model the minimization of sum of inverse functions

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Miranda

2021年05月09日 04:46

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Hello,

The problem I am solving is to minimize sum of 1/x[k] for k in range(n), under a set of linear constraints. I set the lower and upper bounds for each x[k] be greater than 0; therefore, 1/x is a convex function over the intervals.

Here is how I write the model in Gurobi,

x = model.addVars(n, lb = -GRB.INFINITY, ub = GRB.INFINITY, vtype=GRB.CONTINUOUS, name = 'x')

for i in range(n):
  model.addConstr(x[i] <= upper[i]) # upper bounds are greater than 0
  model.addConstr(x[i] >= lower[I]) # lower bounds are also greater than 0

# add some additional linear constraints here

model.setObjective(quicksum(1/x[i] for i in range(n)), GRB.MINIMIZE)

Then Gurobi gives me the following error

unsupported operand type(s) for /: 'int' and 'Var'

My first question is, can Gurobi recognize this model as a convex program as I expected?

Then, I tried to reformulate the problem by introducing additional variables z[k] for k in range(n), and adding constraints 

for i in range(n):
  model.addConstr(1 <= z[i]*x[i])

The new objective function is 

model.setObjective(quicksum(z[i] for i in range(n)), GRB.MINIMIZE).

My expectation is to solve my original problem as a convex optimization by Gurobi. Even though I could get outputs by introducing variables z, the set of additional constraints 1 <= z[i]x[i] are not convex constraints. 

How could I model my original problem as a convex program?

Thank you.

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• 正式なコメント

  

  Simranjit Kaur

  • Gurobi Staff

  2025年06月23日 02:55 編集済み

  This post is more than three years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum. Or why not try our AI Gurobot?.
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  Jaromił Najman

  • Gurobi Staff

  2021年05月10日 10:32 編集済み

  Hi Miranda,

  Indeed, your problem can be solved as a second order cone problem. You can reformulate

  \[ x \cdot z \geq 1\]

  as

  \[ \begin{align}x^2 + x\cdot z + z^2 - x^2 - z^2 &\geq 1\\
  (x + z)^2 &\geq x^2 + z^2 + 1\\
  x + z & \geq \sqrt{x^2 + z^2 + 1} \end{align}\]

  which is a rotated second order cone. Thus, Gurobi reformulates your problem and can solve it as a convex problem. You do not have to set the NonConvex parameter. Note that this is possible as long as \(x,z \geq 0\)

  Best regards,
  Jaromił

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