model the minimization of sum of inverse functions
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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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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,
Jaromił0 -
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