Problems solving a mixed integer quadratic program (MIQP) in Gurobi

Question

I need your help. For my thesis i need to solve a mixed integer quadratic problem (MIQP) with quadratic constraints using Gurobi. When I write the problem into a file the implementation is fine, the solving part is the problem because the best bound and objective for it is 0....... which can't be! Definition of the problem:

          maximize: \sum_{i \in A, j \in Q} c_ij*x_ij

          \sum_{i \in A} c_ij*x_ij <= B_i
                              c_ij <= b_ij 
                        x_ij, c_ij >=0

Implementation Using Java interface:

    public class Gurobi_mod {
public static int m = 10; //number of items
public static int n = 5; //number of agents 
public static double b_ij[][] = new double [n][m];
public static double B_i[] = new double [n];

public static void main(String[] args) throws IOException {

    try {

    GRBEnv env = new GRBEnv();
    GRBModel model = new GRBModel(env);

      GRBVar[][] xij = new GRBVar[n][m];
      for (int i = 0; i < n; i++){
          for (int j = 0; j < m; j++){
              xij[i][j] =
                        model.addVar(0.0, 1.0, 1, GRB.BINARY, "x" + i + "," + j);
          }
      }
      model.update();
      GRBVar[][] cij = new GRBVar[n][m];
        for (int i = 0; i < n; i++){
          for (int j = 0; j < m; j++){
              cij[i][j] =
                        model.addVar(0.0, GRB.INFINITY, 1, GRB.CONTINUOUS, "c" + i + "," + j); 
          }
        }

        model.update();
        double coeff = 1;

        GRBQuadExpr linearobj = new GRBQuadExpr();
        for (int i = 0; i < n; ++i){
            GRBQuadExpr obj = new GRBQuadExpr();
              for (int j = 0; j < m; ++j){
                  obj.addTerm(1, xij[i][j], cij[i][j]);
              }
              linearobj.multAdd (coeff, obj);//addTerm(coeff, var);add(obj);
        }

        model.setObjective(linearobj, GRB.MAXIMIZE);
        model.update();    


    for (int i = 0; i < n; i++){
        GRBQuadExpr thexpr1 = new GRBQuadExpr();
        for (int j = 0; j < m; j++){
            thexpr1.addTerm(1, cij[i][j], xij[i][j]);   
        }
        model.addQConstr(thexpr1, GRB.LESS_EQUAL, B_i[i], "Budget"+ i); 
    }
    model.update();  

    for (int j = 0; j < m; ++j){    
        GRBLinExpr thexpr = new GRBLinExpr();
        for (int i = 0; i < n; ++i){            
            thexpr.addTerm(1, xij[i][j]);               
        }
        model.addConstr(thexpr, GRB.LESS_EQUAL, 1, "Item"+j);
    }
    model.update();  

    for (int i = 0; i < n; i++){    
        for (int j = 0; j < m; j++){
            GRBLinExpr thexprcij = new GRBLinExpr();
            thexprcij.addTerm(1, cij [i][j]);   
            model.addConstr(thexprcij, GRB.LESS_EQUAL, b_ij[i][j], "Bid"+ i + j);   
        }
    }

    // Solve 
    model.optimize();

    }catch (GRBException e){
        System.out.println("Error code: " + e.getErrorCode() + ". " +
                e.getMessage());
    }
  }
 }

Can Gurobi solve this kind of mixed integer quadratic problem, since the variable x_ij is BINARY and c_ij is CONTINUOUS. If I set c_ij to be also BINARY i get a plausible result. Does this mean that the problem is not a concave maximisation problem??? (As far as i know Gurobi can solve only this kind of special MIQP). Thanks in advance!!

Answer 1

A new reformulation-linearization technique for bilinear programming problems goes through a reformulation technique that would be useful for your problem. Assuming I understand you right, the below is your optimization problem

This can be reformulated to

where

This reformulated problem is a MILP and should be easy to solve in Gurobi.

EDIT: As b is the upper bound of c the problem could be written more simply as: