In this work we deene a trust-region feasible-point algorithm for approximating solutions of the nonlin-ear system of equalities and inequalities F (x; y) = 0; y 0, where F : IR n IR m ! IR p is continuously diierentiable. This formulation is quite general; the Karush-Kuhn-Tucker conditions of a general nonlinear programming problem are an obvious example, and a set of equalities and inequalities can be transformed, using slack variables, into such form. We will be concerned with the possibility that n, m, and p may be large and that the Jacobian matrix may be sparse and rank deecient. Exploiting the convex structure of the local model trust-region subproblem, we propose a globally convergent inexact trust-region feasible-point algorithm to minimize an arbitrary norm of the residual, say kF(x;y)ka, subject to the nonnegativity constraints. This algorithm uses a trust-region globalization strategy to determine a descent direction as an inexact solution of the local model trust-region subproblem and then, it uses linesearch techniques to obtain an acceptable steplength. We demonstrate that, under rather weak hypotheses, any accumulation point of the iteration sequence is a constrained stationary point for f = kFka, and that the sequence of constrained residuals converges to zero.
[1]
K. Anstreicher,et al.
On the convergence of an infeasible primal-dual interior-point method for convex programming
,
1994
.
[2]
P. Huard.
Background to point-to-set maps in mathematical programming
,
1979
.
[3]
R. Tapia,et al.
On the Formulation of the Primal-Dual Newton Interior-Point Method for Nonlinear Programming
,
1992
.
[4]
M. El Hallabi.
A Global Convergence Theory for Arbitrary Norm Trust-Region Algorithms for Equality Constrained Optimization
,
1993
.
[5]
Roy E. Marsten,et al.
On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming
,
1992,
SIAM J. Optim..
[6]
An Inexact Hybrid Algorithm for Nonlinear Systems of Equations an Inexact Hybrid Algorithm for Nonlinear Systems of Equations 1
,
1994
.
[7]
Yin Zhang,et al.
On the Superlinear and Quadratic Convergence of Primal-Dual Interior Point Linear Programming Algorithms
,
1992,
SIAM J. Optim..
[8]
A Global Convergence Theory for Sequential Linear Programming Inexact Hybrid Algorithms a Global Convergence Theory for Sequential Linear Programming Inexact Hybrid Algorithms 1
,
1994
.
[9]
R. Tapia,et al.
Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited
,
1992
.
[10]
Hiroshi Yamashita,et al.
Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization
,
1996,
Math. Program..
[11]
F. Clarke.
Optimization And Nonsmooth Analysis
,
1983
.
[12]
Stephen J. Wright.
An Interior-Point Algorithm for Linearly Constrained Optimization
,
1992,
SIAM J. Optim..
[13]
R. Tapia,et al.
A Global Convergence Theory for Arbitrary Norm Trust-Region Methods for Nonlinear Equations
,
1995
.