A condition-based algorithm for solving polyhedral feasibility problems

It is known that Polyhedral Feasibility Problems can be solved via interior-point methods whose real number complexity is polynomial in the dimension of the problem and the logarithm of a condition number of the problem instance. A limitation of these results is that they do not apply to ill-posed instances, for which the condition number is infinite. We propose an algorithm for solving polyhedral feasibility problems in homogeneous form that is applicable to all problem instances, and whose real number complexity is polynomial in the dimension of the problem instance and in the logarithm of an “extended condition number” that is always finite.