HELP: How to transform the continuous variable matrix x into binary variable x', and then create constraints about the summation of x' ?

Answered

mmc hee

• April 30, 2024 09:19

such as I have a MVars x = [x1，x2,  x3, x4]

i want create a constraint like: np.where(x>0, 1, 0).sum() > 5 , actually, i hope the count of nonzero vars must >= 5

How can I linearize this constraint?

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  mmc hee

  • April 30, 2024 09:34

  

  

  oh, i've got a solution:

  create y as a binary MVars

  and  x = bigM * y

          y.sum() >= 5

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  Gurobot AI-generated response

  Gurobi Staff

  • April 30, 2024 09:50
  • Edited

  To linearize the constraint that the count of non-zero elements in the continuous variable matrix must be greater than or equal to 5, you can follow these steps using Gurobi:

  1. Binary Variables: Introduce binary variables \(x_i'\)​ for each \(x_i\)​. Each ​\(x_i'\)​ will be 1 if \(x_i \neq 0\) and 0 otherwise.

  2. Linking Constraints: For each \(i\), add constraints to link \(x_i\)​​ and \(x_i'\)​​. Since Gurobi does not support "not equal" directly in a linear model, you use a small tolerance \(\epsilon\) to define nonzero:

     • \(x_i \leq M x_i'\)​ where \(M\) is a sufficiently large number.
     • \(x_i \geq \epsilon x_i'\)​

     Here, \(M\) should be an upper bound on the possible values of \(x_i\)​ (this can sometimes be set based on the problem context), and \(\epsilon\) is a small positive value, like \(10^{−5}\), assuming \(x_i\)​ cannot naturally be this small if it is considered "nonzero."

  3. Summation Constraint: Add a constraint that the sum of all binary variables \(x_i'\) should be greater than or equal to 5:

     • \(\sum_{i=0}^n x_i' \geq 5\)

  This way, you are enforcing that at least five of the \(x_i\) variables must be non-zero, translating that into a linear constraint suitable for Gurobi's solver. This method does not count how many are exactly zero but rather ensures at least a certain number are nonzero, which matches your requirements.

  Here's a small example in Python using Gurobi:

  import gurobipy as gp
  from gurobipy import GRB
  
  # Create a new model
  m = gp.Model("model")
  
  # Add continuous variables x1, x2, x3, x4
  x = m.addVars(4, lb=-GRB.INFINITY, name="x")
  
  # Add binary variables x1', x2', x3', x4'
  x_prime = m.addVars(4, vtype=GRB.BINARY, name="x_prime")
  
  # Big M and small epsilon
  M = 100
  epsilon = 1e-5
  
  # Add linking constraints
  for i in range(4):
      m.addConstr(x[i] <= M * x_prime[i], name=f"link_upper_{i}")
      m.addConstr(x[i] >= epsilon * x_prime[i], name=f"link_lower_{i}")
  
  # Add summation constraint
  m.addConstr(x_prime.sum() >= 5, "count_nonzero")
  
  # Set objective (if any specific objective is needed)
  m.setObjective(1, GRB.MAXIMIZE)  # Dummy objective
  
  # Optimize model
  m.optimize()

  This code creates variables and constraints in a Gurobi model to ensure that at least five of the continuous variables x are non-zero according to the linearized constraint approach.

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