Cplex/Gurobi options to quickly check the feasibility of quadratic/conic problem

[Ran Ji]

Dear All,

We are running a bisection procedure to find the optimal value of a variable (say VarX) iteratively. At each iteration, given a fixed value of VarX (as a parameter), we need to check the feasibility of a problem with quadratic/conic constraints. The objective function could be set arbitrarily, such as maximizing a constant number (e.g. maximize 0). The primary purpose is to only check the feasibility of the problem, the indication of feasible or infeasible status is good enough for us to conduct further procedure. The optimality is not required in this case. To possibly increase the computational efficiency, we are wondering if there are any specific options for the solver Cplex and Gurobi that they can quickly return the feasibility status of the problem. Alternatively speaking, we are trying to use the options (if any) that the solver stops the solving process once a feasible solution is found, and it does not necessarily to search for other solutions. 

We are greatly appreciated for any insights and advice. Thank you.

Best regards,

Ran   

[Victor Zverovich]

I think it should be enough to set objective expression to a constant. In CPLEX you can also set mipemphasis=1 to emphasize finding feasible solutions, but it only applies to MIP.

If you already have a solution you can check its feasibility using AMPL script without calling a solver.

HTH,

Victor

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[e.d.an...@mosek.com]

In general checking feasibility of conic optimization problems is just as hard as finding the optimal solution.

I cannot say for sure what the optimizers you are mentioning are doing, but it is quite common to use the so called homogeneous model + an interior-point algorithm to find the optimal solution or detect the infeasibility. The paper

https://scholar.google.dk/citations?view_op=view_citation&hl=da&user=3I3Gs2cAAAAJ&citation_for_view=3I3Gs2cAAAAJ:9yKSN-GCB0IC

presents the idea. 

This algorithm has the feature that feasibility and optimality is reached at the exactly the same time. And hence the feasibility status cannot be determined early.

[Ran Ji]

Thank you all for the valuable insights and advice.

[duncan...@gmail.com]

That link is expired but I would love to read the paper. Do you know it's name?

[e.d.an...@mosek.com]

https://link.springer.com/article/10.1007/s10107-002-0349-3

[duncan...@gmail.com]

Thank you very much!