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Analysis of Inexact Trust-Region Interior-Point SQP Algorithms

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M. Heinkenschloss

Department of Computational and Applied Mathematics

Rice University

L. N. Vicente

Department of Mathematics

University of Coimbra

June 1995 (revised April 1996)

### Abstract

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.
#### Keywords

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
(gzipped) PostScript file.

#### Note

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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