Primal-Dual Interior Methods for Nonconvex Nonlinear Programming

Recently, infeasibility issues in interior methods for nonconvex nonlinear programming have been studied. In particular, it has been shown how many line-search interior methods may converge to an infeasible point which is on the boundary of the feasible region with respect to the inequality constraints. The convergence is such that the search direction does not tend to zero, but the step length does. Such “false” convergence may occur even for a fixed value of the barrier parameter. In this paper, two commonly used reformulations for handling infeasibility are studied. It is shown that when the above-mentioned convergence difficulty occurs, the multiplier search directions diverge along a direction that can be characterized in terms of the gradients of the equality constraints and the asymptotically active inequality constraints. Further, for a penalty-barrier interior method, similar convergence difficulties along with diverging multiplier search directions can occur when the penalty and barrier parameters are driven to zero. The divergence is motivated through an analysis of the penalty-barrier trajectory. Further, the direction of divergence is characterized.