A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

Abstract

With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, not only the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point as the original problem is feasible, but also it can be superlinearly or quadratically convergent to the infeasible stationary point when a problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems and some standard test problems from the CUTE collection, where the superlinear convergence is demonstrated when we solve two infeasible problems and one well-posed feasible counterexample presented in the literature.