\title{ A Global Convergence Theory for General Trust-Region-Based
 Algorithms for Equality Constrained Optimization \footnotemark[2]}

\author{J. E. DENNIS, Jr. \footnotemark[3]
\and MAHMOUD~EL-ALEM\footnotemark[4]
\and Maria C. MACIEL\footnotemark[5]}

\begin{document}
\maketitle

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\footnotetext[2]{Research was supported by DOE DE-FG005-86ER25017, 
CRPC CCR-9120008, AFOSR-F49620-9310212, and the
 REDI Foundation.}

\footnotetext[3]{Department of Computational and Applied Mathematics \&
Center for Research on Parallel Computation, Rice 
University, P. O. Box 1892, Houston TX 77251.}

\footnotetext[4]{Department of Mathematics, Faculty of Science,
Alexandria University, Alexandria, Egypt.}

\footnotetext[5]{Departamento de Matematica,         
Universidad Nacional del Sur,      
Avenida Alem 1253,              
8000 Bahia Blanca,   
Argentina.}

\renewcommand{\thefootnote}{\arabic{footnote}}

\begin{abstract}
This work presents a global convergence theory for a broad class of 
trust-region algorithms for the smooth nonlinear programming
problem with equality
constraints. The main result generalizes Powell's 1975 result for 
unconstrained trust-region algorithms.

The trial step is characterized by very mild conditions on its normal and
tangential components. The normal component need not be computed accurately.
The theory requires a quasi-normal component to 
satisfy a fraction of Cauchy decrease condition on the quadratic model
of the linearized constraints. The tangential component then must satisfy
a fraction of Cauchy decrease condition on a quadratic model of the Lagrangian
function in the translated tangent space of the constraints determined by the
quasi-normal component.
The Lagrange multipliers estimates and the Hessian estimates
are assumed only to be bounded.

The other main characteristic of this class of algorithms is that the step is
evaluated by using the augmented Lagrangian as a merit function and the penalty 
parameter is updated using the El-Alem scheme.
The properties of the step together with the way that the
penalty parameter is chosen are sufficient to establish global convergence.

As an example, an algorithm is presented
which can be viewed as a generalization of the Steihaug-Toint dogleg algorithm for 
the unconstrained case. It is based on a quadratic programming algorithm that
uses a step in a quasi-normal direction to the tangent space of
the constraints and then does feasible conjugate reduced-gradient steps to
solve the reduced quadratic program. This algorithm should cope quite well with
large problems for which effective preconditioners are known.
\end{abstract}

{ \bf{Key Words:}} Constrained Optimization, Global Convergence, Trust Regions, 
Equality Constrained, Nonlinear Programming, Conjugate Gradient, Inexact Newton
Method.

\begin{AMS}
65K05, 49D37.
\end{AMS}

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\markboth{J.~DENNIS, M.~EL-ALEM , AND M.~MACIEL}
{A THEORY FOR GENERAL TRUST-REGION-BASED ALGORITHMS}