Analysis of Inexact Trust-Region Interior-Point SQP Algorithms

In this paper we analyze inexact trust–region interior–point (TRIP) sequential quadra– tic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trust–region radius, then the TRIP SQP algorithms are globally convergent to a stationary point. Numerical experiments with two optimal control problems governed by nonlinear partial differential equations are reported.