Approximation of non-linear functions

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Lasse D

2025年07月10日 15:41

• 

Hello, I have the following question. I have integrated various non-linear functions into my model as follows. Unfortunately, the calculation time is significantly longer than when I leave out the constraints. Is there perhaps another way to approximate these curves linearly? Unfortunately, it is not clear how many support points I need for this.

I want to linearize this relationship:

\(f(r)=base+(1−base)⋅(1−e ^{−kr})\)

These are my constraints in the code:

a = Model.addVars(I, D, R, vtype=GRB.BINARY, name="a")

b = Model.addVars(I, D, vtype=GRB.CONTINUOUS, lb=0, ub=1, name="b")

k = 0.1

exp_values = {r: base + (1 - base) * (1 - math.exp(-k * r)) for r in R}

for i in I:

    for d in D:

        Model.addLConstr(1 == quicksum(a[i, d, r] for r in R))

        Model.addLConstr(S[i, d] == quicksum(y[i, t] for t in range(Q[i], d + 1)))

Model.addLConstr(S[i, d] == quicksum(r * a[i, d, r] for r in R))

        Model.addLConstr(C[i, d] == quicksum(exp_values[r] * a[i, d, r] for r in R))

Model.addLConstr(b[i, d] <= y[i, d])

        Model.addLConstr(b[i, d] <= C[i, d])

        Model.addLConstr(b[i, d] >= C[i, d] - (1 - y[i, d]))

Model.addConstr(l[i, d] * R_i[i] <= quicksum(quicksum(x[i, t, j, itr] for j in range(Q[i], d + 1)) for t in T) + quicksum(b[i, j] for j in range(Q[i], d + 1)))

関連

• Nonlinear

0

• 

  Riley Clement

  • Gurobi Staff

  2025年07月13日 00:07

  Hi Lasse,

  You can actually get Gurobi to use a static PWL approximation using the FuncNonLinear parameter.

  Can you try this first?

  - Riley

   

  0
• 

  Jaromił Najman

  • Gurobi Staff

  2025年07月14日 09:34

  Hi Lasse,

  If you are using version 12+ then you should try using Gurobi's nonlinear expressions to write your constraints.

  For your particular function this would be something like

  import gurobipy as gp
  from gurobipy import nlfunc
  
  resultvar = model.addVar(lb=-gp.GRB.INFINITY)
  model.addConstr(resultvar == base + (1 + base)*(1 - gp.nlfunc.exp(-k*r))

  Note that every nonlinear constraint has to be an equality constraint and be equal to an optimization variable.

  If this still does not solve your problem, then you can follow Riley's recommendation about the PWL approximation.

  Best regards, 
  Jaromił

  0

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