Analysis of Inexact Trust-Region Interior-Point SQP Algorithms

M. Heinkenschloss
Department of Computational and Applied Mathematics
Rice University

L. N. Vicente
Department of Mathematics
University of Coimbra

June 1995 (revised April 1996)


In this paper we analyze inexact trust-region interior-point (TRIP) sequential quadratic 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 first-order convergent. Numerical experiments with two optimal control problems governed by nonlinear partial differential equations are reported.


nonlinear programming, trust-region methods, interior-point algorithms, Coleman and Li scaling, simple bounds, inexact linear systems solvers, Krylov subspace methods, optimal control

AMS subject classifications

49M37, 90C06, 90C30

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This paper extends the analysis of SQP TRIP methods developed in J. E. Dennis, M. Heinkenschloss, and L. N. Vicente Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems .

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