How to model this "nested constraint" in gurobipy?

Answered

Rihot Gusron

• December 03, 2022 09:32
• Edited

I have this formulation that I called it "nested constraint" (don't know if its right). What is the best way to formulate this in python gurobipy.

 

My current approach is that decompose these with three separate constraint below,

m.addConstrs(((gp.quicksum(y[i,j,l] for i in vertex if i!=j)) - (gp.quicksum(y[j,i,l] for i in vertex if i!=j)) == demand[l]
            for j in vertex for l in customer if j==l))
m.addConstrs(((gp.quicksum(y[i,j,l] for i in vertex if i!=j)) - (gp.quicksum(y[j,i,l] for i in vertex if i!=j)) == 0
            for j in customer for l in customer if j!=l))
m.addConstrs(((gp.quicksum(y[i,j,l] for i in vertex if i!=j)) - (gp.quicksum(y[j,i,l] for i in vertex if i!=j)) == -demand[l]
            for j in vertex for l in customer if j==0))

Related to

• Python
• Modeling

0

• 

  Jaromił Najman

  Gurobi Staff

  • December 05, 2022 09:26
  • Edited

  Hi Rihot,

  The probably easiest way is to use auxiliary variables \(z_{j,l}\) and auxiliary equality constraints.

  \[\begin{align*}
  \sum_{i\in V}y_{ij}^l - \sum_{i \in V} y_{ji}^l &= z_{j,l} \quad \forall j, l \in V'\\
  z_{j,l} &= q_l \quad \forall j,l \in V' \text{ with } j = l\\
  z_{j,l} &= 0 \quad \forall j,l \in V' \text{ with } j \neq l\\
  z_{j,l} &= -q_l \quad \forall j, l \in V' \text{ with } j = 0
  \end{align*}\]

  A pseudo-code might look something like

  m.addConstrs(qp.quicksum(y[i,j,l] - y[j,i,l] for i in I) == z[j,l] for j in V, for l in V)
  for j in V:
    for l in V:
      if j == 0:
        m.addConstr(z[j,l] == -q[l])    
      else if j == l:
        m.addConstr(z[j,l] == q[l])
      else:
        m.addConstr(z[j,l] == 0)

  Best regards, 
  Jaromił

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