Hi,

at https://github.com/AlexFleischerParis/howtowithopl/blob/master/multiplybinarybydecisionvariable.mod

I shared a few ways to do that with the OPL CPLEX API but you can do the same with pyomo

// suppose we want b * x <= 7 

    dvar int x in 2..10;
    dvar boolean b;

    dvar int bx;

    maximize x;
    subject to
    {
      
    // Linearization  
    bx<=7;

     

    2*b<=bx;
    bx<=10*b;

    bx<=x-2*(1-b);
    bx>=x-10*(1-b);
    
    // if we use CP we could write directly
    // b*x<=7
    
    // or rely on logical constraints within CPLEX
    // (b==1) => (bx==x);
    // (b==0) => (bx==0);
    }

Dear Alex,

Thank you for your valuable insights on handling binary-continuous variable products in CPLEX. I would like to share additional information based on my own experience.

I am using CPLEX Studio v22.1, and when I attempt to solve my model with this version, after applying linearization, I encounter an optimality gap of 160% within the 10-hour time limit. In contrast, when I run the same model with Gurobi (using Gurobipy), it reaches a gap of approximately 60%, without the need for linearization.

Could you provide any insights into why Gurobi performs better without linearization? Additionally, what methods or techniques does Gurobi employ that might explain this discrepancy in performance?

I would greatly appreciate any advice or suggestions you can offer.

Thank you again for your assistance!

Best regards,
Aysan

// suppose we want b * x <= 7     dvar int x in 2..10;    dvar boolean b;    dvar int bx;    maximize x;    subject to    {          // Linearization      bx<=7;         2*b<=bx;    bx<=10*b;    bx<=x-2*(1-b);    bx>=x-10*(1-b);        // if we use CP we could write directly    // b*x<=7        // or rely on logical constraints within CPLEX    // (b==1) => (bx==x);    // (b==0) => (bx==0);    }

The general case can be formulated this way :
Assume L <= x < = U and b in {0, 1}, the formulation of u = x * b is 

L (1-b) <= x - u <= U (1-b)
Lb <= u <= Ub

This formulation is tight: it gives the convex hull.

I assumed the variable u is a brand new one introduced for this product.

Thank you for your valuable insights on handling binary-continuous variable products in CPLEX. I would like to share additional information based on my own experience.

I am using CPLEX Studio v22.1, and when I attempt to solve my model with this version, after applying linearization, I encounter an optimality gap of 160% within the 10-hour time limit. In contrast, when I run the same model with Gurobi (using Gurobipy), it reaches a gap of approximately 60%, without the need for linearization.

Could you provide any insights into why Gurobi performs better without linearization? Additionally, what methods or techniques does Gurobi employ that might explain this discrepancy in performance?

I would greatly appreciate any advice or suggestions you can offer.

Thank you again for your assistance!

Best regards,
Aysan