item(Routes[f],j-1)][v] == 1;
                        
  forall (f in F, j in 1..2, l in Lout[item(Routes[f],j-1)], v in V) {
    3 * z[l][f][j][item(Routes[f],j-1)][v] <= ( IL[l][f][j][item(Routes[f],j-1)] + I[v][item(Routes[f],j-1)][f][j] );     
    z[l][f][j][item(Routes[f],j-1)][v] >= ( IL[l][f][j][item(Routes[f],j-1)] + I[v][item(Routes[f],j-1)][f][j] ) - 2;  
  }
}

内容的提问来源于stack exchange，提问作者Mahsa