An Inexact Trust-Region Feasible-Point Algorithm for Nonlinear Systems of Equalities and Inequalitie

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.