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Trust-Region Interior-Point SQP Algorithms for a Class of
Nonlinear Programming Problems

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J. E. Dennis

Department of Computational and Applied Mathematics

Rice University

M. Heinkenschloss

Department of Computational and Applied Mathematics

Rice University

L. N. Vicente

Department of Mathematics

University of Coimbra

December 1994 (revised November 1995 and December 1996)

### Abstract

In this paper a family of trust--region interior--point SQP algorithms
for the solution of minimization problems with
nonlinear equality constraints and
simple bounds on some of the variables is described and analyzed.
Such nonlinear programs arise e.g.\ from the discretization of optimal
control problems.
The algorithms treat states and controls as independent variables.
They are designed to take advantage of the structure of the problem.
In particular they do not rely on matrix factorizations of the
linearized constraints, but use solutions of the linearized state
equation and the adjoint equation. They are well suited for large
scale problems arising from optimal control problems governed
by partial differential equations.
The algorithms keep strict feasibility with respect to the
bound constraints by using a primal--dual affine scaling method proposed
for a different class of problems by Coleman and Li and they
exploit trust--region techniques for equality--constrained optimization.
Thus, they allow the computation of the steps using a variety of
methods, including many iterative techniques.
Global convergence of these algorithms to a first--order KKT limit point is proved under very mild conditions on the trial steps.
Under reasonable, but more stringent conditions on the quadratic model
and on the trial steps,
the sequence of iterates generated by the algorithms is shown to have a
limit point satisfying the second--order necessary KKT conditions.
The local rate of convergence to a nondegenerate strict local minimizer
is q--quadratic.
The results given here include as special cases current results for
only equality constraints and for only simple bounds.
Numerical results for the solution of an optimal control problem governed by
a nonlinear heat equation are reported.
#### Keywords

Nonlinear programming, SQP methods, trust--region methods,
interior--point algorithms,
Dikin--Karmarkar ellipsoid, Coleman--Li affine scaling, simple bounds,
optimal control problems.
#### AMS subject classifications

49M37, 90C06, 90C30
(gzipped) PostScript file.

** See also **
** M. Heinkenschloss and L. N. Vicente **
* Analysis of Inexact Trust-Region Interior-Point SQP Algorithms *
.

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